Linear phase transition in random linear constraint satisfaction problems

Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and a constraint is chosen from also uniformly at random. This procedure is repeated m times independently. We are interested in whether the resulting linear programming problem is feasible. We prove that the feasibility property experiences a linear phase transition, when n and m = cn for a constant c. Namely, there exists a critical value c^* such that, when c < c^*, the problem is feasible or is asymptotically almost feasible, as n, but, when c>c^*, the distance to feasibility is at least a positive constant independent of n. Our result is obtained using the combination of a powerful local weak convergence method developed in Aldous [Ald92], [Ald01], Aldous and Steele [AS03], Steele [Ste02] and martingale techniques. By exploiting a linear programming duality, our theorem implies the following result in the context of sparse random graphs G(n, cn) on n nodes with cn edges, where edges are equipped with randomly generated weights. Let (n, c) denote maximum weight matching in G(n, cn). We prove that when c is a constant and n , the limit lim_n (n, c)/n, exists, with high probability. We further extend this result to maximum weight b-matchings also in G(n, cn).     

A scaling algorithm for polynomial constraint satisfaction problems

Good scaling is an essential requirement for the good behavior of many numerical algorithms. In particular, for problems involving multivariate polynomials, a change of scale in one or more variable may have drastic effects on the robustness of subsequent calculations. This paper surveys scaling algorithms for systems of polynomials from the literature, and discusses some new ones, applicable to arbitrary polynomial constraint satisfaction problems.     

Sharp thresholds for constraint satisfaction problems and homomorphisms

We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the (d,k,t)‐model, and binary constraint satisfaction problems with domain size three. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The SAT-UNSAT transition for random constraint satisfaction problems

We study threshold phenomena for a large class of random constraint satisfaction problems over finite domains. Our main contribution is a complete classification of the nature (sharp or coarse) of the SAT-UNSAT transition for random Boolean CSPs, which is based on easily decidable properties.     

Property testers for dense constraint satisfaction programs on finite domains

Many NP‐hard languages can be “decided” in subexponential time if the definition of “decide” is relaxed only slightly. Rubinfeld and Sudan introduced the notion of property testers, probabilistic algorithms that can decide, with high probability, if a function has a certain property or if it is far from any function having this property. Goldreich, Goldwasser, and Ron constructed property testers with constant query complexity for dense instances of a large class of graph problems. Since many graph problems can be viewed as special cases of the Constraint Satisfaction Problem on Boolean domains, it is natural to try to construct property testers for more general cases of the Constraint Satisfaction Problem. In this paper, we give explicit constructions of property testers using a constant number of queries for dense instances of Constraint Satisfaction Problems where the constraints have constant arity and the variables assume values in some domain of finite size. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 14–32, 2002     

The complexity of recursive constraint satisfaction problems

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint satisfaction problems where the problem of finding a solution to the infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through finitely branching recursive trees or recursive binary trees. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to give equivalent results about the complexity of the problem of finding a solution to a recursive constraint satisfaction problem.     

Conditional lexicographic orders in constraint satisfaction problems

The lexicographically-ordered CSP (“lexicographic CSP” or “LO-CSP” for short) combines a simple representation of preferences with the feasibility constraints of ordinary CSPs. Preferences are defined by a total ordering across all assignments, such that a change in assignment to a given variable is more important than any change in assignment to any less important variable. In this paper, we show how this representation can be extended to handle conditional preferences in two ways. In the first, for each conditional preference relation, the parents have higher priority than the children in the original lexicographic ordering. In the second, the relation between parents and children need not correspond to the importance ordering of variables. In this case, by obviating the “overwhelming advantage” effect with respect to the original variables and values, the representational capacity is significantly enhanced. For problems of the first type, any of the algorithms originally devised for ordinary LO-CSPs can also be used when some of the domain orderings are dependent on assignments to “parent” variables. For problems of the second type, algorithms based on lexical orders can be used if the representation is augmented by variables and constraints that link preference orders to assignments. In addition, the branch-and-bound algorithm originally devised for ordinary LO-CSPs can be extended to handle CSPs with conditional domain orderings.     

Linear programs with an additional reverse convex constraint

A constraintg(x)0 is said to be a reverse convex constraint if the functiong is continuous and strictly quasi-convex. The feasible regions for linear programs with an additional reverse convex constraint are generally non-convex and disconnected. It is shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytope defined by only the linear constraints. The only possible edges which can contain such an optimal solution are characterized in relation to the best feasible vertex of the polytope defined by only the linear constraints. This characterization then provides a finite algorithm for finding a globally optimal solution.Research supported by NSF Grant ENG76-12250 and ONR Contract N00014-78-C-0428.     

The satisfiability threshold for randomly generated binary constraint satisfaction problems

We study two natural models of randomly generated constraint satisfaction problems. We determine how quickly the domain size must grow with n to ensure that these models are robust in the sense that they exhibit a non‐trivial threshold of satisfiability, and we determine the asymptotic order of that threshold. We also provide resolution complexity lower bounds for these models. One of our results immediately yields a theorem regarding homomorphisms between two random graphs. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Improving degree-based variable ordering heuristics for solving constraint satisfaction problems

In this paper, we improved two classical degree-based variable ordering heuristics, \(\frac{\textit{Dom}}{\textit{Ddeg}}\) and \(\frac{\textit{Dom}}{\textit{Wdeg}}\). We propose a method using the summation of constraint tightness in degree-based heuristics. We also propose two methods to calculate dynamic constraint tightness for binary extensional constraints and non-binary intensional constraints respectively. Our work shows how constraint tightness can be practically used to guide search. We performed a number of experiments on some benchmark instances. The results have shown that, the new heuristics improve the classical ones by both computational time and search tree nodes and they are more efficient than some other successful heuristics on the instances where the classical heuristics work well.     

Linear programs with an additional rank two reverse convex constraint

An efficient algorithm is developed for solving linear programs with an additional reverse convex constraint having a special structure. Computational results are presented and discussed.     

The property of being polynomial for Mal’tsev constraint satisfaction problems

A combinatorial constraint satisfaction problem aims at expressing in unified terms a wide spectrum of problems in various branches of mathematics, computer science, and AI. The generalized satisfiability problem is NP-complete, but many of its restricted versions can be solved in a polynomial time. It is known that the computational complexity of a restricted constraint satisfaction problem depends only on a set of polymorphisms of relations which are admitted to be used in the problem. For the case where a set of such relations is invariant under some Mal’tsev operation, we show that the corresponding constraint satisfaction problem can be solved in a polynomial time. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 655–686, November–December, 2006.     

Heuristics and look-ahead integration to solve constraint satisfaction problems efficiently

Logic programming languages, such asProlog, allow a declarative specification of Constraint Satisfaction Problems (CSPs), freeing the user from specifying more or less complex control directives. However, the price to pay for such flexibility is a loss of efficiency, which makes Logic Programming inadequate to solve CSPs of even moderate size and complexity. To extend the range of applicability of logic programming, several improvements have been proposed. The use of heuristics is one such improvement. Although this usually forces the user to specify some form of control (thus abandoning the pure declarative nature of a logic program), these specifications can be made declarative by making use of some appropriate meta-predicates. Another extension to logic programming that improves its efficiency, is the active use of constraints, as done in the various formulations of constraint logic programming languages. In particular, the use of finite domains is quite adequate to implement look-ahead schemes to efficiently solve several types of CSPs. In this paper, we discuss the complementary nature of heuristics and look-ahead schemes and present a constraint logic programming framework that integrates both these techniques. Results obtained with a time-tabling problem executed on a prototype that implements such a framework are presented, and show that significant efficiency improvements can be achieved when compared with the separate use of the two techniques.     

On the solution‐space geometry of random constraint satisfaction problems

For various random constraint satisfaction problems there is a significant gap between the largest constraint density for which solutions exist and the largest density for which any polynomial time algorithm is known to find solutions. Examples of this phenomenon include random k‐SAT, random graph coloring, and a number of other random constraint satisfaction problems. To understand this gap, we study the structure of the solution space of random k‐SAT (i.e., the set of all satisfying assignments viewed as a subgraph of the Hamming cube). We prove that for densities well below the satisfiability threshold, the solution space decomposes into an exponential number of connected components and give quantitative bounds for the diameter, volume, and number.© 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 251–268, 2011     

On approximate constraint satisfaction

We consider an abstract problem on fulfilling asymptotic constraints. We propose a very general approach to constructing “nonsequential” attraction sets in the space of generalized elements formalizable as finitely additive measures. We study the existence and the structure of the asymptote universal in the range of “asymptotic constraints” not requiring the compactifiability of the space of ordinary solutions.     

Boosting distributed constraint satisfaction

Competition and cooperation can boost the performance of a combinatorial search process. Both can be implemented with a portfolio of algorithms which run in parallel, give hints to each other and compete for being the first to finish and deliver the solution. In this paper we present a new generic framework for the application of algorithms for distributed constraint satisfaction that makes use of both cooperation and competition. This framework improves the performance of two different standard algorithms by one order of magnitude. Furthermore, it can reduce the risk of poor performance by up to three orders of magnitude diminishing the heavy-tailed behaviour of complete distributed search. Moreover it greatly reduces the classical idleness flaw usually observed in distributed tree-based searches. We expect our new methods to be similarly beneficial for any tree-based distributed search and describe ways on how to incorporate them. Remarkably, our ideas while applied to a parallel SAT setting were able to beat divide-and-conquers approaches, and win the gold medal of the parallel track of the 2008 SAT-Race.     

Algebra complexity problems involving graph homomorphism,semigroups and the constraint satisfaction problem

In this paper, we connect the constraint satisfaction problem with other complexity problems, like the polynomial equivalence problem for combinatorial 0-simple semigroups, the graph retraction problem and the geometry problem. We show that every constraint satisfaction problem is polynomially equivalent to an easily formulated algebra complexity problem. As an application we prove that the polynomial equivalence problem (word problem) for the 2×2 matrices over the two element field is co-NP-complete.     

Existences theorems of systems of vector quasi-equillibrium problems and mathematical programs with equilibrium constraint

In this paper, we introduce systems of vector quasi-equilibrium problems and prove the existence of their solutions. As applications of our results, we derive the existence theorems for solution of system of vector quasi-saddle point problem, the existences theorems of a solution of system of generalized quasi-minimax inequalities, the mathematical program with equilibrium constraint, semi-infinite and bilevel problems.     

Scaled constraint qualifications for generalized equation constrained problems and application to nonsmooth mathematical programs with equilibrium constraints

Positivity - In this paper, the notion of graphical derivatives is applied to define a new class of several well-known constraint qualifications for a nonconvex multifunction M at a point of its...     

Generalizations of Slater's constraint qualification for infinite convex programs

In this paper we study constraint qualifications and duality results for infinite convex programs (P) = inf{f(x): g(x) – S, x C}, whereg = (g ₁,g ₂) andS = S ₁ ×S ₂,S _i are convex cones,i = 1, 2,C is a convex subset of a vector spaceX, andf andg _i are, respectively, convex andS _i -convex,i = 1, 2. In particular, we consider the special case whenS ₂ is in afinite dimensional space,g ₂ is affine andS ₂ is polyhedral. We show that a recently introduced simple constraint qualification, and the so-called quasi relative interior constraint qualification both extend to (P), from the special case thatg = g ₂ is affine andS = S ₂ is polyhedral in a finite dimensional space (the so-called partially finite program). This provides generalized Slater type conditions for (P) which are much weaker than the standard Slater condition. We exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong ₂ and the finite dimensionality assumption onS ₂, by a local compactness assumption. We then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption. Our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones.