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Partial cover and complete cover inequalities
,,,,
jm
ELSEVIER
Operations Research Letters 15 (1994) 19-33
Partial cover and complete cover inequalities E l e n a F e r n f i n d e z *'a, K u r t J o r n s t e n b aDepartment o[ Estadt'stica i Investigaci60perativa, Universitat Politbcnica de Catalunya. Pau Gargallo 5, 08028 Barcelona, Spain bNorwegian School o[ Economics and Business Administration, Bergen, Norway
(Received 1 October 1993)
Abstract In this paper we use an extension of the well-known cover inequalities to obtain valid inequalities for 0-1 integer problems with two different sided knapsack constraints and, in general, for any kind of 0-1 integer problems having at least two different sided constraints. These inequalities cannot be derived from any of the individual constraints alone and are stronger than the cover inequalities derived from knapsack constraints individually. Given a solution, we state the conditions under which violated inequalities of this type exist, and we also propose heuristics to identify them. The application of these inequalities to different classes of 0-1 integer problems is also studied. Key words: 0 1 Integer problems; Cover inequalities; Knapsack constraints
O. Introduction
In this paper we will use an extension of the well-known cover inequalities (see, for instance, [2]) to obtain valid inequalities for 0-1 integer problems with two different sided knapsack constraints and, in general, for any kind of 0-1 integer problems having at least two different sided constraints. These inequalities cannot be derived from any of the individual constraints alone and are stronger than the cover inequalities derived from knapsack constraints individually. The problems we have considered are of the form: (IP)
max
cx
s.t.
a x <~ ao, bx >~ bo,
x {o, 1)", where c, a, b ¢ Z"~ a n d ao, bo are positive integers.
* Corresponding author 0167-6377/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 3 7 7 ( 9 4 ) 0 0 0 0 3 - 0
E. Fernandez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
20
Programs with this structure appear when solving the separation problem for 0-1 knapsacks after a feasible solution is known. Also, they might appear as subproblems of general 0-1 programs when the original problems have different sided constraints or when solving the separation problem for the original problem after a feasible solution is known. The convex hull of the feasible solutions of (IP) can be defined in terms of the convex hulls of the integer points satisfying one of the constraints. Let P = { x • R " / ~ a t x t < < . ao, ~ b t x t >~bo, 0~< xt~< 1, j • N = { 1. . . . . n} } be the polytope of the solutions to the linear problem associated to (IP) and P~ = c o n v { x • P / x j • { O , 1}, j • N } the convex hull of the integer vertices of P, P ~ = P~ c~ PIE, where P ~ = conv {x • P1/xt • {0, 1}, j • N}, P~ = c o n v {x • Pz/Xj • {0, 1}, j • N}, P, = {x • R " / 2 a j x j <~ ao, 0 ~ Xj ~ 1, j • N } , P2 = { x • R " / 2 bjxt >l bo, 0<~ xt <~. 1 , j • N}. Note that (0,0 . . . . . 0 ) ¢ P ~ and (1,1 . . . . . 1 ) ¢ P ] . Furthermore, in general V k • N bk>-0 are not facets of p i . Similarly, in general Vk • N ak < Y~a t -- ao, i.e. ~k = (1,..., 1, 0, 1, ..., 1) ¢ PI1; therefore, Xk ~< 1 are not facets of P~. Hence, P[ has not, in general, the full dimension property; furthermore, in general, we do not know its dimension so we cannot derive facets for P'. Instead, what we will do is generate valid inequalities for P~ that cut off some vertices of the unit hypercube which are not in P~. In what follows, we suppose that Vk • N ak ~ ao, bk ~ ~j ~ k bj - bo, bk < bo, ak < ~j ~ k aj -- ao. This paper has 5 sections: in Section 1 partial covers and complete covers are defined and the conditions under which they define valid inequalities for (IP) are stated. In Section 2 we present the conditions under which these inequalities are violated by a given solution; heuristics to identify inequalities violated by a given solution are proposed in Section 3. In Section 4 we address some applications of these inequalities: the separation problem for 0-1 knapsack problems and the general 0-1 problem, and the case of problems with cardinality constraints. Finally, in Section 5 we report on the computational experiments we have performed.
1. Partial covers and complete covers
Definition 1.1. (i) A subset C ~_ N is a partial Cover of N if ~'.t~c at <- ao, and Zt~c bt < bo, (ii) A subset T ~_ N is a Complete Cover of N if ~t~T bt >~ bo, and )~t~T at > ao, A partial cover C can be associated with a 0-1 vector x C • {0, 1}" (xc -- 1, j • C; x c = 0, j ¢ C), which is feasible with respect to the first constraint but is not feasible with respect to the second one. In order to obtain feasible solutions for (IP) from a partial cover, the set C of components fixed at one must be enlarged to satisfy the second constraint. Similarly, a complete cover Tcan be associated with a 0-1 vector x T • {0, 1}" (xf = I, j • T; x T = 0,j ~ T), which is feasible with respect to the second constraint but is not feasible with respect to the first one. In order to obtain feasible solutions for (IP) from a complete cover, the set T of components fixed at one must be reduced to satisfy the first constraint. Proposition 1.2. No enlargement of C provides a feasible solution to the problem ( I P ) / f and only if the partial
cover problem: (PCP)
Z p c = m i n ~ at + jeC
s.t.
~ bt + jEC
~
at s t
jEN\C
~
bts t >~bo,
jeN\,C
st e {0, 1}, j ~ X
has a value Zpc strictly greater than ao.
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
21
Proof. =~: Ifzpc ~< ao, the vector x given by xj = 1 j • C w C', x~ = 0 otherwise, where C' = {j • N/sj = 1} in an optimal solution to (PCP), is an enlargement of C that provides a feasible solution for (IP). : Obviously, since any enlargement of C satisfying the second constraint violates the first one. A partial cover C corresponds to a vertex Vc of the unit hypercube which does not belong to P~; when the value of the associated (PCP) is strictly greater than ao, none of the vertices that can be obtained by lifting Vc belong to P~, since they will violate either the first or the second constraint. In this case, at least one of the variables of C must be zero in any feasible solution for (IP) and, therefore, the partial cover inequality (PCI) Zj~cXj<~ ICI- 1 (1), is valid for PX. PCIs are valid inequalities for (IP) that are stronger than any of the ordinary cover inequalities that can be derived from constraint 1 individually, since partial covers are contained in covers of constraint 1. Obviously, the strongest PCIs will be derived from minimal partial covers, i.e. partial covers that do not contain any partial cover.
Proposition 1.3. No reduction of T provides a feasible solution to the problem (IP) if and only if the complete cover problem: (CCP)
zcc = max ~ b~sj jeT s.t.
2 ajsj~ jeT
a o,
s j • {0, 1}, j E N has a value Zcc strictly smaller than bo.
Proof. =~ : Ifzcc >~ bo, the vector x given by xj = l j • T', xi = 0 otherwise, where T' = {j E T/s~ = 1} in an optimal solution to (CCP), is a reduction of T that provides a feasible solution for (IP). : Obvious, since any reduction of T satisfying the first constraint violates the second one. Again, a complete cover T corresponds to a vertex vr of the unit hypercube which is not in P~; now, when the value of the associated (CCP) is strictly smaller than b0, none of the vertices obtained projecting vr belong to P~ since they violate either the first or the second constraint. In this case, at least one of the variables which are not in T must be 1 in any feasible solution for (IP) and, therefore, the complete cover inequality (CCI) ~jEN\TXj >/ 1 (2) is valid for pi. CCIs are also valid inequalities valid for (IP) stronger than any of the ordinary cover inequalities that can be derived from constraint 2 individually. Now the strongest CCIs will be derived from maximal complete covers, i.e. complete covers that are not contained in any complete cover.
2. Partial and complete cover inequalities violated by a given solution x* In this section we will see under which conditions it is possible to derive valid inequalities (1) and (2) violated by a given solution x* and how these inequalities can be obtained. From an algorithmic point of view, this will allow us to define procedures for identifying such inequalities and embedding them in an algorithmic scheme for solving (IP) using an LP solver. The following results can be considered as extensions of the well-known similar result for standard 0-1 knapsack problems [3]; this is not surprising, given that (IP) can be considered as a standard 0-1 knapsack problem with an additional constraint.
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
22
Proposition 2.1. Given a vector x*, 0 <~x* <<.1 j ~ N, a PCI violated by x* exists if and only if the solution to the program (P1)
z* = min y (1 - x*) sj j~N
s.t.
aj sj <. ao,
(3)
b~si < bo,
(4)
aj(sj + tj) > ao,
(5)
jeN jeN teN
sje
{0, 1}, Vj ~ N, and t solving the problem
min
(6)
~ aitj jeN
s.t.
bj(sj + tj) >1 bo,
(7)
sj + tj <~ 1 Vj e N
(8)
jeN
t i e {0, 1}
has a value strictly smaller than one. Moreover, in this case, the inequality y,j~c Xj <~ I C [ - 1, where C = {j ~ N/sj = 1} is a PCI violated by x*. Constraints (3) and (4) ensure that C is a partial cover. On the other hand, (5)-(8) guarantee that the solution to the (PCP) associated with C has a value greater than ao and that, therefore, the corresponding PCI is valid for (IP). z* < 1 ensures that this PCI is violated by x*.
Proposition 2.2. Given a solution x*, 0 <<.x* <~ l j ~ N, a CCI violated by x* exists if and only if the solution to the program (P2)
x* sj
z* = max jeN
s.t.
(9)
a j s j > ao, jeN
bjsj >1 bo
(lO)
bj(sj - t j) < bo,
(11)
j~N jsN
sje {0, 1 }, min
Vj ~ N, and t solving the problem
(12)
Z bjtj j~N
s.t.
Z aj(sj -- tj) <~ ao, jeN
si--ti>~O
Vj~N
(13) (14)
tj~{o, 1}, has a value strictly greater than Y~j~Nx* - 1. Moreover, in this case, ~ j~s~r xj >. 1 is a CCI violated by x* where T = {j ~ N/sj = 1}.
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
23
Now, constraints (9) and (10) ensure that Tis a complete cover. On the other hand, (11)-(14) guarantee the solution to the (CCP) associated with T has a value smaller than bo and, therefore, the corresponding CCI is valid for (IP). The value z* of the objective function guarantees that this CCI is violated by x*. The above two programs are bilevel problems which, in general, are difficult to solve. Nevertheless, feasible solutions to these problems with objective function values smaller than one or greater than ~j~N x* - 1, respectively, provide valid PCIs or CCIs violated by x*. In the following section we propose two heuristics to obtain such feasible solutions.
3. Heuristics for identifying partial cover and complete cover inequalities violated by a given solution x* In this section we present a general scheme for heuristics to identify violated PCIs and CCIs; the goal of these procedures is finding feasible solutions to (P1) and (P2) that fulfill the conditions of Propositions 2.1 and 2.2, respectively. This framework can be utilized by any heuristics for identifying partial (or complete) covers. We also present the greedy procedures which we are currently using.
3.1. Heuristic for identifying a PCI violated by a given solution x* Given x*, the heuristic is as follows: (1) Select a suitable C _~ N such that (i) ~.j~caj<<.ao,
(ii) Zj~cbj
and
(iii) ~ c
x* > I C [ -
1.
(2) Solve the associated (PCP) defined in Section 1. (3) If Zpc > ao, the PCI ~j~c xj ~< [CI - 1 is valid for (IP) and violated by x*. In the first step of this heuristic the partial cover C must be chosen to have as few elements as possible (to provide a strong inequality) but these elements must be taken as to make C derive a valid PCI for (IP). As we want to have a value of (PCP) strictly bigger than ao, we are currently using a greedy procedure in which the variables are included in C according to decreasing values of aj/bj. Note that this ordering favours having a high fixed value in the objective function of (PCP) relative to a small fixed value in the constraint of (PCP). Of course, only variables with values preserving condition (iii) can be included.
3.2. Heuristic for identifying a CCI violated by a given solution x* Given x*, the heuristic is as follows: (1) Select a suitable T ~_ N such that (i) Z ~ T a j > a o , (ii) ~j~Tbj>~ bo and (iii) ~jEN\TX~ • 1 (2) Solve the associated complete cover problem (CCP) defined in Section 1. (3) Ifzcc < bo, the CCI ~j~N\rxj ~> 1 is valid for (IP) and violated by x*. In the first step of this heuristic the set T must be chosen to have as many variables as possible in order to keep N \ T small and, therefore, have an inequality as strong as possible. As we want to have the value of (CCP) smaller than bo we are currently using a greedy procedure in which the variables are included in T according to increasing values of bJa~. Note that with this ordering the variables that appear in (CCP) have small coefficients in the objective function with respect to their coefficient in the constraint of (CCP). Now we must include in T all the variables with value 1 as well as enough of the remaining variables as to attain condition (iii).
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E. Ferndmdez, K. Jornsten / Operations Research Letters 15 (1994) 19 33
4. Applications
As we have already mentioned in the Introduction, PCIs and CCIs can be used for solving the separation problem for different classes of 0-1 integer programs. In this section we will see how these inequalities can be applied to some specific 0-1 problems. 4.1. Knapsack problems
Consider a standard 0-1 knapsack problem of the form (KP)
max
cx
s.t.
a x <<.ao,
x {0, m} If we know a feasible solution x* for (KP) with value Co, in order to prove optimality, or to find feasible solutions with a better value, we can formulate (KP1)
max
cx
s.t.
ax ~ ao, c x >1 Co + 1,
which provides an equivalent formulation of (KP) in the sense that any feasible solution to (KP) better than x*, if it exists, must be feasible for (KP1). (KP1) has the same structure as (IP) and, therefore PCIs and CCIs can be derived. 4.2. General case
PCIs and CCIs can also be derived for general 0-1 integer programs having at least one inequality constraint, as soon as a feasible solution to the problem is known. This solution may be obtained, for instance, with the Pivot and Complement heuristic [1] applied to the solution to the LP relaxation. 4.3. Cardinality constraint
The results presented in the above sections can be easily adapted to the case of 0-1 maximization programs with a cardinality constraint of the form Y ~ N X j = k. After we know a feasible solution with value Co, we can add the constraint c x t> Co + 1, and obtain an equivalent formulation of the original problem. In this case, a partial cover is defined to be a set C ~ N, such that IC[ < k and Y.j~c cj < Co + 1. Now, when the value of the partial cover problem
min ICl+ ~ sj jeN\C
s.t.
~ cj + ~ jeC
c j s j >l Co +
1,
jeN\C
si~ {0, 1}, j a N has a value strictly greater than k, the associated PCI is valid for the original problem.
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E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
Similarly, a complete cover is defined to be a set T ~_ N, such that ~ j e r cj >1 Co + 1 and value of the complete cover problem max
ICI > k. When the
~ cs sj jeT
~
s.t.
Sj = k,
jeT
sj e {0, 1}, j e N has a value strictly smaller than Co + 1 the associated CCI is valid for the original problem. 4.4. Example
Consider the following 0-1 knapsack problem example [5-1 (KP)
Max
16xl + 1 2 x 2 + 1 4 x 3 + 1 7 X a + 2 0 x s + 2 7 x 6 + 4 x T + 6 x 8 + 8 x 9 + 2 0 X 1 o +
11xll
+ 10X12 + 7X13 , s.t.
(15)
7xl + 6x2 + 5X 3 -t- 6X4 + 7X5 + 10X6 -F 2X7 -4- 3xs + 3X9 + 9Xlo 4- 3Xll "+- 5X12 + 5x13 ~< 48 x~E{0,1},
j=
(16) 1 . . . . . 13.
The LP solution to this problem is x* = 1, j = 1,3,4,5,6,9, 11, X*o = 7/9 with value Z~p = 128 + 5/9. The greedy heuristic gives the solution xj = 1, j = 1, 2, 3, 4, 5, 6, 9, 11 with value Zheur = 125. Therefore, the constraint 16xx + 12x2 + 14x3 + 17x4 + 20x5 + 27x6 + 4 x 7 + 6x8 + 8X 9 "~ 20X10 "t- 11XXX + 10Xxz + 7X13 ~> 126
(17)
must be fulfilled by any feasible solution to (KP) better than the one provided by the heuristic. If we add it to the formulation of (KP) we get the problem (IP) (IP)
Max
(15)
s.t.
(16),(17),
x~{0,1},
j=
1 . . . . . 13.
If we now apply the heuristic proposed in Section 3 to identify a PCI violated by the solution to the LP x*, we get C = {10, 1,9,6} which is a partial cover since (i) al + a6 + a9 + alo 7 + 10 + 3 + 9 = 29 < 48 = ao, (ii) bl + b6 + b9 + blo = 16 + 27 + 8 + 20 = 71 < 126 = bo. The solution of the associated (PCP) is given by xj = 1, j = 3,4, 5, 7, x~ = 0 otherwise, and gives a value Zpc = 49 > 48 = ao. The value in (P1) for the feasible solution associated to C is =
(1-x*)+(1-x~)+(1-x~)+(1-X*o)=(1-
1)+(1-
1)+(1-1)+(1-7/9)=2/9<
1
Therefore, the PCI violated by x* is (18)
X 1 + X 6 "]- X 9 "~ X10 ~ 3.
Applying the heuristic proposed in Section 3 to T = {1,3,4,5,6,9, 10, 11} which is a complete cover since
identify
a
CCI
violated
by
x*,
we
get
E. Ferndmdez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
26
(i) a~ + a 3 + a4 + a s + a6 + a 9 + alo + a ~ = 7 + 5 + 6 + 7 + 1 0 + 3 + 9 + 3 = 50 > 4 8 , (ii) b~ + b 3 + b 4 + b 5 + b 6 + b 9 + b l o + b l l = 16 + 14 + 17 + 20 + 27 + 8 + 20 + 11 = 71 < 126. The solution of the associated (CCP) is given by xj = 1, j = 1,3, 4, 5, 6, 10, 11, xj = 0 otherwise, and gives a value Zcc = 125 < 126 = bo. The value in (P2) for the feasible solution associated to T is
x*+x*+x*+x~+x*+x*+x*o+X*~=
=
Z
1 + 1 + 1 + 1 + 1 + 1 + 7/9+ 1 =7~>6
7
x*-I
j~N',,T
Therefore, the CCI violated by x* is
xz+xv+Xs+Xlz+x13~
1
(19)
The resulting LP relaxation after adding (18) and (19) to (IP) is Max
(15)
s.t.
(16),(17),(18) and (19) O<~xj<~ 1, j ~ N .
The optimal solution to this problem is given by xj = 1, j = 1, 3, 4, 5, 6, 7, 9, 11, x~ = 0, otherwise with value Zlp~ = 127. As this solution is feasible for (IP), it is the optimal solution for (KP).
5. Computational experience In this section we present the results obtained in the computational experiments we have performed on a set of 90 0-1 knapsack test problems of different dimensions. The problems have been randomly generated as explained in [41 . The dimensions of the problems are 50, 100, 200 and 500. For each dimension, we have generated two data sets of 10 problems each. For n = 50, 100, 200 the first data set contains uncorrelated problems with cj and aj uniformly random in [1, v], with v = 100, and the second data set contains weakly correlated problems with cj uniformly random in [1, v], and aj uniformly random in [cj - r, cj + r]; v = 100 and r = 10. For n = 500, the first data set contains uncorrelated problems with cj and aj uniformly random in [1, v], with v = 1000 and the second one contains weakly correlated problems with cj uniformly random in 1-1, v], and aj uniformly random in [c~ - r, cj + r], where v = 1000 and r = 100. The capacity value we have taken is ao = 0.5 ~ at; as explained in [4], this value gives, in general, instances that are expected to be more difficult than the value ao = 2v. For n = 500 we have also generated a third set of 10 weakly correlated problems with this latter capacity value. We have applied two different LP solution schemes to the problems. In the first one, (LPSI), in each iteration we used the greedy heuristics proposed in Section 3 to derive a PCI and/or a CCI violated by the current LP solution. Previously, we performed the greedy heuristic to obtain an equivalent formulation as problems with two different sided knapsack constraints, as explained in Section 4.1. In the second scheme, (LPS2), in each iteration we derived the classical cover inequality most violated by the current LP solution and performed the lifting procedure [2] to obtain facets of the knapsack polytope. We did not choose any specific order for the lifting so the variables were lifted according to increasing values of their indices. The results we present compare the behaviour of the above LP solution schemes. In 84 out of 90 problems PCIs and CCIs gave better results than the classical cover inequalities together with the lifting procedure, in 5 problems both schemes gave the same result and only in one problem, $49, (LPS2) turned out to be better than (LPS1).
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
27
In Tables 1-9 we present the results obtained. The first four columns give results related to the original problems. Heur is the value of the greedy heuristic, Opt the value of the optimal solution, LP the value of the linear programming relaxation and Gap the value LP-Opt. The next five columns contain the results related to (LPS1) (the LP solution scheme using PCIs and CCIs). LP1 is the value of the linear programming relaxation in the final iteration of the scheme, itl the number of iterations performed, pc and cc are, respectively, the number of PCIs and CCIs obtained (since in some iterations only one type of inequality could be derived), and Gapl is the value LPl-Opt. Columns 10-12 give the results related to (LPS2) (the LP solution scheme using cover inequalities plus the lifting procedure). LP2 is the value of the linear programming relaxation in the final iteration of the scheme, ci the number of cover inequalities derived which, in this case, coincides with the number of iterations performed, and Gap2 is the value LP2-Opt. Finally, in the last two columns % 1 and %2 give the percentage of reduction of the original gap (Gap) attained with (LPS1) and (LPS2), respectively. We find that these two columns are significant since they permit to compare the behaviour of the above schemes on the same problem. We have not included any information about the computational effort needed to generate the different types of inequalities. The reason is that our aim was to compare the strength of PCIs and CCIs with that of classical cover inequalities together with a lifting procedure. In this sense, we consider that choosing the most violated cover inequality instead of choosing the cover heuristically makes our comparison more valuable. Nevertheless, in our experience PCIs and CCIs were, in most cases, easier to find than classical lifted inequalities. The reason is quite simple: a 0-1 knapsack problem must be solved each time a PCI or a CCI is obtained, but this is also the case for generating the most violated cover inequality where, in addition, the lifting procedure also requires solving a collection of 0-1 knapsack problems. As we can see, most of the test problems have a small gap between the value of the linear programming relaxation and the value of the optimal 0-1 solution; nevertheless, this gap has been difficult to reduce. For this reason we consider that the percentage of the reduction of this gap is a good measure of the efficiency of the LP solution schemes, especially if we want to compare the behaviour of different procedures. We would like to stress the fact that, in our computational experience, the proposed PCIs and CCIs give results that are considerably better than those obtained with the classical cover inequalities plus the lifting procedure. As we have already mentioned, this has been so in all the test problems except one, $49, for the different dimensions and kind of problems (uncorrelated or weakly correlated). In particular, the percentage of the reduction of the gap achieved with (LPS1) is in most of the problems twice or three times the improvement attained with the classical scheme. The values of LP relaxations marked with an asterisk correspond to problems for which a 0-1 optimal solution has been found. As we can see, this has happened with 15 problems using (LPSI) and only with three of these problems with (LPS2). Again, PCIs and CCIs have given better results than cover inequalities for finding an optimal solution. We would like to mention problems $23 and $27 of Table 6 for which the original linear programming relaxation already gave an optimal value; nevertheless, since this value corresponds to solutions which are not integral, optimality could not be proved. For both problems (LPS1) identified an optimal 0-1 solution, hence proving optimality; only one of them was optimally solved with (LPS2). Also, problem $28 was optimally solved by (LPS1); for this problem (LPS2) gave an optimal value, but did not identify an optimal 0 1 solution and, therefore, we cannot consider that it was solved optimally with (LPS2). With respect to the number of PCIs and CCIs derived in (LPS1), we can see that, in general, the number of PCIs is higher than the number of CCIs. In fact, in our experience, very seldom the procedure identified a CCI in one iteration in which a PCI was not derived. We have not appreciated significant differences between both schemes with respect to the number of iterations performed. Finally, we would like to point out that, although the aim of this paper was not to give an exhaustive computational study of the behaviour of PCIs and CCIs, we consider that the results obtained in the experiments we have performed confirm that, in the case of 0-1 knapsack problems, the use of PCIs and CCIs is clearly worthwhile with respect to the classical cover inequalities plus the lifting procedure.
1816 2284 2143 2007 2141 1965 2031 2114 2229 1971
1819 2295 2146 2014 2143 1966 2032 2117 2231 1984
Opt 1822.76 2303.46 2155.24 2022.32 2157.66 1969.23 2037.16 2118.09 2237.61 1990.22
LP 3.76 8.46 9.24 8.32 14.66 3.23 5.16 1.09 6.61 6.22
Gap 1820.61 2299.40 2152.65 2014.71 2144.60 1966.71 2034.08 2117.00" 2234.98 1985.45
LP1 5 12 28 12 4 29 30 2 4 4
itl 5 12 28 12 4 29 30 2 4 4
pc 1 2 6 2 2 27 25 1 1 1
cc
1424 1319 1427 1458 1244 1370 1305 1472 1289 1394
1427 1323 1429 1464 1249 1371 1306 1473 1291 1400
Opt 1428.25 1324.84 1430.89 1466.13 1249.20 1372.81 1311.00 1474.38 1292.84 1400.72
LP 1.25 1.84 1.89 2.13 0.20 1.81 5.00 1.38 1.84 0.72
Gap 1427.00" 1324.00 1429.94 1465.30 1249.00" 1371.00" 1307.02 1473.00" 1291.50 1400.38
LP 1 4 5 20 13 1 4 4 5 14 5
it I
3 5 20 13 1 3 3 5 10 4
pc
3 2 17 9 0 4 4 2 4 4
cc
0.00 1.00 0.94 1.30 0.00 0.00 1.02 0.00 0.50 0.38
Gap 1
1.61 4.40 6.65 0.71 1.60 0.71 2.08 0.00 3.98 1.45
Gapl
1428.13 1324.69 1430.85 1465.92 1249.00" 1372.60 1310.68 1474.05 1292.50 1400.66
LP2
1821.26 2300.73 2153.43 2020.53 2150.72 1968.64 2035.86 2117.00* 2235.84 1985.45
LP2
5 13 3 15 2 4 18 7 4 3
ci
14 9 2 11 6 6 19 4 5 9
ci
1.13 1.69 1.85 1.92 0.00 1.60 4.68 1.05 1.50 0.66
Gap2
2.26 5.73 7.43 6.53 7.72 2.64 3.86 0.00 4.84 1.45
Gap2
n = 50 Weakly correlated: cj uniformly random in [1, v], a r uniformly random in [cj - r, c~ + r]; v = 100, r = 10; ao = 0.5 E a r.
SI $2 $3 $4 $5 $6 $7 $8 $9 S10
Heur
Table 2
n = 50 Uncorrelated: cj and a~ uniformly random in [1, v], v = 100; ao = 0.5 Ea r.
UI U2 U3 U4 U5 U6 U7 U8 U9 U10
Heur
Table 1
100.00 45.65 50.26 38.67 100.00 100.00 79.60 100.00 72.82 47.22
%1
57.18 47.99 28.03 91.46 89.08 78.01 59.68 100.00 39.78 76.68
%1
9.60 8.15 2.11 9.43 100.00 11.60 6.40 23.91 18.47 8.33
%2
39.89 32.26 19.58 21.51 47.33 18.26 25.19 100.00 26.77 76.68
%2
I
¢<
e~
t~
3830 4118 4277 4107 3922 4371 4117 4047 3883 4165
3840 4121 4280 4109 3927 4375 4118 4048 3890 4172
Opt
2865 2910 2490 2796 2650 2841 2694 2631 2915 2966
2869 2916 2492 2798 2651 2843 2695 2632 2920 2970
Opt
3840.52 4121.25 4281.43 4111.56 3927.60 4375.00* 4118.00" 4049.18 3893.52 4173.74
LP1 8 5 6 8 6 3 8 10 4 2
itl 8 4 6 8 6 3 8 10 4 1
pc 1 5 5 3 1 2 7 6 1 2
cc
2869.93 2916.71 2493.43 2800.14 2652.00 2845.00 2695.99 2633.43 2920.59 2972.00
LP 0.93 0.71 1.43 2.14 1.00 2.00 0.99 1.43 0.59 2.00
Gap 2869.59 2916.46 2492.93 2799.24 2651.46 2844.42 2695.78 2633.08 2920.16 2971.31
LPI 14 8 5 1 4 20 12 5 3 3
itl 14 8 5 1 4 20 12 5 3 2
pc 0 0 0 1 4 4 6 2 2 3
cc
0.59 0.46 0.93 1.24 0.46 1.42 0.78 1.08 0.16 1.31
Gapl
0.52 0.25 1.43 2.56 0.60 0.00 0.00 1.18 3.52 1.74
Gapl
2869.66 2916.51 2493.08 2799.73 2651.95 2844.93 2695.95 2633.32 2920.39 2971.91
LP2
3841.02 4121.84 4282.04 4114.08 3928.28 4376.14 4122.12 4054.12 3897.90 4174.72
LP2
6 8 9 5 26 5 3 4 5 5
ci
10 3 3 6 13 9 9 18 17 5
ci
0.66 0.51 1.08 1.73 0.95 1.93 0.95 1.32 0.39 1.91
Gap2
1.02 0.84 2.04 5.08 1.28 1.14 4.12 6.12 7.90 2.72
Gap2
n = 100 Weakly correlated: cj uniformly random in [1, v], a t uniformly random in [c~ - r, c~ + r]; v = 100, r = 10; a0 = 0.5 Zaj.
SI 1 S12 S13 S14 S15 S16 S17 S18 S19 $20
Heur
Table 4
2.17 1.36 3.24 5.54 2.75 1.57 5.79 7.02 9.34 3.63
Gap
aj uniformly random in [1, v], v = 100; ao = 0.5 Y~aj.
3842.17 4122.36 4283.24 4114.54 3929.75 4376.57 4123.79 4055.02 3899.34 4175.63
LP
n = 100 Uncorrelated: cj and
Ull U12 UI3 U14 U15 U16 UI7 U18 U19 U20
Heur
Table 3
36.55 35.21 34.96 42.05 54.00 29.00 21.21 24.47 72.88 34.50
%1
76.03 81.61 55.86 53.79 78.18 100.00 100.00 83.19 62.31 52.06
%1
29.03 28.16 24.47 19.15 5.00 3.50 4.04 7.69 33.89 4.50
%2
52.99 38.23 37.03 8.30 53.45 27.38 28.84 12.82 15.41 25.06
%2
I
,2
e~
5
t~
8413 8575 8813 8439 7759 8360 8001 8520 8391 8395
8414 8579 8814 8441 7761 8365 8007 8523 8392 8400
Opt 8415.96 8582.31 8817.70 8443.09 7762.50 8365.83 8007.94 8526.19 8397.09 8400.34
LP 1.96 3.31 3.70 2.09 1.50 0.83 0.94 3.19 5.09 0.34
Gap 8414.87 8580.83 8815.45 8442.44 7761.00" 8365.00* 8007.52 8524.78 8392.78 8400.00*
LP1 4 8 8 4 4 3 6 10 15 4
it 1 4 8 4 4 4 3 6 10 15 4
pc 3 4 8 2 3 3 1 6 10 3
cc
5665 5749 5917 5077 5465 5655 5574 5810 5167 5123
5667 5755 5919 5078 5466 5656 5575 5811 5169 5124
Opt 5667.21 5755.17 5919.00 5979.11 5466.85 5656.78 5575.00 5811.77 5169.45 5125.03
LP 0.21 0.17 0.00 1.11 0.85 0.78 0.00 0.77 0.45 1.03
Gap 5667.00* 5755.08 5919.00" 5078.55 5466.59 5656.58 5575.00* 5811.00" 5169.20 5124.58
LP1 3 2 6 14 9 7 11 2 6 11
itl
3 2 6 14 9 7 11 2 6 11
pc
2 0 1 1 9 4 6 3 1 8
cc
0.00 0.08 0.00 0.55 0.59 0.58 0.00 0.00 0.20 0.58
Gapl
0.87 1.83 1.45 1.44 0.00 0.00 0.52 1.78 0.78 0.00
Gapl
5667.01 5755.08 5919.00 5078.94 5466.83 5656.76 5575.00* 5811.00 5169.37 5124.95
LP2
8415.62 8581.72 8816.56 8442.76 7761.91 8365.09 8007.57 8526.08 8396.65 8400.25
LP2
3 2 4 32 4 2 2 2 9 6
ci
4 15 5 7 10 4 17 6 8 4
ci
0.01 0.08 0.00 0.94 0.83 0.76 0.00 0.00 0.37 0.95
Gap2
1.62 2.72 2.56 1.76 0.91 0.09 0.57 3.08 4.65 0.25
Gap2
n = 200 Weakly correlated: ('j uniformly random in [1, v], aj uniformly random in [G - r, cj + r]; v = 100, r = 10; ao = 0.5 E a r.
$21 $22 $23 $24 $25 $26 $27 $28 $29 $30
Heur
Table 6
n = 200 Uncorrelated: cj and aj uniformly random in [1, v], v = 100; ao = 0.5 Eai.
U21 U22 U23 U24 U25 U26 U27 U28 U29 U30
Heur
Table 5
100.00 52.94 100.00 50.45 30.58 25.64 100.00 100.00 55.55 43.68
%1
55.61 44.71 60.81 31.10 100.00 100.00 44.68 44.20 84.67 100.00
%1
95.23 52.94 0.00 15.31 2.35 2.65 100.00 100.00 17.77 7.76
%2
17.34 17.82 30.81 15.79 39.33 89.15 39.36 3.44 8.64 26.47
%2
I
e~
tl
200872 200298 204001 199095 202087 203252 200694 192911 204424 198940
200873 200324 204041 199140 202104 203297 200696 192921 204425 198944
Opt 200887.82 200335.39 204045.23 199153.92 202122.95 203318.78 200710.11 192938.14 204451.51 198961.53
LP 14.82 11.39 4.23 13.92 18.95 21.78 14.11 17.14 26.51 17.53
Gap 200878.70 200331.94 204042.80 199150.00 202114.64 203307.36 200699.06 192931.14 204437.42 198946.99
LP1 13 2 2 5 5 7 15 18 10 6
itl 13 1 2 5 4 7 12 18 10 4
pc 9 2 1 1 5 1 15 4 10 6
cc
138197 133753 142597 135169 132918 136294 140569 134675 140187 137355
138234 133764 142605 135173 132926 136308 140570 134690 140211 137361
Opt 138238.92 133765.00 142610.95 135178.68 132931.34 136313.26 140575.37 134694.46 140213.00 137365.89
LP 4.92 1.00 5.95 5.68 5.34 5.26 5.37 4.46 2.00 4.89
Gap 138237.46 133764.00* 142609.23 135177.57 132930.00 136312.10 140574.06 134693.18 140212.50 137363.92
LP 1 6 3 4 9 4 2 4 2 2 6
it I 6 3 4 9 4 2 4 2 2 6
pc 0 3 4 3 3 1 4 1 2 5
cc
3.46 0.00 4.23 4.57 4.00 4.10 4.06 3.18 1.50 2.92
Gap 1
5.70 7.94 1.80 10.00 10.64 10.36 3.06 10.14 12.42 2.99
Gapl
138237.87 133764.85 142609.71 135178.06 132930.92 136312.75 140575.27 134693.97 140212.57 137365.83
LP2
200887.61 200332.76 204043.74 199153.20 202120.45 203310.37 200708.87 192935.95 204446.71 198960.40
LP2
14 9 6 19 11 4 2 4 8 17
ci
26 4 3 11 16 5 3 3 20 3
ci
3.87 0.85 4.71 5.06 4.92 4.75 5.27 3.97 1.57 4.83
Gap2
14.61 8.76 2.74 13.20 16.45 13.37 12.87 14.95 21.71 16.40
Gap2
29.67 100.00 28.90 19.54 25.09 22.05 24.39 28.69 25.00 40.28
%1
61.53 30.28 57.44 28.16 43.85 52.43 78.31 40.84 53.14 82.94
%1
n = 500 Weakly correlated: cj uniformly random in [1, v], a i uniformly random in [c i - r, c'j + r]; v = 1000, r = 100; ao = 0.5 E a~.
$31 $32 $33 $34 $35 $36 $37 $38 $39 $40
Heur
Table 8
n = 500 Uncorrelated: G and aj uniformly random in [1, v], v = 1000; ao = 0.5 Z a r.
U31 U32 U33 U34 U35 U36 U37 U38 U39 U40
Heur
Table 7
21.34 15.00 20.84 10.91 7.86 9.69 1.86 10.98 21.50 1.22
%2
1.41 23.09 35.22 5.17 13.19 38.61 8.78 12.77 18.10 6.44
%2
t~a
t.i
3685 3815 3904 4058 3683 3327 3715 3901 3862 3701
3704 3822 3908 4069 3700 3336 3717 3906 3885 3728
Opt 3718.13 3828.53 3914.06 4082.31 3705.25 3351.84 3734.19 3928.42 3890.91 3737.81
LP 14.13 6.53 6.06 13.31 5.25 15.84 17.19 22.42 5.81 9.81
Gap 3713.00 3824.53 3909.42 4077.13 3701.43 3345.54 3718.80 3908.38 3887.60 3730.02
LP1 4 3 14 9 5 3 5 11 15 9
itl 3 3 14 9 5 3 5 11 15 9
pc 4 1 5 2 5 3 5 11 0 1
cc 9.00 2.53 1.42 8.13 1.43 9.54 1.80 2.38 2.60 2.02
Gapl 3716.63 3825.54 3913.47 4079.39 3704.29 3351.34 3733.56 3918.58 3887.47 3733.87
LP2 5 12 4 7 6 1 15 8 5 9
ci
12.63 3.54 5.47 10.39 4.29 15.34 16.56 12.58 2.47 5.87
Gap2
n = 500 Weakly correlated: ci uniformly random in [I, v], ai uniformly random in [cj - r, cj + r]; v = 1000, r = 100; ao = 2v
$41 $42 $43 $44 $45 $46 $47 $48 $49 $50
Heur
Table 9
36.30 61.25 76.56 38.91 72.76 39.77 89.52 89.38 55.24 79.40
%1
10.61 45.78 9.73 21.93 18.28 3.15 3.66 43.88 57.48 40.16
%2
I
~n
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19 33
33
Therefore, we consider that the application of these inequalities to more general 0-1 problems, as we have already pointed out in Section 4, seems very promising especially if we take into account that PCIs and CCIs can be derived in the same cases that cover inequalities are already being used.
Acknowledgements The authors are very grateful to the anonymous referee whose detailed comments contributed to improve the paper considerably. This work has been partially supported by the C.I.R.I.T, Generalitat de Catalunya.
References [1] E. Balas and R. Martin, "Pivot and complement: A heuristic for 0 1 programming", Mgmt Sci., 26, pp. 86-89, 1980. [2] E. Balas and E. Zemel, "Facets of the knapsack polytope from minimal covers", SIAM J. Appl. Math. 34, pp. 119-148, 1978. [3] H.P. Crowder, E.L. Johnson and M. Padberg, "Solving large scale zero-one linear programming problems", Oper. Res. 31, pp. 803 834, 1983. [4] S. Martello and P. Toth, Knapsack Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York, 1990. [5] G.I. Nemhauser and L.A. Wolsey, Inteoer and Combinatorial Optimization, Wiley-lnterscience Series in Discrete Mathematics and Optimization, Wiley, New York, 1988.
jm
ELSEVIER
Operations Research Letters 15 (1994) 19-33
Partial cover and complete cover inequalities E l e n a F e r n f i n d e z *'a, K u r t J o r n s t e n b aDepartment o[ Estadt'stica i Investigaci60perativa, Universitat Politbcnica de Catalunya. Pau Gargallo 5, 08028 Barcelona, Spain bNorwegian School o[ Economics and Business Administration, Bergen, Norway
(Received 1 October 1993)
Abstract In this paper we use an extension of the well-known cover inequalities to obtain valid inequalities for 0-1 integer problems with two different sided knapsack constraints and, in general, for any kind of 0-1 integer problems having at least two different sided constraints. These inequalities cannot be derived from any of the individual constraints alone and are stronger than the cover inequalities derived from knapsack constraints individually. Given a solution, we state the conditions under which violated inequalities of this type exist, and we also propose heuristics to identify them. The application of these inequalities to different classes of 0-1 integer problems is also studied. Key words: 0 1 Integer problems; Cover inequalities; Knapsack constraints
O. Introduction
In this paper we will use an extension of the well-known cover inequalities (see, for instance, [2]) to obtain valid inequalities for 0-1 integer problems with two different sided knapsack constraints and, in general, for any kind of 0-1 integer problems having at least two different sided constraints. These inequalities cannot be derived from any of the individual constraints alone and are stronger than the cover inequalities derived from knapsack constraints individually. The problems we have considered are of the form: (IP)
max
cx
s.t.
a x <~ ao, bx >~ bo,
x {o, 1)", where c, a, b ¢ Z"~ a n d ao, bo are positive integers.
* Corresponding author 0167-6377/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 3 7 7 ( 9 4 ) 0 0 0 0 3 - 0
E. Fernandez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
20
Programs with this structure appear when solving the separation problem for 0-1 knapsacks after a feasible solution is known. Also, they might appear as subproblems of general 0-1 programs when the original problems have different sided constraints or when solving the separation problem for the original problem after a feasible solution is known. The convex hull of the feasible solutions of (IP) can be defined in terms of the convex hulls of the integer points satisfying one of the constraints. Let P = { x • R " / ~ a t x t < < . ao, ~ b t x t >~bo, 0~< xt~< 1, j • N = { 1. . . . . n} } be the polytope of the solutions to the linear problem associated to (IP) and P~ = c o n v { x • P / x j • { O , 1}, j • N } the convex hull of the integer vertices of P, P ~ = P~ c~ PIE, where P ~ = conv {x • P1/xt • {0, 1}, j • N}, P~ = c o n v {x • Pz/Xj • {0, 1}, j • N}, P, = {x • R " / 2 a j x j <~ ao, 0 ~ Xj ~ 1, j • N } , P2 = { x • R " / 2 bjxt >l bo, 0<~ xt <~. 1 , j • N}. Note that (0,0 . . . . . 0 ) ¢ P ~ and (1,1 . . . . . 1 ) ¢ P ] . Furthermore, in general V k • N bk
1. Partial covers and complete covers
Definition 1.1. (i) A subset C ~_ N is a partial Cover of N if ~'.t~c at <- ao, and Zt~c bt < bo, (ii) A subset T ~_ N is a Complete Cover of N if ~t~T bt >~ bo, and )~t~T at > ao, A partial cover C can be associated with a 0-1 vector x C • {0, 1}" (xc -- 1, j • C; x c = 0, j ¢ C), which is feasible with respect to the first constraint but is not feasible with respect to the second one. In order to obtain feasible solutions for (IP) from a partial cover, the set C of components fixed at one must be enlarged to satisfy the second constraint. Similarly, a complete cover Tcan be associated with a 0-1 vector x T • {0, 1}" (xf = I, j • T; x T = 0,j ~ T), which is feasible with respect to the second constraint but is not feasible with respect to the first one. In order to obtain feasible solutions for (IP) from a complete cover, the set T of components fixed at one must be reduced to satisfy the first constraint. Proposition 1.2. No enlargement of C provides a feasible solution to the problem ( I P ) / f and only if the partial
cover problem: (PCP)
Z p c = m i n ~ at + jeC
s.t.
~ bt + jEC
~
at s t
jEN\C
~
bts t >~bo,
jeN\,C
st e {0, 1}, j ~ X
has a value Zpc strictly greater than ao.
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
21
Proof. =~: Ifzpc ~< ao, the vector x given by xj = 1 j • C w C', x~ = 0 otherwise, where C' = {j • N/sj = 1} in an optimal solution to (PCP), is an enlargement of C that provides a feasible solution for (IP). : Obviously, since any enlargement of C satisfying the second constraint violates the first one. A partial cover C corresponds to a vertex Vc of the unit hypercube which does not belong to P~; when the value of the associated (PCP) is strictly greater than ao, none of the vertices that can be obtained by lifting Vc belong to P~, since they will violate either the first or the second constraint. In this case, at least one of the variables of C must be zero in any feasible solution for (IP) and, therefore, the partial cover inequality (PCI) Zj~cXj<~ ICI- 1 (1), is valid for PX. PCIs are valid inequalities for (IP) that are stronger than any of the ordinary cover inequalities that can be derived from constraint 1 individually, since partial covers are contained in covers of constraint 1. Obviously, the strongest PCIs will be derived from minimal partial covers, i.e. partial covers that do not contain any partial cover.
Proposition 1.3. No reduction of T provides a feasible solution to the problem (IP) if and only if the complete cover problem: (CCP)
zcc = max ~ b~sj jeT s.t.
2 ajsj~ jeT
a o,
s j • {0, 1}, j E N has a value Zcc strictly smaller than bo.
Proof. =~ : Ifzcc >~ bo, the vector x given by xj = l j • T', xi = 0 otherwise, where T' = {j E T/s~ = 1} in an optimal solution to (CCP), is a reduction of T that provides a feasible solution for (IP). : Obvious, since any reduction of T satisfying the first constraint violates the second one. Again, a complete cover T corresponds to a vertex vr of the unit hypercube which is not in P~; now, when the value of the associated (CCP) is strictly smaller than b0, none of the vertices obtained projecting vr belong to P~ since they violate either the first or the second constraint. In this case, at least one of the variables which are not in T must be 1 in any feasible solution for (IP) and, therefore, the complete cover inequality (CCI) ~jEN\TXj >/ 1 (2) is valid for pi. CCIs are also valid inequalities valid for (IP) stronger than any of the ordinary cover inequalities that can be derived from constraint 2 individually. Now the strongest CCIs will be derived from maximal complete covers, i.e. complete covers that are not contained in any complete cover.
2. Partial and complete cover inequalities violated by a given solution x* In this section we will see under which conditions it is possible to derive valid inequalities (1) and (2) violated by a given solution x* and how these inequalities can be obtained. From an algorithmic point of view, this will allow us to define procedures for identifying such inequalities and embedding them in an algorithmic scheme for solving (IP) using an LP solver. The following results can be considered as extensions of the well-known similar result for standard 0-1 knapsack problems [3]; this is not surprising, given that (IP) can be considered as a standard 0-1 knapsack problem with an additional constraint.
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
22
Proposition 2.1. Given a vector x*, 0 <~x* <<.1 j ~ N, a PCI violated by x* exists if and only if the solution to the program (P1)
z* = min y (1 - x*) sj j~N
s.t.
aj sj <. ao,
(3)
b~si < bo,
(4)
aj(sj + tj) > ao,
(5)
jeN jeN teN
sje
{0, 1}, Vj ~ N, and t solving the problem
min
(6)
~ aitj jeN
s.t.
bj(sj + tj) >1 bo,
(7)
sj + tj <~ 1 Vj e N
(8)
jeN
t i e {0, 1}
has a value strictly smaller than one. Moreover, in this case, the inequality y,j~c Xj <~ I C [ - 1, where C = {j ~ N/sj = 1} is a PCI violated by x*. Constraints (3) and (4) ensure that C is a partial cover. On the other hand, (5)-(8) guarantee that the solution to the (PCP) associated with C has a value greater than ao and that, therefore, the corresponding PCI is valid for (IP). z* < 1 ensures that this PCI is violated by x*.
Proposition 2.2. Given a solution x*, 0 <<.x* <~ l j ~ N, a CCI violated by x* exists if and only if the solution to the program (P2)
x* sj
z* = max jeN
s.t.
(9)
a j s j > ao, jeN
bjsj >1 bo
(lO)
bj(sj - t j) < bo,
(11)
j~N jsN
sje {0, 1 }, min
Vj ~ N, and t solving the problem
(12)
Z bjtj j~N
s.t.
Z aj(sj -- tj) <~ ao, jeN
si--ti>~O
Vj~N
(13) (14)
tj~{o, 1}, has a value strictly greater than Y~j~Nx* - 1. Moreover, in this case, ~ j~s~r xj >. 1 is a CCI violated by x* where T = {j ~ N/sj = 1}.
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
23
Now, constraints (9) and (10) ensure that Tis a complete cover. On the other hand, (11)-(14) guarantee the solution to the (CCP) associated with T has a value smaller than bo and, therefore, the corresponding CCI is valid for (IP). The value z* of the objective function guarantees that this CCI is violated by x*. The above two programs are bilevel problems which, in general, are difficult to solve. Nevertheless, feasible solutions to these problems with objective function values smaller than one or greater than ~j~N x* - 1, respectively, provide valid PCIs or CCIs violated by x*. In the following section we propose two heuristics to obtain such feasible solutions.
3. Heuristics for identifying partial cover and complete cover inequalities violated by a given solution x* In this section we present a general scheme for heuristics to identify violated PCIs and CCIs; the goal of these procedures is finding feasible solutions to (P1) and (P2) that fulfill the conditions of Propositions 2.1 and 2.2, respectively. This framework can be utilized by any heuristics for identifying partial (or complete) covers. We also present the greedy procedures which we are currently using.
3.1. Heuristic for identifying a PCI violated by a given solution x* Given x*, the heuristic is as follows: (1) Select a suitable C _~ N such that (i) ~.j~caj<<.ao,
(ii) Zj~cbj
and
(iii) ~ c
x* > I C [ -
1.
(2) Solve the associated (PCP) defined in Section 1. (3) If Zpc > ao, the PCI ~j~c xj ~< [CI - 1 is valid for (IP) and violated by x*. In the first step of this heuristic the partial cover C must be chosen to have as few elements as possible (to provide a strong inequality) but these elements must be taken as to make C derive a valid PCI for (IP). As we want to have a value of (PCP) strictly bigger than ao, we are currently using a greedy procedure in which the variables are included in C according to decreasing values of aj/bj. Note that this ordering favours having a high fixed value in the objective function of (PCP) relative to a small fixed value in the constraint of (PCP). Of course, only variables with values preserving condition (iii) can be included.
3.2. Heuristic for identifying a CCI violated by a given solution x* Given x*, the heuristic is as follows: (1) Select a suitable T ~_ N such that (i) Z ~ T a j > a o , (ii) ~j~Tbj>~ bo and (iii) ~jEN\TX~ • 1 (2) Solve the associated complete cover problem (CCP) defined in Section 1. (3) Ifzcc < bo, the CCI ~j~N\rxj ~> 1 is valid for (IP) and violated by x*. In the first step of this heuristic the set T must be chosen to have as many variables as possible in order to keep N \ T small and, therefore, have an inequality as strong as possible. As we want to have the value of (CCP) smaller than bo we are currently using a greedy procedure in which the variables are included in T according to increasing values of bJa~. Note that with this ordering the variables that appear in (CCP) have small coefficients in the objective function with respect to their coefficient in the constraint of (CCP). Now we must include in T all the variables with value 1 as well as enough of the remaining variables as to attain condition (iii).
24
E. Ferndmdez, K. Jornsten / Operations Research Letters 15 (1994) 19 33
4. Applications
As we have already mentioned in the Introduction, PCIs and CCIs can be used for solving the separation problem for different classes of 0-1 integer programs. In this section we will see how these inequalities can be applied to some specific 0-1 problems. 4.1. Knapsack problems
Consider a standard 0-1 knapsack problem of the form (KP)
max
cx
s.t.
a x <<.ao,
x {0, m} If we know a feasible solution x* for (KP) with value Co, in order to prove optimality, or to find feasible solutions with a better value, we can formulate (KP1)
max
cx
s.t.
ax ~ ao, c x >1 Co + 1,
which provides an equivalent formulation of (KP) in the sense that any feasible solution to (KP) better than x*, if it exists, must be feasible for (KP1). (KP1) has the same structure as (IP) and, therefore PCIs and CCIs can be derived. 4.2. General case
PCIs and CCIs can also be derived for general 0-1 integer programs having at least one inequality constraint, as soon as a feasible solution to the problem is known. This solution may be obtained, for instance, with the Pivot and Complement heuristic [1] applied to the solution to the LP relaxation. 4.3. Cardinality constraint
The results presented in the above sections can be easily adapted to the case of 0-1 maximization programs with a cardinality constraint of the form Y ~ N X j = k. After we know a feasible solution with value Co, we can add the constraint c x t> Co + 1, and obtain an equivalent formulation of the original problem. In this case, a partial cover is defined to be a set C ~ N, such that IC[ < k and Y.j~c cj < Co + 1. Now, when the value of the partial cover problem
min ICl+ ~ sj jeN\C
s.t.
~ cj + ~ jeC
c j s j >l Co +
1,
jeN\C
si~ {0, 1}, j a N has a value strictly greater than k, the associated PCI is valid for the original problem.
25
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
Similarly, a complete cover is defined to be a set T ~_ N, such that ~ j e r cj >1 Co + 1 and value of the complete cover problem max
ICI > k. When the
~ cs sj jeT
~
s.t.
Sj = k,
jeT
sj e {0, 1}, j e N has a value strictly smaller than Co + 1 the associated CCI is valid for the original problem. 4.4. Example
Consider the following 0-1 knapsack problem example [5-1 (KP)
Max
16xl + 1 2 x 2 + 1 4 x 3 + 1 7 X a + 2 0 x s + 2 7 x 6 + 4 x T + 6 x 8 + 8 x 9 + 2 0 X 1 o +
11xll
+ 10X12 + 7X13 , s.t.
(15)
7xl + 6x2 + 5X 3 -t- 6X4 + 7X5 + 10X6 -F 2X7 -4- 3xs + 3X9 + 9Xlo 4- 3Xll "+- 5X12 + 5x13 ~< 48 x~E{0,1},
j=
(16) 1 . . . . . 13.
The LP solution to this problem is x* = 1, j = 1,3,4,5,6,9, 11, X*o = 7/9 with value Z~p = 128 + 5/9. The greedy heuristic gives the solution xj = 1, j = 1, 2, 3, 4, 5, 6, 9, 11 with value Zheur = 125. Therefore, the constraint 16xx + 12x2 + 14x3 + 17x4 + 20x5 + 27x6 + 4 x 7 + 6x8 + 8X 9 "~ 20X10 "t- 11XXX + 10Xxz + 7X13 ~> 126
(17)
must be fulfilled by any feasible solution to (KP) better than the one provided by the heuristic. If we add it to the formulation of (KP) we get the problem (IP) (IP)
Max
(15)
s.t.
(16),(17),
x~{0,1},
j=
1 . . . . . 13.
If we now apply the heuristic proposed in Section 3 to identify a PCI violated by the solution to the LP x*, we get C = {10, 1,9,6} which is a partial cover since (i) al + a6 + a9 + alo 7 + 10 + 3 + 9 = 29 < 48 = ao, (ii) bl + b6 + b9 + blo = 16 + 27 + 8 + 20 = 71 < 126 = bo. The solution of the associated (PCP) is given by xj = 1, j = 3,4, 5, 7, x~ = 0 otherwise, and gives a value Zpc = 49 > 48 = ao. The value in (P1) for the feasible solution associated to C is =
(1-x*)+(1-x~)+(1-x~)+(1-X*o)=(1-
1)+(1-
1)+(1-1)+(1-7/9)=2/9<
1
Therefore, the PCI violated by x* is (18)
X 1 + X 6 "]- X 9 "~ X10 ~ 3.
Applying the heuristic proposed in Section 3 to T = {1,3,4,5,6,9, 10, 11} which is a complete cover since
identify
a
CCI
violated
by
x*,
we
get
E. Ferndmdez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
26
(i) a~ + a 3 + a4 + a s + a6 + a 9 + alo + a ~ = 7 + 5 + 6 + 7 + 1 0 + 3 + 9 + 3 = 50 > 4 8 , (ii) b~ + b 3 + b 4 + b 5 + b 6 + b 9 + b l o + b l l = 16 + 14 + 17 + 20 + 27 + 8 + 20 + 11 = 71 < 126. The solution of the associated (CCP) is given by xj = 1, j = 1,3, 4, 5, 6, 10, 11, xj = 0 otherwise, and gives a value Zcc = 125 < 126 = bo. The value in (P2) for the feasible solution associated to T is
x*+x*+x*+x~+x*+x*+x*o+X*~=
=
Z
1 + 1 + 1 + 1 + 1 + 1 + 7/9+ 1 =7~>6
7
x*-I
j~N',,T
Therefore, the CCI violated by x* is
xz+xv+Xs+Xlz+x13~
1
(19)
The resulting LP relaxation after adding (18) and (19) to (IP) is Max
(15)
s.t.
(16),(17),(18) and (19) O<~xj<~ 1, j ~ N .
The optimal solution to this problem is given by xj = 1, j = 1, 3, 4, 5, 6, 7, 9, 11, x~ = 0, otherwise with value Zlp~ = 127. As this solution is feasible for (IP), it is the optimal solution for (KP).
5. Computational experience In this section we present the results obtained in the computational experiments we have performed on a set of 90 0-1 knapsack test problems of different dimensions. The problems have been randomly generated as explained in [41 . The dimensions of the problems are 50, 100, 200 and 500. For each dimension, we have generated two data sets of 10 problems each. For n = 50, 100, 200 the first data set contains uncorrelated problems with cj and aj uniformly random in [1, v], with v = 100, and the second data set contains weakly correlated problems with cj uniformly random in [1, v], and aj uniformly random in [cj - r, cj + r]; v = 100 and r = 10. For n = 500, the first data set contains uncorrelated problems with cj and aj uniformly random in [1, v], with v = 1000 and the second one contains weakly correlated problems with cj uniformly random in 1-1, v], and aj uniformly random in [c~ - r, cj + r], where v = 1000 and r = 100. The capacity value we have taken is ao = 0.5 ~ at; as explained in [4], this value gives, in general, instances that are expected to be more difficult than the value ao = 2v. For n = 500 we have also generated a third set of 10 weakly correlated problems with this latter capacity value. We have applied two different LP solution schemes to the problems. In the first one, (LPSI), in each iteration we used the greedy heuristics proposed in Section 3 to derive a PCI and/or a CCI violated by the current LP solution. Previously, we performed the greedy heuristic to obtain an equivalent formulation as problems with two different sided knapsack constraints, as explained in Section 4.1. In the second scheme, (LPS2), in each iteration we derived the classical cover inequality most violated by the current LP solution and performed the lifting procedure [2] to obtain facets of the knapsack polytope. We did not choose any specific order for the lifting so the variables were lifted according to increasing values of their indices. The results we present compare the behaviour of the above LP solution schemes. In 84 out of 90 problems PCIs and CCIs gave better results than the classical cover inequalities together with the lifting procedure, in 5 problems both schemes gave the same result and only in one problem, $49, (LPS2) turned out to be better than (LPS1).
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19-33
27
In Tables 1-9 we present the results obtained. The first four columns give results related to the original problems. Heur is the value of the greedy heuristic, Opt the value of the optimal solution, LP the value of the linear programming relaxation and Gap the value LP-Opt. The next five columns contain the results related to (LPS1) (the LP solution scheme using PCIs and CCIs). LP1 is the value of the linear programming relaxation in the final iteration of the scheme, itl the number of iterations performed, pc and cc are, respectively, the number of PCIs and CCIs obtained (since in some iterations only one type of inequality could be derived), and Gapl is the value LPl-Opt. Columns 10-12 give the results related to (LPS2) (the LP solution scheme using cover inequalities plus the lifting procedure). LP2 is the value of the linear programming relaxation in the final iteration of the scheme, ci the number of cover inequalities derived which, in this case, coincides with the number of iterations performed, and Gap2 is the value LP2-Opt. Finally, in the last two columns % 1 and %2 give the percentage of reduction of the original gap (Gap) attained with (LPS1) and (LPS2), respectively. We find that these two columns are significant since they permit to compare the behaviour of the above schemes on the same problem. We have not included any information about the computational effort needed to generate the different types of inequalities. The reason is that our aim was to compare the strength of PCIs and CCIs with that of classical cover inequalities together with a lifting procedure. In this sense, we consider that choosing the most violated cover inequality instead of choosing the cover heuristically makes our comparison more valuable. Nevertheless, in our experience PCIs and CCIs were, in most cases, easier to find than classical lifted inequalities. The reason is quite simple: a 0-1 knapsack problem must be solved each time a PCI or a CCI is obtained, but this is also the case for generating the most violated cover inequality where, in addition, the lifting procedure also requires solving a collection of 0-1 knapsack problems. As we can see, most of the test problems have a small gap between the value of the linear programming relaxation and the value of the optimal 0-1 solution; nevertheless, this gap has been difficult to reduce. For this reason we consider that the percentage of the reduction of this gap is a good measure of the efficiency of the LP solution schemes, especially if we want to compare the behaviour of different procedures. We would like to stress the fact that, in our computational experience, the proposed PCIs and CCIs give results that are considerably better than those obtained with the classical cover inequalities plus the lifting procedure. As we have already mentioned, this has been so in all the test problems except one, $49, for the different dimensions and kind of problems (uncorrelated or weakly correlated). In particular, the percentage of the reduction of the gap achieved with (LPS1) is in most of the problems twice or three times the improvement attained with the classical scheme. The values of LP relaxations marked with an asterisk correspond to problems for which a 0-1 optimal solution has been found. As we can see, this has happened with 15 problems using (LPSI) and only with three of these problems with (LPS2). Again, PCIs and CCIs have given better results than cover inequalities for finding an optimal solution. We would like to mention problems $23 and $27 of Table 6 for which the original linear programming relaxation already gave an optimal value; nevertheless, since this value corresponds to solutions which are not integral, optimality could not be proved. For both problems (LPS1) identified an optimal 0-1 solution, hence proving optimality; only one of them was optimally solved with (LPS2). Also, problem $28 was optimally solved by (LPS1); for this problem (LPS2) gave an optimal value, but did not identify an optimal 0 1 solution and, therefore, we cannot consider that it was solved optimally with (LPS2). With respect to the number of PCIs and CCIs derived in (LPS1), we can see that, in general, the number of PCIs is higher than the number of CCIs. In fact, in our experience, very seldom the procedure identified a CCI in one iteration in which a PCI was not derived. We have not appreciated significant differences between both schemes with respect to the number of iterations performed. Finally, we would like to point out that, although the aim of this paper was not to give an exhaustive computational study of the behaviour of PCIs and CCIs, we consider that the results obtained in the experiments we have performed confirm that, in the case of 0-1 knapsack problems, the use of PCIs and CCIs is clearly worthwhile with respect to the classical cover inequalities plus the lifting procedure.
1816 2284 2143 2007 2141 1965 2031 2114 2229 1971
1819 2295 2146 2014 2143 1966 2032 2117 2231 1984
Opt 1822.76 2303.46 2155.24 2022.32 2157.66 1969.23 2037.16 2118.09 2237.61 1990.22
LP 3.76 8.46 9.24 8.32 14.66 3.23 5.16 1.09 6.61 6.22
Gap 1820.61 2299.40 2152.65 2014.71 2144.60 1966.71 2034.08 2117.00" 2234.98 1985.45
LP1 5 12 28 12 4 29 30 2 4 4
itl 5 12 28 12 4 29 30 2 4 4
pc 1 2 6 2 2 27 25 1 1 1
cc
1424 1319 1427 1458 1244 1370 1305 1472 1289 1394
1427 1323 1429 1464 1249 1371 1306 1473 1291 1400
Opt 1428.25 1324.84 1430.89 1466.13 1249.20 1372.81 1311.00 1474.38 1292.84 1400.72
LP 1.25 1.84 1.89 2.13 0.20 1.81 5.00 1.38 1.84 0.72
Gap 1427.00" 1324.00 1429.94 1465.30 1249.00" 1371.00" 1307.02 1473.00" 1291.50 1400.38
LP 1 4 5 20 13 1 4 4 5 14 5
it I
3 5 20 13 1 3 3 5 10 4
pc
3 2 17 9 0 4 4 2 4 4
cc
0.00 1.00 0.94 1.30 0.00 0.00 1.02 0.00 0.50 0.38
Gap 1
1.61 4.40 6.65 0.71 1.60 0.71 2.08 0.00 3.98 1.45
Gapl
1428.13 1324.69 1430.85 1465.92 1249.00" 1372.60 1310.68 1474.05 1292.50 1400.66
LP2
1821.26 2300.73 2153.43 2020.53 2150.72 1968.64 2035.86 2117.00* 2235.84 1985.45
LP2
5 13 3 15 2 4 18 7 4 3
ci
14 9 2 11 6 6 19 4 5 9
ci
1.13 1.69 1.85 1.92 0.00 1.60 4.68 1.05 1.50 0.66
Gap2
2.26 5.73 7.43 6.53 7.72 2.64 3.86 0.00 4.84 1.45
Gap2
n = 50 Weakly correlated: cj uniformly random in [1, v], a r uniformly random in [cj - r, c~ + r]; v = 100, r = 10; ao = 0.5 E a r.
SI $2 $3 $4 $5 $6 $7 $8 $9 S10
Heur
Table 2
n = 50 Uncorrelated: cj and a~ uniformly random in [1, v], v = 100; ao = 0.5 Ea r.
UI U2 U3 U4 U5 U6 U7 U8 U9 U10
Heur
Table 1
100.00 45.65 50.26 38.67 100.00 100.00 79.60 100.00 72.82 47.22
%1
57.18 47.99 28.03 91.46 89.08 78.01 59.68 100.00 39.78 76.68
%1
9.60 8.15 2.11 9.43 100.00 11.60 6.40 23.91 18.47 8.33
%2
39.89 32.26 19.58 21.51 47.33 18.26 25.19 100.00 26.77 76.68
%2
I
¢<
e~
t~
3830 4118 4277 4107 3922 4371 4117 4047 3883 4165
3840 4121 4280 4109 3927 4375 4118 4048 3890 4172
Opt
2865 2910 2490 2796 2650 2841 2694 2631 2915 2966
2869 2916 2492 2798 2651 2843 2695 2632 2920 2970
Opt
3840.52 4121.25 4281.43 4111.56 3927.60 4375.00* 4118.00" 4049.18 3893.52 4173.74
LP1 8 5 6 8 6 3 8 10 4 2
itl 8 4 6 8 6 3 8 10 4 1
pc 1 5 5 3 1 2 7 6 1 2
cc
2869.93 2916.71 2493.43 2800.14 2652.00 2845.00 2695.99 2633.43 2920.59 2972.00
LP 0.93 0.71 1.43 2.14 1.00 2.00 0.99 1.43 0.59 2.00
Gap 2869.59 2916.46 2492.93 2799.24 2651.46 2844.42 2695.78 2633.08 2920.16 2971.31
LPI 14 8 5 1 4 20 12 5 3 3
itl 14 8 5 1 4 20 12 5 3 2
pc 0 0 0 1 4 4 6 2 2 3
cc
0.59 0.46 0.93 1.24 0.46 1.42 0.78 1.08 0.16 1.31
Gapl
0.52 0.25 1.43 2.56 0.60 0.00 0.00 1.18 3.52 1.74
Gapl
2869.66 2916.51 2493.08 2799.73 2651.95 2844.93 2695.95 2633.32 2920.39 2971.91
LP2
3841.02 4121.84 4282.04 4114.08 3928.28 4376.14 4122.12 4054.12 3897.90 4174.72
LP2
6 8 9 5 26 5 3 4 5 5
ci
10 3 3 6 13 9 9 18 17 5
ci
0.66 0.51 1.08 1.73 0.95 1.93 0.95 1.32 0.39 1.91
Gap2
1.02 0.84 2.04 5.08 1.28 1.14 4.12 6.12 7.90 2.72
Gap2
n = 100 Weakly correlated: cj uniformly random in [1, v], a t uniformly random in [c~ - r, c~ + r]; v = 100, r = 10; a0 = 0.5 Zaj.
SI 1 S12 S13 S14 S15 S16 S17 S18 S19 $20
Heur
Table 4
2.17 1.36 3.24 5.54 2.75 1.57 5.79 7.02 9.34 3.63
Gap
aj uniformly random in [1, v], v = 100; ao = 0.5 Y~aj.
3842.17 4122.36 4283.24 4114.54 3929.75 4376.57 4123.79 4055.02 3899.34 4175.63
LP
n = 100 Uncorrelated: cj and
Ull U12 UI3 U14 U15 U16 UI7 U18 U19 U20
Heur
Table 3
36.55 35.21 34.96 42.05 54.00 29.00 21.21 24.47 72.88 34.50
%1
76.03 81.61 55.86 53.79 78.18 100.00 100.00 83.19 62.31 52.06
%1
29.03 28.16 24.47 19.15 5.00 3.50 4.04 7.69 33.89 4.50
%2
52.99 38.23 37.03 8.30 53.45 27.38 28.84 12.82 15.41 25.06
%2
I
,2
e~
5
t~
8413 8575 8813 8439 7759 8360 8001 8520 8391 8395
8414 8579 8814 8441 7761 8365 8007 8523 8392 8400
Opt 8415.96 8582.31 8817.70 8443.09 7762.50 8365.83 8007.94 8526.19 8397.09 8400.34
LP 1.96 3.31 3.70 2.09 1.50 0.83 0.94 3.19 5.09 0.34
Gap 8414.87 8580.83 8815.45 8442.44 7761.00" 8365.00* 8007.52 8524.78 8392.78 8400.00*
LP1 4 8 8 4 4 3 6 10 15 4
it 1 4 8 4 4 4 3 6 10 15 4
pc 3 4 8 2 3 3 1 6 10 3
cc
5665 5749 5917 5077 5465 5655 5574 5810 5167 5123
5667 5755 5919 5078 5466 5656 5575 5811 5169 5124
Opt 5667.21 5755.17 5919.00 5979.11 5466.85 5656.78 5575.00 5811.77 5169.45 5125.03
LP 0.21 0.17 0.00 1.11 0.85 0.78 0.00 0.77 0.45 1.03
Gap 5667.00* 5755.08 5919.00" 5078.55 5466.59 5656.58 5575.00* 5811.00" 5169.20 5124.58
LP1 3 2 6 14 9 7 11 2 6 11
itl
3 2 6 14 9 7 11 2 6 11
pc
2 0 1 1 9 4 6 3 1 8
cc
0.00 0.08 0.00 0.55 0.59 0.58 0.00 0.00 0.20 0.58
Gapl
0.87 1.83 1.45 1.44 0.00 0.00 0.52 1.78 0.78 0.00
Gapl
5667.01 5755.08 5919.00 5078.94 5466.83 5656.76 5575.00* 5811.00 5169.37 5124.95
LP2
8415.62 8581.72 8816.56 8442.76 7761.91 8365.09 8007.57 8526.08 8396.65 8400.25
LP2
3 2 4 32 4 2 2 2 9 6
ci
4 15 5 7 10 4 17 6 8 4
ci
0.01 0.08 0.00 0.94 0.83 0.76 0.00 0.00 0.37 0.95
Gap2
1.62 2.72 2.56 1.76 0.91 0.09 0.57 3.08 4.65 0.25
Gap2
n = 200 Weakly correlated: ('j uniformly random in [1, v], aj uniformly random in [G - r, cj + r]; v = 100, r = 10; ao = 0.5 E a r.
$21 $22 $23 $24 $25 $26 $27 $28 $29 $30
Heur
Table 6
n = 200 Uncorrelated: cj and aj uniformly random in [1, v], v = 100; ao = 0.5 Eai.
U21 U22 U23 U24 U25 U26 U27 U28 U29 U30
Heur
Table 5
100.00 52.94 100.00 50.45 30.58 25.64 100.00 100.00 55.55 43.68
%1
55.61 44.71 60.81 31.10 100.00 100.00 44.68 44.20 84.67 100.00
%1
95.23 52.94 0.00 15.31 2.35 2.65 100.00 100.00 17.77 7.76
%2
17.34 17.82 30.81 15.79 39.33 89.15 39.36 3.44 8.64 26.47
%2
I
e~
tl
200872 200298 204001 199095 202087 203252 200694 192911 204424 198940
200873 200324 204041 199140 202104 203297 200696 192921 204425 198944
Opt 200887.82 200335.39 204045.23 199153.92 202122.95 203318.78 200710.11 192938.14 204451.51 198961.53
LP 14.82 11.39 4.23 13.92 18.95 21.78 14.11 17.14 26.51 17.53
Gap 200878.70 200331.94 204042.80 199150.00 202114.64 203307.36 200699.06 192931.14 204437.42 198946.99
LP1 13 2 2 5 5 7 15 18 10 6
itl 13 1 2 5 4 7 12 18 10 4
pc 9 2 1 1 5 1 15 4 10 6
cc
138197 133753 142597 135169 132918 136294 140569 134675 140187 137355
138234 133764 142605 135173 132926 136308 140570 134690 140211 137361
Opt 138238.92 133765.00 142610.95 135178.68 132931.34 136313.26 140575.37 134694.46 140213.00 137365.89
LP 4.92 1.00 5.95 5.68 5.34 5.26 5.37 4.46 2.00 4.89
Gap 138237.46 133764.00* 142609.23 135177.57 132930.00 136312.10 140574.06 134693.18 140212.50 137363.92
LP 1 6 3 4 9 4 2 4 2 2 6
it I 6 3 4 9 4 2 4 2 2 6
pc 0 3 4 3 3 1 4 1 2 5
cc
3.46 0.00 4.23 4.57 4.00 4.10 4.06 3.18 1.50 2.92
Gap 1
5.70 7.94 1.80 10.00 10.64 10.36 3.06 10.14 12.42 2.99
Gapl
138237.87 133764.85 142609.71 135178.06 132930.92 136312.75 140575.27 134693.97 140212.57 137365.83
LP2
200887.61 200332.76 204043.74 199153.20 202120.45 203310.37 200708.87 192935.95 204446.71 198960.40
LP2
14 9 6 19 11 4 2 4 8 17
ci
26 4 3 11 16 5 3 3 20 3
ci
3.87 0.85 4.71 5.06 4.92 4.75 5.27 3.97 1.57 4.83
Gap2
14.61 8.76 2.74 13.20 16.45 13.37 12.87 14.95 21.71 16.40
Gap2
29.67 100.00 28.90 19.54 25.09 22.05 24.39 28.69 25.00 40.28
%1
61.53 30.28 57.44 28.16 43.85 52.43 78.31 40.84 53.14 82.94
%1
n = 500 Weakly correlated: cj uniformly random in [1, v], a i uniformly random in [c i - r, c'j + r]; v = 1000, r = 100; ao = 0.5 E a~.
$31 $32 $33 $34 $35 $36 $37 $38 $39 $40
Heur
Table 8
n = 500 Uncorrelated: G and aj uniformly random in [1, v], v = 1000; ao = 0.5 Z a r.
U31 U32 U33 U34 U35 U36 U37 U38 U39 U40
Heur
Table 7
21.34 15.00 20.84 10.91 7.86 9.69 1.86 10.98 21.50 1.22
%2
1.41 23.09 35.22 5.17 13.19 38.61 8.78 12.77 18.10 6.44
%2
t~a
t.i
3685 3815 3904 4058 3683 3327 3715 3901 3862 3701
3704 3822 3908 4069 3700 3336 3717 3906 3885 3728
Opt 3718.13 3828.53 3914.06 4082.31 3705.25 3351.84 3734.19 3928.42 3890.91 3737.81
LP 14.13 6.53 6.06 13.31 5.25 15.84 17.19 22.42 5.81 9.81
Gap 3713.00 3824.53 3909.42 4077.13 3701.43 3345.54 3718.80 3908.38 3887.60 3730.02
LP1 4 3 14 9 5 3 5 11 15 9
itl 3 3 14 9 5 3 5 11 15 9
pc 4 1 5 2 5 3 5 11 0 1
cc 9.00 2.53 1.42 8.13 1.43 9.54 1.80 2.38 2.60 2.02
Gapl 3716.63 3825.54 3913.47 4079.39 3704.29 3351.34 3733.56 3918.58 3887.47 3733.87
LP2 5 12 4 7 6 1 15 8 5 9
ci
12.63 3.54 5.47 10.39 4.29 15.34 16.56 12.58 2.47 5.87
Gap2
n = 500 Weakly correlated: ci uniformly random in [I, v], ai uniformly random in [cj - r, cj + r]; v = 1000, r = 100; ao = 2v
$41 $42 $43 $44 $45 $46 $47 $48 $49 $50
Heur
Table 9
36.30 61.25 76.56 38.91 72.76 39.77 89.52 89.38 55.24 79.40
%1
10.61 45.78 9.73 21.93 18.28 3.15 3.66 43.88 57.48 40.16
%2
I
~n
E. Fernhndez, K. Jornsten / Operations Research Letters 15 (1994) 19 33
33
Therefore, we consider that the application of these inequalities to more general 0-1 problems, as we have already pointed out in Section 4, seems very promising especially if we take into account that PCIs and CCIs can be derived in the same cases that cover inequalities are already being used.
Acknowledgements The authors are very grateful to the anonymous referee whose detailed comments contributed to improve the paper considerably. This work has been partially supported by the C.I.R.I.T, Generalitat de Catalunya.
References [1] E. Balas and R. Martin, "Pivot and complement: A heuristic for 0 1 programming", Mgmt Sci., 26, pp. 86-89, 1980. [2] E. Balas and E. Zemel, "Facets of the knapsack polytope from minimal covers", SIAM J. Appl. Math. 34, pp. 119-148, 1978. [3] H.P. Crowder, E.L. Johnson and M. Padberg, "Solving large scale zero-one linear programming problems", Oper. Res. 31, pp. 803 834, 1983. [4] S. Martello and P. Toth, Knapsack Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York, 1990. [5] G.I. Nemhauser and L.A. Wolsey, Inteoer and Combinatorial Optimization, Wiley-lnterscience Series in Discrete Mathematics and Optimization, Wiley, New York, 1988.