The approximability of MAX CSP with fixed-value constraints (original) (raw)

Supermodular functions and the complexity of MAX CSP

Discrete Applied Mathematics, 2005

In this paper we study the complexity of the maximum constraint satisfaction problem (Max CSP) over an arbitrary finite domain. An instance of Max CSP consists of a set of variables and a collection of constraints which are applied to certain specified subsets of these variables; the goal is to find values for the variables which maximize the number of simultaneously satisfied constraints. Using the theory of sub-and supermodular functions on finite lattice-ordered sets, we obtain the first examples of general families of efficiently solvable cases of Max CSP for arbitrary finite domains. In addition, we provide the first dichotomy result for a special class of non-Boolean Max CSP, by considering binary constraints given by supermodular functions on a totally ordered set. Finally, we show that the equality constraint over a non-Boolean domain is non-supermodular, and, when combined with some simple unary constraints, gives rise to cases of Max CSP which are hard even to approximate.

The Approximability of Three-valued MAX CSP

SIAM Journal on Computing, 2006

In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, two-valued) Max CSP problem with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APX-complete (and hence can have no polynomial time approximation scheme unless P = NP). It has been an open problem for several years whether this result can be extended to non-Boolean Max CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for Max CSP over a three-element domain. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known algebraic combinatorial property of supermodularity. We also show that every hard three-valued Max CSP problem contains, in a certain specified sense, one of the two basic hard Max CSP problems which are the Maximum k-colourable subgraph problems for k = 2, 3.

A Decomposition Technique for Max-CSP

European Conference on Artificial Intelligence, 2008

Valued Constraint Satisfaction Problems: Hard and Easy Problems

1995

In order to deal with over-constrained Constraint Satisfaction Problems, various extensions of the CSP framework have been considered by taking into account costs, uncertainties, preferences, priorities...Each extension uses a specific mathematical operator (+, max...) to aggregate constraint violations. In this paper, we consider a simple algebraic framework, related to Partial Constraint Satisfaction, which subsumes most of these proposals and use it to characterize existing proposals in terms of rationality and computational complexity. We exhibit simple relationships between these proposals, try to extend some traditional CSP algorithms and prove that some of these extensions may be computationally expensive.

On the PLS-complexity of maximum constraint assignment

Theoretical Computer Science, 2013

In this paper, we investigate the complexity of computing locally optimal solutions for the local search problem MAXIMUM CONSTRAINT ASSIGNMENT (MCA). For our investigation, we use the framework of PLS, as defined in Johnson et al. [9]. In a nutshell, the MCA-problem is a local search version of weighted, generalized MAXSAT on constraints (functions mapping assignments to integers) over variables with higher valence; additional parameters in (p, q, r)-MCA simultaniously limit the maximum length p of each constraint, the maximum appearance q of each variable and its valence r. We focus on hardness results and show PLScompleteness of (3,2,3)-MCA and (2,3,6)-MCA using tight reductions from CIRCUIT/FLIP. Our results are optimal in the sense that (2, 2, r)-MCA is solvable in polynomial time for every r ∈ N. We also pay special attention to the case of binary variables and show that (6, 2, 2)-MCA is tight PLS-complete. For our results, we extend and refine a technique from Krentel [10].

A characterisation of the complexity of forbidding subproblems in binary max-CSP

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2012

Tractable classes of binary CSP and binary Max-CSP have recently been discovered by studying classes of instances defined by excluding subproblems. In this paper we characterise the complexity of all classes of binary Max-CSP instances defined by forbidding a single subproblem. In the resulting dichotomy, the only non-trivial tractable class defined by a forbidden subproblem corresponds to the set of instances satisfying the so-called joint-winner property.

An algebraic approach to constraint satisfaction problems

A constraint satisfaction problem, or CSP, can be reformulated as an integer linear programming problem. The reformulated problem can be solved via polynomial multiplication. If the CSP has n variables whose domain size is m, and if the equivalent programming problem involves M equations, then the number of solutions can be determined in time O(nm2 M −n ). This surprising link between search problems and algebraic techniques allows us to show improved bounds for several constraint satisfaction problems, including new simply exponential bounds for determining the number of solutions to the n-queens problem. We also address the problem of minimizing M for a particular CSP.

Dynamic algorithms for classes of constraint satisfaction problems

Theoretical computer …, 2001

Many fundamental tasks in arti cial intelligence and in combinatorial optimization can be formulated as a Constraint Satisfaction Problem (CSP) 19]. The problem consists in nding an assignment of values for a set of n variables, each de ned on a nite domain of feasible values of size at most k, subject to a given collection of constraints. Each constraint is de ned over a set of variables and speci es the set of allowed combinations of values as a collection of tuples. In general the problem of nding a solution to a CSP is NP-complete, even if restricted to binary constraints. As an example, the graph coloring problem 11] can be formulated as a binary CSP, where each edge in the graph is associated to a constraint consisting of the collection of C 2 ?C pairs of allowed di erent colorings of the two endpoints with C colors: the resulting CSP is solvable if and only if the graph is colorable with C colors.

A Decomposition Technique for Solving {Max-CSP}

The objective of the Maximal Constraint Satisfaction Problem (Max-CSP) is to find an instantiation which minimizes the number of constraint violations in a constraint network. In this paper, inspired from the concept of inferred disjunctive constraints introduced by Freuder and Hubbe, we show that it is possible to exploit the arc-inconsistency counts, associated with each value of a network, in order to avoid exploring useless portions of the search space. The principle is to reason from the distance between the two best values in the domain of a variable, according to such counts. From this reasoning, we can build a decomposition technique which can be used throughout search in order to decompose the current problem into easier sub-problems. Interestingly, this approach does not depend on the structure of the constraint graph, as it is usually proposed. Alternatively, we can dynamically post hard constraints that can be used locally to prune the search space. The practical interest of our approach is illustrated, using this alternative, with an experimentation based on a classical branch and bound algorithm, namely PFC-MRDAC.

Polynomial time algorithms for hard constraint satisfaction problems

2023

A novel representation is described that models some important NP-hard problems, such as the propositional satisfiability problem (SAT), the Traveling Salesperson Problem (TSP), and the Minimal Set Covering Problem (MSCP) by means of only two types of constraints:'choiceconstraints'and 'exclusion constraints'. In its main section the paper presents an approach for solving a m-CNF-SAT problem (Conjunctive Normal Form Satisfaction: n variables, p clauses, clause length m) by integer programming. The paper presents a 0/1 Simplex for solving the obtained integer program. A main theorem of the paper is that this algorithm always finds a 0-1 integer solution. A solution of the integer program corresponds to a solution of the m-CNF-SAT and vice versa. The same modelling technique is then used for the Traveling Salesperson Problem and for the Minimal Set Covering: it is shown that a uniform approach is thus useful. Black A., J.A. De Loera, S. Kafer and Laura Sanità [Bla21] present new pivot rules for the Simplex method for LP over 0/1 polytopes such as ours, that require only polynomial steps in the number of variables, and give the proof. Thus, based on this result and using these pivot rules for our CNF-SAT solver Simplex algorithm, we find a solution in polynomial time. The complexity of CNF-SAT is NP-complete.

The approximability of MAX CSP with fixed-value constraints (original) (raw)

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