- Email: [email protected]
Approximating linear programming is log-space complete for P
Information Processing Letters 37 (1991) 233-236 North-Holland
Deportment of Appiied Mathematics I and Department ofLmguages and I~~~~~at~~ Systems Polytechnic university of Catalonia, Av. Dr. Gregorio MaraM s/n, 0802%Barcelona, Spain Coruscate by T. Lengauer Received IS June 1990
Abstract Sema, M., Approximating linear programming is log-space complete for P, Information Processing Letters 37 (199I) 23~-23~. We consider here two approximations of the general linear programmin g problem. A solution approximation requires a vector close to an optimal vector solution in some suitable norm. A value approximation seeks for a vector at which the objective function attains a value near to the optimum. We show that a~pro~at~g within any factor c > 0 any of those problems is P-complete under log-space reductions. In order to show the above result we prove the nc: parallel appro~ab~ty of computing the number of true gates in a Boolean circuit.
Kqwords: P-compiete probIems, parallel approximabiiity, linear programming, computation II complexity
Given a F-complete problem (under NC or log-space reductions, see e.g., [2]), there is no MC algorithm to solve it unless P = NC. However, even assuming P # NC, parallelism can be of some use for P-complete problems when we focus our attention in fast parallel algorithms that give approximate solutions. In fact it is known that Pcomplete problems present a different behaviour with respect to the degree of difficufty of their approbation. For some of them, like ~ig~ ~~~~~~~ degree s~~gr~p~ [I], high vertex ( connected s~~gr~~~[S], there are values 0 < ~~ -=zq < 1 for which an NC approbation is possible * This research was done during the visit of the author to Patras diversity, and it is supported by a Spanish Research ~hol~s~p and by the ESHUT Basic Research Action No. 3075 (ALCOM).
within a factor less than co. owever for any CE > q the approximzAon is a P-complete problem. For others, as cirmit depth of ones [93 or distance PA 227 [12], we know that there is no approximation algorithm within any factor e E (0, I) of the optimal solution, unless P = MC. Thus an evidence of strong parahel ~tractab~ty, for a approximations.
In this paper we show
Volume 37. Number 4
INFORMATION
PROCESSING LEITERS
x A, an integer n-vector b, and an integer &vector c. Find a rational d-vector x such that Ax < b and cTx is maximized.
28 February 1991
We use the circuit value problem [l l] (the problem of computing ([Y,‘soutput) to prove the nonapproximability of CTG -complete for any c E (0, 11.
Looking at the definition of LP we can consider two approximations, as pointed out in [6]. The first one is an approximation on the solution space. We try to obtain a vector near to an optimal solution in some suitable norm (p-norm), independently of the value of the objective function on it. The second is an approximation on the objective function space. We seek a vector such that the objective function on it attains a value close to the optimal value. We show that both approximation problems cannot be solved in NC within any factor E:E (0, l), unless P = NC. In order to prove this result we introduce the circuit true gates problem; this problem takes as input an encoding of a Boolean circuit, and outputs the number of gates such that their output value is true. Once we prove the nonapproximability of the true gates problem, we translate the result to LP.
(Y n+k arn+k
e gates In this section we consider a function defined over a Boolean circuit, the number of true gates. oolean circuit A together with its ent is encoded by a sequence of equations A = (a,, q, . . . , a,), specifying the outputs of the gates in the circuit. The first two equations are a0 = 0 and eyl= 1, and for each i = 2,a**9n there is an equation of the form (Ye= lo+ or ak = (xiV or Lyk= ai A aj where i and j are nonnegative indices less than k. The output of the circuit is the value of Q,. Let TG(A) denote the number of gates such that their output value is 1 (true). For any particular E E (0, l] the c-true gates problem is: o”j
es ea cTG(A)Q-=TG(A).
First notice that c-CTGP is trivially in P. We now reduce CVP to E-CTGP, we provide a log-space transformation of a circuit A on n gates to a bigger circuit A’ such that if the output of A is 1, then in A’ the number of true gates is greater than In,41 and if the output of A is 0, then this number is less than n. Thus, if we had an algorithm in NC to approximate TG(A’) within a factor E, then we would get an algorithm in NC that would decide the output of A. Let A =(a,, q,..., (II,) and let I = [H/E], the new circuit is obtained from A adding ! gates, these gates will alternate between AND an OR, propagating in a trivial way the output value of circuit A. Formally, A’ = ((Ye, al, . . . , a,, a,, I’, . . . ,ay,+,) where
). Given an encoding an integer d such that
=
%+k-1
Aa
=
Ly,+k_,
v
n+k-1 a,,+k_I
for
k= 1, 3,...,
for
k=2,
As the circuit added to ,4 only propagates A’s output, we have (a) if the circuit A outputs 0, then TG(A’) = TG( A) < n; (b) if the circuit A outputs 1, then TG(A’) = TG(A) + Ia [n/cl. 0 Note that the circuit constructed in the previous theorem is planar and alternating, further all gates have fan-in two and fan-out two. So using whatever kind of circuits that follow the above restrictions, the reduction provides a log-space reduction from the specific circuit value problem to the corresponding E-true gages problem, then we have: core
.
E-CTGP is P-complete for any CE E (0, 11,
for any of the following circuit classes: binary,
monotone, fan-out two monotone, planar, alternating and fan-out two alternating.
234
4 ,....
Volume 37, Number 4
3.
FOXi
ons
INFORMATION
to ii
PROCESSING LETTERS
ing
3. The solution of the LP instance constructed above verifies
Let x0 be an optimal solution to an stance, and x an approximate solution, tinguish between two types of approximations. A solution approximation requires an x close to x0 in some suitable norm. A value approximation requires only cTx close to cTxo. Actually, we have dropped for approximate solutions the condition “ satisfy all the inequalities simultaneously”. Formally, for any particular 6 E (0, l] we consider the following approximation problems. ation to linear pro
ing (E-
). Compute a rational d-vector x such that
IIx0 IIp 3 II x II p 2 6 IIx0 II p where ]Ix ]Ip = (c~__,~ip)~/~. ation to Hi a rational crx, >, cTx 3 rcTxo. We show that both approximation problems are P-complete for any value of C. We first provide a log-space transformation from a Boolean circuit A to an LP instance. The transformation is a slight modification of the one used in [5] to show that the linear inequalities problem is P-complete. We only add an adequate objective function. Let A = (%, al,..., (Y,J be an encoding of a Boolean circuit. We consider the following transformation: (1) generate the inequalities xk < 1 and xk >, 0 forO
28 February 1991
.Y = 1
iff
a, outputs 1.
a~:“.Note that considering the set of inequalities obtabned in steps (1) to (S), we have, for any gate, if the inputs are 0 or 1, then the output (in the system) will be 0 or 1. Further the computed value corresponds to the value computed by the corresponding gate. Thus the (unique) solution of the subsystem verifies xi = 1 iff CY,has value 1. 0 The above transformation gives us a reduction from CVP to LP in such a way that the maximum possible value of the objective function is just the number of true gates. Extending this property to the p-norm of solutions we have the desired result. The following hold: (1) e-LPV is P-complete under log-space reductions for any E E (0, 11. (2) E-LPS is P-complete under log-space reductions for any E E (0, l] and any p >, 1.
f. We consider the transformation given above. By noting that the unique (optimal) system solution verifies xi = 1 iff ai has value 1, and that x is a O/l vector, we have: l/P
= II x II 1 = TG(A) for all pa
1.
Thus the above construction is a log-space transformation from r-CTGP to E-LPS or E-LPV (for any EE (0, 11). q
I want to thank Paul Spirakis for very useful comments.
‘xi)
ayr, Approkat& P-mm[l] R.J. Anderson and E.W. plete problems, Tech. Rept., Stanford University (1986). 235
Volume 37, Number 4
INFORMATION
[2] S.A. Cook, A taxonomy of problems with fast parallel algorithms, Inform and Control 64 (1985) 2-22. [3] D. Dobkin, R.J. Lipton and S. Reiss, Linear programming (1979) is log space hard for P, Inform. Process. Lett. 96-97. [4] D. Dobkin and S. Reiss, The complexity of linear programming, Theoret. Comput. Sci. 11 (1980) P-18. [5] H.J. Hoover and W.L. Ruzzo, A compedium of problems complete for P, Manuscript, 1984. [6] N. Karmarkar, A new polynomial time algorithm in linear programming, in: Proc. 16th Annual ACM Symposium on Theory of Computing (1984) 302-311. [7] L.G. Khachian, A polynomial algorithm in linear programming, D&l. Akad Nauk SSSR 20 (1) (1979) 10931096; also: Soviet Math Dokl. 20 (1) (1979) 191-194 (in English).
236
PROCESSING
LETTERS
28 February
1991
[8] L. Kirousis, M. Sema and P. Spiral&, The parallel complexity of the subgraph connectivity problem, in: Proc. 30th Annual IEEE Symposium on Foundations of Computer Science (1989) 294-299. [9] L. Kirousis and P. Spiral+, tions ad problems probabi First Scandinavian Workshop on Algorithm Theory (1988). [lo] v. ee and G.L. Minty, How good is the simplex algorithm? , in: 0. Shisha, 4. Inequalities III (Academic Press, New York, 1972) 159-179. [ll] R.E. Ladner, The circuit value problem is log-space complete for P, SIGACT News 7 (1) (1975) 18-20. [12] M. Sema and P. Spiral&, The approximability of problems complete for P, in: Proc. International Symposium on OptimaI Algorithms, Lecture Notes in Computer Science 401 (Springer, Berlin, 1989) 193-204.
Deportment of Appiied Mathematics I and Department ofLmguages and I~~~~~at~~ Systems Polytechnic university of Catalonia, Av. Dr. Gregorio MaraM s/n, 0802%Barcelona, Spain Coruscate by T. Lengauer Received IS June 1990
Abstract Sema, M., Approximating linear programming is log-space complete for P, Information Processing Letters 37 (199I) 23~-23~. We consider here two approximations of the general linear programmin g problem. A solution approximation requires a vector close to an optimal vector solution in some suitable norm. A value approximation seeks for a vector at which the objective function attains a value near to the optimum. We show that a~pro~at~g within any factor c > 0 any of those problems is P-complete under log-space reductions. In order to show the above result we prove the nc: parallel appro~ab~ty of computing the number of true gates in a Boolean circuit.
Kqwords: P-compiete probIems, parallel approximabiiity, linear programming, computation II complexity
Given a F-complete problem (under NC or log-space reductions, see e.g., [2]), there is no MC algorithm to solve it unless P = NC. However, even assuming P # NC, parallelism can be of some use for P-complete problems when we focus our attention in fast parallel algorithms that give approximate solutions. In fact it is known that Pcomplete problems present a different behaviour with respect to the degree of difficufty of their approbation. For some of them, like ~ig~ ~~~~~~~ degree s~~gr~p~ [I], high vertex ( connected s~~gr~~~[S], there are values 0 < ~~ -=zq < 1 for which an NC approbation is possible * This research was done during the visit of the author to Patras diversity, and it is supported by a Spanish Research ~hol~s~p and by the ESHUT Basic Research Action No. 3075 (ALCOM).
within a factor less than co. owever for any CE > q the approximzAon is a P-complete problem. For others, as cirmit depth of ones [93 or distance PA 227 [12], we know that there is no approximation algorithm within any factor e E (0, I) of the optimal solution, unless P = MC. Thus an evidence of strong parahel ~tractab~ty, for a approximations.
In this paper we show
Volume 37. Number 4
INFORMATION
PROCESSING LEITERS
x A, an integer n-vector b, and an integer &vector c. Find a rational d-vector x such that Ax < b and cTx is maximized.
28 February 1991
We use the circuit value problem [l l] (the problem of computing ([Y,‘soutput) to prove the nonapproximability of CTG -complete for any c E (0, 11.
Looking at the definition of LP we can consider two approximations, as pointed out in [6]. The first one is an approximation on the solution space. We try to obtain a vector near to an optimal solution in some suitable norm (p-norm), independently of the value of the objective function on it. The second is an approximation on the objective function space. We seek a vector such that the objective function on it attains a value close to the optimal value. We show that both approximation problems cannot be solved in NC within any factor E:E (0, l), unless P = NC. In order to prove this result we introduce the circuit true gates problem; this problem takes as input an encoding of a Boolean circuit, and outputs the number of gates such that their output value is true. Once we prove the nonapproximability of the true gates problem, we translate the result to LP.
(Y n+k arn+k
e gates In this section we consider a function defined over a Boolean circuit, the number of true gates. oolean circuit A together with its ent is encoded by a sequence of equations A = (a,, q, . . . , a,), specifying the outputs of the gates in the circuit. The first two equations are a0 = 0 and eyl= 1, and for each i = 2,a**9n there is an equation of the form (Ye= lo+ or ak = (xiV or Lyk= ai A aj where i and j are nonnegative indices less than k. The output of the circuit is the value of Q,. Let TG(A) denote the number of gates such that their output value is 1 (true). For any particular E E (0, l] the c-true gates problem is: o”j
es ea cTG(A)Q-=TG(A).
First notice that c-CTGP is trivially in P. We now reduce CVP to E-CTGP, we provide a log-space transformation of a circuit A on n gates to a bigger circuit A’ such that if the output of A is 1, then in A’ the number of true gates is greater than In,41 and if the output of A is 0, then this number is less than n. Thus, if we had an algorithm in NC to approximate TG(A’) within a factor E, then we would get an algorithm in NC that would decide the output of A. Let A =(a,, q,..., (II,) and let I = [H/E], the new circuit is obtained from A adding ! gates, these gates will alternate between AND an OR, propagating in a trivial way the output value of circuit A. Formally, A’ = ((Ye, al, . . . , a,, a,, I’, . . . ,ay,+,) where
). Given an encoding an integer d such that
=
%+k-1
Aa
=
Ly,+k_,
v
n+k-1 a,,+k_I
for
k= 1, 3,...,
for
k=2,
As the circuit added to ,4 only propagates A’s output, we have (a) if the circuit A outputs 0, then TG(A’) = TG( A) < n; (b) if the circuit A outputs 1, then TG(A’) = TG(A) + Ia [n/cl. 0 Note that the circuit constructed in the previous theorem is planar and alternating, further all gates have fan-in two and fan-out two. So using whatever kind of circuits that follow the above restrictions, the reduction provides a log-space reduction from the specific circuit value problem to the corresponding E-true gages problem, then we have: core
.
E-CTGP is P-complete for any CE E (0, 11,
for any of the following circuit classes: binary,
monotone, fan-out two monotone, planar, alternating and fan-out two alternating.
234
4 ,....
Volume 37, Number 4
3.
FOXi
ons
INFORMATION
to ii
PROCESSING LETTERS
ing
3. The solution of the LP instance constructed above verifies
Let x0 be an optimal solution to an stance, and x an approximate solution, tinguish between two types of approximations. A solution approximation requires an x close to x0 in some suitable norm. A value approximation requires only cTx close to cTxo. Actually, we have dropped for approximate solutions the condition “ satisfy all the inequalities simultaneously”. Formally, for any particular 6 E (0, l] we consider the following approximation problems. ation to linear pro
ing (E-
). Compute a rational d-vector x such that
IIx0 IIp 3 II x II p 2 6 IIx0 II p where ]Ix ]Ip = (c~__,~ip)~/~. ation to Hi a rational crx, >, cTx 3 rcTxo. We show that both approximation problems are P-complete for any value of C. We first provide a log-space transformation from a Boolean circuit A to an LP instance. The transformation is a slight modification of the one used in [5] to show that the linear inequalities problem is P-complete. We only add an adequate objective function. Let A = (%, al,..., (Y,J be an encoding of a Boolean circuit. We consider the following transformation: (1) generate the inequalities xk < 1 and xk >, 0 forO
28 February 1991
.Y = 1
iff
a, outputs 1.
a~:“.Note that considering the set of inequalities obtabned in steps (1) to (S), we have, for any gate, if the inputs are 0 or 1, then the output (in the system) will be 0 or 1. Further the computed value corresponds to the value computed by the corresponding gate. Thus the (unique) solution of the subsystem verifies xi = 1 iff CY,has value 1. 0 The above transformation gives us a reduction from CVP to LP in such a way that the maximum possible value of the objective function is just the number of true gates. Extending this property to the p-norm of solutions we have the desired result. The following hold: (1) e-LPV is P-complete under log-space reductions for any E E (0, 11. (2) E-LPS is P-complete under log-space reductions for any E E (0, l] and any p >, 1.
f. We consider the transformation given above. By noting that the unique (optimal) system solution verifies xi = 1 iff ai has value 1, and that x is a O/l vector, we have: l/P
= II x II 1 = TG(A) for all pa
1.
Thus the above construction is a log-space transformation from r-CTGP to E-LPS or E-LPV (for any EE (0, 11). q
I want to thank Paul Spirakis for very useful comments.
‘xi)
ayr, Approkat& P-mm[l] R.J. Anderson and E.W. plete problems, Tech. Rept., Stanford University (1986). 235
Volume 37, Number 4
INFORMATION
[2] S.A. Cook, A taxonomy of problems with fast parallel algorithms, Inform and Control 64 (1985) 2-22. [3] D. Dobkin, R.J. Lipton and S. Reiss, Linear programming (1979) is log space hard for P, Inform. Process. Lett. 96-97. [4] D. Dobkin and S. Reiss, The complexity of linear programming, Theoret. Comput. Sci. 11 (1980) P-18. [5] H.J. Hoover and W.L. Ruzzo, A compedium of problems complete for P, Manuscript, 1984. [6] N. Karmarkar, A new polynomial time algorithm in linear programming, in: Proc. 16th Annual ACM Symposium on Theory of Computing (1984) 302-311. [7] L.G. Khachian, A polynomial algorithm in linear programming, D&l. Akad Nauk SSSR 20 (1) (1979) 10931096; also: Soviet Math Dokl. 20 (1) (1979) 191-194 (in English).
236
PROCESSING
LETTERS
28 February
1991
[8] L. Kirousis, M. Sema and P. Spiral&, The parallel complexity of the subgraph connectivity problem, in: Proc. 30th Annual IEEE Symposium on Foundations of Computer Science (1989) 294-299. [9] L. Kirousis and P. Spiral+, tions ad problems probabi First Scandinavian Workshop on Algorithm Theory (1988). [lo] v. ee and G.L. Minty, How good is the simplex algorithm? , in: 0. Shisha, 4. Inequalities III (Academic Press, New York, 1972) 159-179. [ll] R.E. Ladner, The circuit value problem is log-space complete for P, SIGACT News 7 (1) (1975) 18-20. [12] M. Sema and P. Spiral&, The approximability of problems complete for P, in: Proc. International Symposium on OptimaI Algorithms, Lecture Notes in Computer Science 401 (Springer, Berlin, 1989) 193-204.