The Approximability of Constraint Satisfaction Problems

We study optimization problems that may be expressed as "Boolean constraint satisfaction problems." An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Different computational problems arise from constraint satisfaction problems depending on the nature of the "underlying" constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used. Tight bounds on the approximability of every problem in Max CSP were obtained by Creignou [ J. Comput. System Sci., 51 (1995), pp. 511--522]. In this work we determine tight bounds on the "approximability" (i.e., the ratio to within which each problem may be approximated in polynomial time) of every problem in Max Ones, Min CSP, and Min Ones. Combined with the result of Creignou, this completely classifies all optimization problems derived from Boolean constraint satisfaction. Our results capture a diverse collection of optimization problems such as MAX 3-SAT, Max Cut, Max Clique, Min Cut, Nearest Codeword, etc. Our results unify recent results on the (in-)approximability of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically.

[1]  David P. Williamson,et al.  New 3⁄4 - Approximation Algorithms for MAX SAT , 2001 .

[2]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[3]  Nadia Creignou A Dichotomy Theorem for Maximum Generalized Satisfiability Problems , 1995, J. Comput. Syst. Sci..

[4]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[5]  Philip N. Klein,et al.  Approximation Algorithms for Steiner and Directed Multicuts , 1997, J. Algorithms.

[6]  Jacques Stern,et al.  The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations , 1997, J. Comput. Syst. Sci..

[7]  M. Bellare,et al.  Efficient probabilistic checkable proofs and applications to approximation , 1994, STOC '94.

[8]  Nadia Creignou,et al.  Complexity of Generalized Satisfiability Counting Problems , 1996, Inf. Comput..

[9]  Mihalis Yannakakis,et al.  On the approximation of maximum satisfiability , 1992, SODA '92.

[10]  Luca Trevisan,et al.  Gadgets, Approximation, and Linear Programming (extended abstract). , 1996, IEEE Annual Symposium on Foundations of Computer Science.

[11]  J. Håstad Clique is hard to approximate within n 1-C , 1996 .

[12]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[13]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[14]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[15]  R. Ravi,et al.  Approximation through multicommodity flow , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[16]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[17]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[18]  Edoardo Amaldi,et al.  The Complexity and Approximability of Finding Maximum Feasible Subsystems of Linear Relations , 1995, Theor. Comput. Sci..

[19]  Rina Panigrahy,et al.  An O(log*n) approximation algorithm for the asymmetric p-center problem , 1996, SODA '96.

[20]  Alessandro Panconesi,et al.  Completeness in Approximation Classes , 1989, Inf. Comput..

[21]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[22]  Takao Asano,et al.  Approximation Algorithms for the Maximum Satisfiability Problem , 1996, Nord. J. Comput..

[23]  David P. Williamson,et al.  New 3/4-Approximation Algorithms for the Maximum Satisfiability Problem , 1994, SIAM J. Discret. Math..

[24]  Mihir Bellare,et al.  Free Bits, PCPs, and Nonapproximability-Towards Tight Results , 1998, SIAM J. Comput..

[25]  Harry B. Hunt,et al.  Generalized CNF satisfiability problems and non-efficient approximability , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[26]  Joseph Naor,et al.  Approximating Minimum Feedback Sets and Multicuts in Directed Graphs , 1998, Algorithmica.

[27]  J. Håstad Clique is hard to approximate withinn1−ε , 1999 .

[28]  Joseph Naor,et al.  Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality , 1993, Math. Program..

[29]  Carsten Lund,et al.  The Approximation of Maximum Subgraph Problems , 1993, ICALP.

[30]  David P. Williamson,et al.  A complete classification of the approximability of maximization problems derived from Boolean constraint satisfaction , 1997, STOC '97.

[31]  Madhu Sudan,et al.  Improved Low-Degree Testing and its Applications , 1997, STOC '97.

[32]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[33]  Carsten Lund,et al.  Efficient probabilistically checkable proofs and applications to approximations , 1993, STOC.

[34]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[35]  Luca Trevisan,et al.  To Weight or Not to Weight: Where is the Question? , 1996, ISTCS.

[36]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[37]  R. Boppana Approximating Maximum Independent Sets by Excluding Subgraphs 1 , 1990 .

[38]  Luca Trevisan,et al.  Structure in Approximation Classes , 1999, Electron. Colloquium Comput. Complex..

[39]  Luca Trevisan,et al.  Constraint satisfaction: the approximability of minimization problems , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[40]  Uri Zwick,et al.  A 7/8-approximation algorithm for MAX 3SAT? , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[41]  Rajeev Motwani,et al.  Towards a syntactic characterization of PTAS , 1996, STOC '96.

[42]  Rajeev Motwani,et al.  On syntactic versus computational views of approximability , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[43]  Satissed Now Consider Improved Approximation Algorithms for Maximum Cut and Satissability Problems Using Semideenite Programming , 1997 .

[44]  Lars Engebretsen,et al.  Clique Is Hard To Approximate Within , 2000 .

[45]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[46]  Pierluigi Crescenzi,et al.  Introduction to the theory of complexity , 1994, Prentice Hall international series in computer science.

[47]  Mihalis Yannakakis,et al.  Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications , 1996, SIAM J. Comput..

[48]  David Zuckerman,et al.  On Unapproximable Versions of NP-Complete Problems , 1996, SIAM J. Comput..

[49]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[50]  Phokion G. Kolaitis,et al.  Approximation properties of NP minimization classes , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[51]  Ravi B. Boppana,et al.  Approximating maximum independent sets by excluding subgraphs , 1990, BIT.

[52]  M. Yannakakis,et al.  Approximate Max--ow Min-(multi)cut Theorems and Their Applications , 1993 .

[53]  Joseph Naor,et al.  Approximating Minimum Feedback Sets and Multi-Cuts in Directed Graphs , 1995, IPCO.

[54]  Luca Trevisan,et al.  Gadgets, Approximation, and Linear Programming , 2000, SIAM J. Comput..