- Email: [email protected]
Strongly polynomial-time approximation for a class of bicriteria problems
Operations Research Letters 32 (2004) 530 – 534
Operations Research Letters www.elsevier.com/locate/dsw
Strongly polynomial-time approximation for a class of bicriteria problems Asaf Levin Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel Received 18 May 2003; received in revised form 3 February 2004; accepted 3 February 2004
Abstract Consider the following problem: given a ground set and two minimization objectives of the same type 0nd a subset from a given subset-class that minimizes the 0rst objective subject to a budget constraint on the second objective. Using Megiddo’s parametric method we improve an earlier weakly polynomial time algorithm. c 2004 Elsevier B.V. All rights reserved. Keywords: Parametric method; Bicriteria optimization; Approximation algorithms
1. Introduction
Assumption 2. f(g1 ; E) ¿ 0
Suppose that a ground set E of m elements, a bound D, a pair of non-negative integer cost functions c and d over E, and a set SOL of feasible subsets (e.g., the edge sets of spanning trees of a given input graph) are given. An element cost function g is a non-negative function de0ned over E. For E ⊆ E, let the g-cost of E denoted as f(g; E) be the cost of E under the element cost function g. In this paper we consider the following type of BICRITERIA COMBINATORIAL OPTIMIZATION PROBLEM: Find a subset E ∈ E such that f(d; E) 6 D and f(c; E) is minimized. Given constants a; b and element cost functions g1 and g2 , ag1 + bg2 denotes the composite function that assigns a cost ag1 (e) + bg2 (e) to each element in E. We assume that g1 =c and g2 =d satisfy the following:
Assumptions 1 and 2 provide the monotonicity property: if g1 ; g2 are element cost functions such that g1 (e) ¿ g2 (e) ∀e ∈ E, then f(g1 ; E) ¿ f(g2 ; E) ∀E ∈ SOL. The interesting case in which Assumptions 1 and 2 hold is when f is a weighted sum of the costs of elements in E (where the weights may depend on E). In Section 4 we discuss the use of these assumptions in the results of this paper and similar assumptions in the paper of Marathe et al. [3]. An (; )-approximation algorithm for a bicriteria combinatorial optimization problem is a polynomial-time algorithm that produces a solution E in which f(d; E) 6 D and f(c; E) is at most times the cost of any solution that satis0es the budget constraint D. For a constant a, we let a+ and a− denote the symbols a + and a − for in0nitesimally small ¿ 0. Marathe et al. [3] presented the parametric search method, a general method for bicriteria combinatorial
Assumption 1. f(ag1 + bg2 ; E) = af(g1 ; E) + bf(g2 ; E) ∀E ∈ SOL. E-mail address: [email protected] (A. Levin).
c 2004 Elsevier B.V. All rights reserved. 0167-6377/$ - see front matter doi:10.1016/j.orl.2004.02.002
∀E ∈ SOL.
A. Levin / Operations Research Letters 32 (2004) 530 – 534
optimization problems. Given an element cost function g, let A denote a -approximation algorithm for the single objective problem de0ned as minimizing f(g; E) subject to E ∈ SOL. We let A(g) denote the g-cost of the solution returned by A when applied with the cost function g. Let ¿ 0 be an accuracy parameter. Let U be an upper bound on the optimal c-cost. The parametric search method (the description of [3] diGers from the description below but both describe the same method) 0nds an approximate solution of the value ∗ which satis0es: ∗+ ∗− A c+ d 6 ∗ (1 + ) 6 A c + d : D D (1) The approximate solution is obtained using a binary search over that 0nds an interval of length that contains ∗ . In Algorithm Parametric-Search of [3], = 1. Marathe et al. proved that given ; ¿ 0 there is a ([1 + 1= + q1 ()]; [1 + + q2 ()])-approximation algorithm where q1 ; q2 are non-negative functions that approach zero as their argument approaches zero. The algorithm is weakly-polynomial with worst-case time complexity of (log U + log1=) times the complexity of A. Let OPT be an optimal solution to the input bicriteria combinatorial optimization problem. We assume that A is a -approximation algorithm for the single objective problem that uses only additions, multiplications by constants, and comparisons. Denote by Ta (m) (Tc (m)) the maximum number of additions and multiplications (comparisons) done by A when it is applied to a ground set with m elements. We 0rst assume that = 1, and present a (1 + 1=; 1 + )-approximation algorithm with time complexity O(Ta (m) + Tc (m)[Ta (m) + Tc (m)]). This approximation algorithm is based on using Megiddo’s parametric method [4] to solve exactly the following equation in the “parametric search method”: ∗ A c+ (2) d = ∗ (1 + ): D Next, we generalize the method to the case where each computation gives an approximate solution, and use this generalization to obtain a ([1 + 1=]; [1 + ])-approximation algorithm with time complexity O(Ta (m) + Tc (m)[Ta (m) + Tc (m)]). These results improve the one of [3] both in the approximation ratio
531
and in the time complexity (if the size of the costs in the input is suLciently large). In particular, if A is a strongly polynomial algorithm, then our method is strongly polynomial whereas the method in [3] is only weakly polynomial. We note that in many cases the time-complexity of our method can be reduced using the methods of Megiddo [5] and Cole [1]. But such uses are problem-speci0c. 2. An approximation algorithm for the case = 1 In this section we assume that A solves (optimally) the single objective problem (i.e., = 1). Assume that ¿ 0. We consider the following function of : d − (1 + ): F() = A c + D F is a piecewise-linear continuous (strictly monotone decreasing) function because for each E ∈ SOL, FE () = f c + d; E − (1 + ) D f(d; E) = f(c; E) + − (1 + ) D is a linear function of , and F is the minimum of FE over E ∈ SOL. We use Megiddo’s method [4] (see also [2,5]) to obtain a value ∗ such that F(∗ ) = 0, i.e., ∗ A c+ (3) d = ∗ (1 + ): D We now describe the use of Megiddo’s method. We execute A with the cost function c + (∗ =D)d without knowing ∗ in advance. In each phase we maintain an interval I = [low ; hi ] such that low A c+ (4) d ¿ low (1 + ) D and hi A c+ d 6 hi (1 + ): D
(5)
Since F is continuous, A(c + (=D)d) is continuous, and therefore, in each phase there is a ∗ ∈ I such that A(c + (∗ =D)d) = ∗ (1 + ). Initially, we set low = 0 that satis0es (4) and hi = ∞ that satis0es (5) because for a solution whose d-cost is at most D, the coeLcient of hi on the left-hand side is less than 1, whereas its
A. Levin / Operations Research Letters 32 (2004) 530 – 534
coeLcient on the right-hand side is 1 + , therefore, property (5) will be satis0ed for a suLciently large value of the parameter . While executing A, we need to compare pairs of linear functions of according to their relative order for = ∗ . To do so, we 0rst compute the break-point br of the two linear functions. If there is no such break-point, then the comparison is independent of (the two functions diGer by a constant), and we can resolve it in constant time. Next, we check if br ∈ I . If not, then the comparison is independent of the value of ∈ I , and we can resolve it in constant time. Assume that br ∈ I . We execute A with cost function c+(br =D)d. If A(c+(br =D)d) ¿ br (1+), then we set low = br , otherwise, we set hi = br . Now the comparison is independent of for ∈ I , and therefore, we can resolve it in constant time. We return the solution HEU that is found by A when it is applied to the cost function c + (∗ =D)d. The time complexity of our algorithm is O(Ta (m) + Tc (m)[Ta (m) + Tc (m)]). The proof of the next theorem is similar to the proof of Claim 6.2 in [3], and it follows by the proof of Theorem 4 below. Theorem 3. The algorithm described above is a (1 + 1=; 1 + )-approximation algorithm. 3. Analysis of the case ¿ 1 Given ¿ 0, denote by OPT the cost of an optimal solution to the single objective problem with the element cost function c + (=D)d. In this section we assume that A is a deterministic -approximation algorithm (i.e., for each , A(c + (=D)d) 6 OPT ). Denote by ∗ a value of that satis0es A(c + (=D)d) = (1 + ). However, A is an approximation algorithm and not an exact algorithm, therefore there may be jumps in the cost of the solution returned by A as a function of . Therefore, we cannot use the method described in the previous section to 0nd ∗ . Instead we 0nd a value ˆ of such that either ˆ ˆ + ) A c + d = (1 (6) D or
ˆ− ˆ+ ˆ + ) 6 A c + A c+ d 6 (1 d : D D
(7)
F(λ)
532
λ jump
λ
Fig. 1. An illustration of F().
ˆ Since A is We now show how to compute . deterministic and it is composed of a 0nite number of additions and comparisons, the function F() = A(c + (=D)d) is a piecewise-linear function of (see Fig. 1 for an illustration of F where F has a single jump-point at jump ). The break-points and the jump-points of F() are points where the solution returned by A changes. Since the cost of each solution E ∈ SOL according to the element cost function c + (=D)d is a non-decreasing function of , OPT is a non-decreasing function (of ). Therefore, since A is a -approximation, for a jump-point j of F, F(j− ) 6 OPT− 6 OPTj 6 OPTj+ : j
(8)
We extend the parametric method of [4] in the following way: we execute A with cost function c + ˆ (=D)d without knowing ˆ in advance. In each phase we maintain an interval I = [low ; hi ] such that low (9) A c+ d ¿ low (1 + ) D and hi d 6 hi (1 + ): A c+ D
(10)
In each phase there exists a value ˆ ∈ I such that either ˆ ˆ ˆ A(c+(=D)d)= (1+) or A(c+(ˆ+ =D)d) 6 (1+ − ˆ ) 6 A(c + ( =D)d). Similar to the = 1 case, initially we set low = 0 that satis0es (9) and hi = ∞ that satis0es (10). While executing A, we need to compare pairs of linear functions of according to their relaˆ To do so, we 0rst compute the tive order for = .
A. Levin / Operations Research Letters 32 (2004) 530 – 534
break-point br of the two linear functions. If there is no such break-point or br ∈ I , then the comparison is independent of the value of ∈ I and we can resolve it in constant time. Assume that br ∈ I . We execute A − − + in br , br and br . If br is a jump-point (i.e., F(br ), + F(br ) and F(br ) are not all equal), then we check − + ) 6 br (1 + ) 6 F(br ). If this holds, then if F(br ˆ =br . In any other case (also if br is not a jump-point) − + the following holds: Either F(br ); F(br ) ¿ br (1 + − + ) or F(br ); F(br ) 6 br (1 + ). In the 0rst case we − + , and in the second case we set hi = br . set low = br Now the comparison is independent of for ∈ I , and therefore, we can resolve it in constant time. The time complexity of our algorithm is O(Ta (m)+ Tc (m)[Ta (m) + Tc (m)]) using the same analysis as in [4]. If (6) is satis0ed, we return the solution that is found by A when it is applied to the element cost function ˆ If (7) is satis0ed, we return the solution c + (=D)d. that is found by A when it is applied to the element cost function c + (ˆ+ =D)d. In both cases let HEU be the returned solution. Theorem 4. The algorithm described above is a ([1+ (1=)]; [1 + ])-approximation algorithm. Proof.
ˆ− ˆ + ) 6 A(c + ( =D)d) (1 6 OPTˆ 6 f(c; OPT ) +
ˆ f(d; OPT ) D
ˆ 6 f(c; OPT ) + ;
(11)
where the 0rst inequality holds by (6) and (7), the second inequality holds because A is a -approximation and by (8), the third inequality holds because OPT is a feasible solution to the single objective problem, and the last inequality holds because f(d; OPT ) 6 D. Therefore, f(c; OPT ) ˆ 6 : (12) We conclude that ˆ f(c; HEU ) 6 f c + d; HEU D
533
6 OPTˆ ˆ 6 [f(c; OPT ) + ] 1 ; 6 f(c; OPT ) 1 + where the 0rst inequality holds because d ¿ 0, the second inequality holds because A is a -approximation algorithm and OPT is continuous (and therefore, OPTˆ = OPTˆ+ ), the third inequality holds by (11), and the last inequality holds by (12). ˆ and if (7) holds, Finally, if (6) holds, then let ˜ = , then let ˜ = ˆ+ . Then, ˜ D f(d; HEU ) 6 A c + d D ˆ 6
D ˆ (1 + ) = D(1 + ); ˆ
where the 0rst inequality holds because c ¿ 0, and the second inequality holds by (6) and (7). 4. Discussion We conclude the paper with a discussion on the use of Assumptions 1 and 2 in the results of this paper, and the use of some assumptions on f in [3]. Assumption 2 is a natural assumption. The more restricting assumption is Assumption 1. The following is an example for a function f that does not satisfy Assumption 1 where the parametric approaches of this paper and [3] fail: Let E = {e1 ; e2 ; e3 } and SOL = {S1 ; S2 }, where S1 = {e1 ; e2 } and S2 = {e2 ; e3 }. c and d are de0ned as c(e1 ) = 1, c(e2 ) = c(e3 ) = 0, d(e1 ) = 0, d(e2 ) = d(e3 ) = 1. The budget D is 1. Let M be a large number, and let f is de0ned as: 0 if g(e1 ) = 0 f(g; S1 ) = M otherwise and f(g; S2 ) = 1. Then, S2 is a feasible solution with a unit c-cost, whereas S1 does not satisfy the budget constraint. However, for all values of the optimal solution with element cost function c + d is S1 with zero cost, but f(d; S1 )=M 1. Therefore, our method and the method of [3] do not 0nd an approximated solution for this problem.
534
A. Levin / Operations Research Letters 32 (2004) 530 – 534
References [1] R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM 34 (1987) 200–208. [2] D. Gus0eld, Parametric combinatorial computing and a problem of program module distribution, J. ACM 30 (1983) 551–563.
[3] M.V. Marathe, R. Ravi, R. Sundaram, S.S. Ravi, D.J. Rosenkrantz, H.B. Hunt III, Bicriteria network design problems, J. Algorithms 28 (1998) 142–171. [4] N. Megiddo, Combinatorial optimization with rational objective functions, Math. Oper. Res. 4 (1979) 414–424. [5] N. Megiddo, Applying parallel computation algorithms in the design of serial algorithms, J. ACM 30 (1983) 852–865.
Operations Research Letters www.elsevier.com/locate/dsw
Strongly polynomial-time approximation for a class of bicriteria problems Asaf Levin Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel Received 18 May 2003; received in revised form 3 February 2004; accepted 3 February 2004
Abstract Consider the following problem: given a ground set and two minimization objectives of the same type 0nd a subset from a given subset-class that minimizes the 0rst objective subject to a budget constraint on the second objective. Using Megiddo’s parametric method we improve an earlier weakly polynomial time algorithm. c 2004 Elsevier B.V. All rights reserved. Keywords: Parametric method; Bicriteria optimization; Approximation algorithms
1. Introduction
Assumption 2. f(g1 ; E) ¿ 0
Suppose that a ground set E of m elements, a bound D, a pair of non-negative integer cost functions c and d over E, and a set SOL of feasible subsets (e.g., the edge sets of spanning trees of a given input graph) are given. An element cost function g is a non-negative function de0ned over E. For E ⊆ E, let the g-cost of E denoted as f(g; E) be the cost of E under the element cost function g. In this paper we consider the following type of BICRITERIA COMBINATORIAL OPTIMIZATION PROBLEM: Find a subset E ∈ E such that f(d; E) 6 D and f(c; E) is minimized. Given constants a; b and element cost functions g1 and g2 , ag1 + bg2 denotes the composite function that assigns a cost ag1 (e) + bg2 (e) to each element in E. We assume that g1 =c and g2 =d satisfy the following:
Assumptions 1 and 2 provide the monotonicity property: if g1 ; g2 are element cost functions such that g1 (e) ¿ g2 (e) ∀e ∈ E, then f(g1 ; E) ¿ f(g2 ; E) ∀E ∈ SOL. The interesting case in which Assumptions 1 and 2 hold is when f is a weighted sum of the costs of elements in E (where the weights may depend on E). In Section 4 we discuss the use of these assumptions in the results of this paper and similar assumptions in the paper of Marathe et al. [3]. An (; )-approximation algorithm for a bicriteria combinatorial optimization problem is a polynomial-time algorithm that produces a solution E in which f(d; E) 6 D and f(c; E) is at most times the cost of any solution that satis0es the budget constraint D. For a constant a, we let a+ and a− denote the symbols a + and a − for in0nitesimally small ¿ 0. Marathe et al. [3] presented the parametric search method, a general method for bicriteria combinatorial
Assumption 1. f(ag1 + bg2 ; E) = af(g1 ; E) + bf(g2 ; E) ∀E ∈ SOL. E-mail address: [email protected] (A. Levin).
c 2004 Elsevier B.V. All rights reserved. 0167-6377/$ - see front matter doi:10.1016/j.orl.2004.02.002
∀E ∈ SOL.
A. Levin / Operations Research Letters 32 (2004) 530 – 534
optimization problems. Given an element cost function g, let A denote a -approximation algorithm for the single objective problem de0ned as minimizing f(g; E) subject to E ∈ SOL. We let A(g) denote the g-cost of the solution returned by A when applied with the cost function g. Let ¿ 0 be an accuracy parameter. Let U be an upper bound on the optimal c-cost. The parametric search method (the description of [3] diGers from the description below but both describe the same method) 0nds an approximate solution of the value ∗ which satis0es: ∗+ ∗− A c+ d 6 ∗ (1 + ) 6 A c + d : D D (1) The approximate solution is obtained using a binary search over that 0nds an interval of length that contains ∗ . In Algorithm Parametric-Search of [3], = 1. Marathe et al. proved that given ; ¿ 0 there is a ([1 + 1= + q1 ()]; [1 + + q2 ()])-approximation algorithm where q1 ; q2 are non-negative functions that approach zero as their argument approaches zero. The algorithm is weakly-polynomial with worst-case time complexity of (log U + log1=) times the complexity of A. Let OPT be an optimal solution to the input bicriteria combinatorial optimization problem. We assume that A is a -approximation algorithm for the single objective problem that uses only additions, multiplications by constants, and comparisons. Denote by Ta (m) (Tc (m)) the maximum number of additions and multiplications (comparisons) done by A when it is applied to a ground set with m elements. We 0rst assume that = 1, and present a (1 + 1=; 1 + )-approximation algorithm with time complexity O(Ta (m) + Tc (m)[Ta (m) + Tc (m)]). This approximation algorithm is based on using Megiddo’s parametric method [4] to solve exactly the following equation in the “parametric search method”: ∗ A c+ (2) d = ∗ (1 + ): D Next, we generalize the method to the case where each computation gives an approximate solution, and use this generalization to obtain a ([1 + 1=]; [1 + ])-approximation algorithm with time complexity O(Ta (m) + Tc (m)[Ta (m) + Tc (m)]). These results improve the one of [3] both in the approximation ratio
531
and in the time complexity (if the size of the costs in the input is suLciently large). In particular, if A is a strongly polynomial algorithm, then our method is strongly polynomial whereas the method in [3] is only weakly polynomial. We note that in many cases the time-complexity of our method can be reduced using the methods of Megiddo [5] and Cole [1]. But such uses are problem-speci0c. 2. An approximation algorithm for the case = 1 In this section we assume that A solves (optimally) the single objective problem (i.e., = 1). Assume that ¿ 0. We consider the following function of : d − (1 + ): F() = A c + D F is a piecewise-linear continuous (strictly monotone decreasing) function because for each E ∈ SOL, FE () = f c + d; E − (1 + ) D f(d; E) = f(c; E) + − (1 + ) D is a linear function of , and F is the minimum of FE over E ∈ SOL. We use Megiddo’s method [4] (see also [2,5]) to obtain a value ∗ such that F(∗ ) = 0, i.e., ∗ A c+ (3) d = ∗ (1 + ): D We now describe the use of Megiddo’s method. We execute A with the cost function c + (∗ =D)d without knowing ∗ in advance. In each phase we maintain an interval I = [low ; hi ] such that low A c+ (4) d ¿ low (1 + ) D and hi A c+ d 6 hi (1 + ): D
(5)
Since F is continuous, A(c + (=D)d) is continuous, and therefore, in each phase there is a ∗ ∈ I such that A(c + (∗ =D)d) = ∗ (1 + ). Initially, we set low = 0 that satis0es (4) and hi = ∞ that satis0es (5) because for a solution whose d-cost is at most D, the coeLcient of hi on the left-hand side is less than 1, whereas its
A. Levin / Operations Research Letters 32 (2004) 530 – 534
coeLcient on the right-hand side is 1 + , therefore, property (5) will be satis0ed for a suLciently large value of the parameter . While executing A, we need to compare pairs of linear functions of according to their relative order for = ∗ . To do so, we 0rst compute the break-point br of the two linear functions. If there is no such break-point, then the comparison is independent of (the two functions diGer by a constant), and we can resolve it in constant time. Next, we check if br ∈ I . If not, then the comparison is independent of the value of ∈ I , and we can resolve it in constant time. Assume that br ∈ I . We execute A with cost function c+(br =D)d. If A(c+(br =D)d) ¿ br (1+), then we set low = br , otherwise, we set hi = br . Now the comparison is independent of for ∈ I , and therefore, we can resolve it in constant time. We return the solution HEU that is found by A when it is applied to the cost function c + (∗ =D)d. The time complexity of our algorithm is O(Ta (m) + Tc (m)[Ta (m) + Tc (m)]). The proof of the next theorem is similar to the proof of Claim 6.2 in [3], and it follows by the proof of Theorem 4 below. Theorem 3. The algorithm described above is a (1 + 1=; 1 + )-approximation algorithm. 3. Analysis of the case ¿ 1 Given ¿ 0, denote by OPT the cost of an optimal solution to the single objective problem with the element cost function c + (=D)d. In this section we assume that A is a deterministic -approximation algorithm (i.e., for each , A(c + (=D)d) 6 OPT ). Denote by ∗ a value of that satis0es A(c + (=D)d) = (1 + ). However, A is an approximation algorithm and not an exact algorithm, therefore there may be jumps in the cost of the solution returned by A as a function of . Therefore, we cannot use the method described in the previous section to 0nd ∗ . Instead we 0nd a value ˆ of such that either ˆ ˆ + ) A c + d = (1 (6) D or
ˆ− ˆ+ ˆ + ) 6 A c + A c+ d 6 (1 d : D D
(7)
F(λ)
532
λ jump
λ
Fig. 1. An illustration of F().
ˆ Since A is We now show how to compute . deterministic and it is composed of a 0nite number of additions and comparisons, the function F() = A(c + (=D)d) is a piecewise-linear function of (see Fig. 1 for an illustration of F where F has a single jump-point at jump ). The break-points and the jump-points of F() are points where the solution returned by A changes. Since the cost of each solution E ∈ SOL according to the element cost function c + (=D)d is a non-decreasing function of , OPT is a non-decreasing function (of ). Therefore, since A is a -approximation, for a jump-point j of F, F(j− ) 6 OPT− 6 OPTj 6 OPTj+ : j
(8)
We extend the parametric method of [4] in the following way: we execute A with cost function c + ˆ (=D)d without knowing ˆ in advance. In each phase we maintain an interval I = [low ; hi ] such that low (9) A c+ d ¿ low (1 + ) D and hi d 6 hi (1 + ): A c+ D
(10)
In each phase there exists a value ˆ ∈ I such that either ˆ ˆ ˆ A(c+(=D)d)= (1+) or A(c+(ˆ+ =D)d) 6 (1+ − ˆ ) 6 A(c + ( =D)d). Similar to the = 1 case, initially we set low = 0 that satis0es (9) and hi = ∞ that satis0es (10). While executing A, we need to compare pairs of linear functions of according to their relaˆ To do so, we 0rst compute the tive order for = .
A. Levin / Operations Research Letters 32 (2004) 530 – 534
break-point br of the two linear functions. If there is no such break-point or br ∈ I , then the comparison is independent of the value of ∈ I and we can resolve it in constant time. Assume that br ∈ I . We execute A − − + in br , br and br . If br is a jump-point (i.e., F(br ), + F(br ) and F(br ) are not all equal), then we check − + ) 6 br (1 + ) 6 F(br ). If this holds, then if F(br ˆ =br . In any other case (also if br is not a jump-point) − + the following holds: Either F(br ); F(br ) ¿ br (1 + − + ) or F(br ); F(br ) 6 br (1 + ). In the 0rst case we − + , and in the second case we set hi = br . set low = br Now the comparison is independent of for ∈ I , and therefore, we can resolve it in constant time. The time complexity of our algorithm is O(Ta (m)+ Tc (m)[Ta (m) + Tc (m)]) using the same analysis as in [4]. If (6) is satis0ed, we return the solution that is found by A when it is applied to the element cost function ˆ If (7) is satis0ed, we return the solution c + (=D)d. that is found by A when it is applied to the element cost function c + (ˆ+ =D)d. In both cases let HEU be the returned solution. Theorem 4. The algorithm described above is a ([1+ (1=)]; [1 + ])-approximation algorithm. Proof.
ˆ− ˆ + ) 6 A(c + ( =D)d) (1 6 OPTˆ 6 f(c; OPT ) +
ˆ f(d; OPT ) D
ˆ 6 f(c; OPT ) + ;
(11)
where the 0rst inequality holds by (6) and (7), the second inequality holds because A is a -approximation and by (8), the third inequality holds because OPT is a feasible solution to the single objective problem, and the last inequality holds because f(d; OPT ) 6 D. Therefore, f(c; OPT ) ˆ 6 : (12) We conclude that ˆ f(c; HEU ) 6 f c + d; HEU D
533
6 OPTˆ ˆ 6 [f(c; OPT ) + ] 1 ; 6 f(c; OPT ) 1 + where the 0rst inequality holds because d ¿ 0, the second inequality holds because A is a -approximation algorithm and OPT is continuous (and therefore, OPTˆ = OPTˆ+ ), the third inequality holds by (11), and the last inequality holds by (12). ˆ and if (7) holds, Finally, if (6) holds, then let ˜ = , then let ˜ = ˆ+ . Then, ˜ D f(d; HEU ) 6 A c + d D ˆ 6
D ˆ (1 + ) = D(1 + ); ˆ
where the 0rst inequality holds because c ¿ 0, and the second inequality holds by (6) and (7). 4. Discussion We conclude the paper with a discussion on the use of Assumptions 1 and 2 in the results of this paper, and the use of some assumptions on f in [3]. Assumption 2 is a natural assumption. The more restricting assumption is Assumption 1. The following is an example for a function f that does not satisfy Assumption 1 where the parametric approaches of this paper and [3] fail: Let E = {e1 ; e2 ; e3 } and SOL = {S1 ; S2 }, where S1 = {e1 ; e2 } and S2 = {e2 ; e3 }. c and d are de0ned as c(e1 ) = 1, c(e2 ) = c(e3 ) = 0, d(e1 ) = 0, d(e2 ) = d(e3 ) = 1. The budget D is 1. Let M be a large number, and let f is de0ned as: 0 if g(e1 ) = 0 f(g; S1 ) = M otherwise and f(g; S2 ) = 1. Then, S2 is a feasible solution with a unit c-cost, whereas S1 does not satisfy the budget constraint. However, for all values of the optimal solution with element cost function c + d is S1 with zero cost, but f(d; S1 )=M 1. Therefore, our method and the method of [3] do not 0nd an approximated solution for this problem.
534
A. Levin / Operations Research Letters 32 (2004) 530 – 534
References [1] R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM 34 (1987) 200–208. [2] D. Gus0eld, Parametric combinatorial computing and a problem of program module distribution, J. ACM 30 (1983) 551–563.
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