There are many complex combinatorial problems which involve searching for an undirected graph satisfying a certain property. These problems are often highly challenging because of the large number of isomorphic representations of a possible solution. In this paper we introduce novel, effective and compact, symmetry breaking constraints for undirected graph search. While incomplete, these prove highly beneficial in pruning the search for a graph. We illustrate the application of symmetry breaking in graph representation to resolve several open instances in extremal graph theory.

We investigate the computational complexity of two global constraints, CUMULATIVE and INTERDISTANCE. These are key constraints in modeling and solving scheduling problems. Enforcing domain consistency on both is NP-hard. However, restricted versions of these constraints are often sufficient in practice. Some examples include scheduling problems with a large number of similar tasks, or tasks sparsely distributed over time. Another example is runway sequencing problems in air-traffic control, where landing periods have a regular pattern. Such cases can be characterized in terms of structural restrictions on the constraints. We identify a number of such structural restrictions and investigate how they impact the computational complexity of propagating these global constraints. In particular, we prove that such restrictions often make propagation tractable.

This paper investigates the manpower allocation problem with time windows and job-teaming constraints(MAPTWTC), a practical scheduling and routing problem that tries to synchronize workers’ schedules to complete all tasks. We first provide an integer programming model for the problem and discuss its properties. Next, we show that tree data structure can be used to represent the MAPTWTC solutions, and its optimal solution can be obtained from one of trees by solving a minimum cost flow model for each worker type. Consequently, we develop for the problem a novel tabu search algorithm employing search operators based on the tree data structure. Finally, we prove the effectiveness of the tabu search algorithm by computational experiments on two sets of instances.

It is widely acknowledged that stochastic local search (SLS) algorithms can efficiently find models of satisfiable formulae for the Boolean Satisfiability (SAT) problem. There has been much interest in studying SLS algorithms on random $k$-SAT instances. Compared to random 3-SAT instances which have special statistical properties rendering them easy to solve, random $k$-SAT instances with long clauses are similar to structured ones and remain very difficult. This paper is devoted to efficient SLS algorithms for random $k$-SAT instances with long clauses. By combining a novel variable property $subscore$ with the commonly used property $score$, we design a scoring function named {\it comprehensive score}, which is utilized to develop a new SLS algorithm called CScoreSAT. The experiments show that CScoreSAT outperforms state-of-the-art SLS solvers, including the winners of recent SAT competitions, by one to two orders of magnitudes on large random 5-SAT and 7-SAT instances. In addition, CScoreSAT significantly outperforms its competitors on random $k$-SAT instances for each $k=4,5,6,7$ from SAT Challenge 2012, which indicates its robustness.

Constraint satisfaction problems may be nearly tractable. For instance, most of the relations in a problem might belong to a tractable language. We introduce a method to take advantage of this fact by computing a backdoor to this tractable language. The method can be applied to many tractable classes for which the membership test is itself tractable. We introduce therefore two polynomial membership testing algorithms, to check if a language is closed under a majority or conservative Mal'tsev polymorphism, respectively. Then we show that computing a minimal backdoor for such classes is fixed parameter tractable (FPT) if the tractable subset of relations is given, and W[2]-complete otherwise. Finally, we report experimental results on the XCSP benchmark set. We identified a few promising problem classes where problems were nearly closed under a majority polymorphism and small backdoors could be computed.

In this paper, we propose the first heuristic approach for the vertex separator problem (VSP), based on Breakout Local Search (BLS). BLS is a recent meta-heuristic that follows the general framework of the popular Iterated Local Search (ILS) with a particular focus  on the perturbation strategy. Based on some relevant information on search history, it tries to introduce the most suitable degree of diversification by determining adaptively the number and type of moves for the next perturbation phase. The proposed heuristic is highly competitive with the exact state-of-art approaches from the literature on the current VSP benchmark. Moreover, we present for the first time computational results for a set of large graphs with up to 3000 vertices, which constitutes a new challenging benchmark for VSP approaches.

Constraint programming techniques are widely used to model and solve interactive decision problems, and especially configuration problems. In this type of application, the configurable product is described by means of a set of constraints bearing on the configuration variables. The user interactively solves the CSP by assigning the variables according to her preferences. The system then has to keep the domains of the other variables consistent with these choices. Since maintaining the global inverse consistency of the domains is not tractable, the domains are instead filtered according to some level of local consistency, e.g. arc-consistency. The present paper aims at offering a more convenient interaction by providing the user with possible alternative values for the already assigned variables, i.e. values that could replace the current ones without leading to a constraint violation. We thus present the new concept of alternative domains in a (possibly) partially assigned CSP. We propose a propagation algorithm that computes all the alternative domains in a single step. Its worst case complexity is comparable to the one of the naive algorithm that would run a full propagation for each variable, but its experimental efficiency is better

Since the first principles of Knowledge Compilation (KC), most of the work    has been focused in finding a good compilation target language in terms of    compromises between compactness and expressiveness. The central idea    remained unchanged in the last fifteen years: an off-line, very hard, stage,    allows to ``compile'' the initial theory in order to guarantee    (theoretically) an efficient on-line stage, on a set of predefined queries    and operations.  We propose a new ``Just-in-Time'' approach    for KC. Here, any Knowledge Base (KB) will be immediately available for    queries, and the effort spent on past queries will be partly amortized for    future ones.  To guarantee efficient answers, we rely on the    tremendous progresses made in the practical solving    of SAT and incremental SAT applicative problems. Even if each query may    be theoretically hard, we  show that our approach outperforms    previous KC approaches on the set of classical problems used in the field,    and allows to handle problems that are out of the scope of current    approaches. 

Motivated by considerations in quantum mechanics, we introduce the class of robust constraint satisfaction problems in which the question is whether every partial assignment of a certain length can be extended to a solution, provided the partial assignment does not violate any of the constraints of the given instance. We explore the complexity of specific robust colorability and robust satisfiability problems, and show that they are NP-complete. We then use these results to establish the computational intractability of detecting local hidden-variable models in quantum mechanics.

The Decentralized Partially Observable Markov Decision Process (Dec-POMDP) is a powerful model for multi-agent planning under uncertainty, but its applicability is hindered by its high complexity -- solving Dec-POMDPs optimally is NEXP-hard. Recently, Kumar et al. introduced the Value Factorization (VF) framework, which exploits decomposable value functions that can be factored into subfunctions. This framework has been shown to be a generalization of several models that leverage sparse agent interactions such as TI-Dec-MDPs, ND-POMDPs and TD-POMDPs. Existing algorithms for these models assume that the interaction graph of the problem is given. In this paper, we introduce three algorithms to automatically generate interaction graphs for models within the VF framework and establish lower and upper bounds on the expected reward of an optimal joint policy. We illustrate experimentally the benefits of these techniques for sensor placement in a decentralized tracking application.