The concept of backdoor was introduced to try to explain the good performances achieved on real world SAT instances by solvers. A backdoor is a set of variables that, once decided, makes the rest of the problem simple (i.e. polynomial-time). In this thesis we provide a comprehensive overview on the state of the art of backdoors for SAT. Moreover, we study the relation between backdoors and para...

Flow cover inequalities are among the most effective valid inequalities for solving capacitated fixed-charge network flow problems. These valid inequalities are implications on the flow quantity on the cut arcs of a two-partitioning of the network, depending on whether some of the cut arcs are open or closed. As the implications are only on the cut arcs, flow cover inequalities can be modeled b...

Typical-case computation complexity is a research topic at the boundary of computer science, applied mathematics, and statistical physics. In the last twenty years the replica-symmetry-breaking mean field theory of spin glasses and the associated message-passing algorithms have greatly deepened our understanding of typical-case computation complexity. In this paper we use the vertex cover probl...

This paper generalizes optimal collective dichotomous choices by including constraints which limit combinations of acceptance and rejection decisions for m projects under consideration. Two types of constraints are examined. The first type occurs when acceptance of some projects requires acceptance of others. This type reduces the choice problem to the tractable (solvable in polynomial time) pr...

Abstract What are the performance guarantees of the algorithms used in practice for clustering and network design problems? We answer this question by showing that the standard local search algorithm returns a nearly-optimal solution for low-dimensional Euclidean instances of the traveling salesman problem, Steiner tree, k-median and k-means. The result also extends to the case of graphs exclud...

In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for graphs with small intersection number and random intersection graphs (a model in which each one of m labels is chosen independently with probability p by each ...

A set P of disjoint paths in a graph G is called a (complete) path cover of G if every vertex of G belongs to one of the paths in P. A path cover of any subgraph of G is called a partial path cover of G. For xed k > 0, a k-blanket of graph G is a partial path cover P of G, consisting of exactly k paths, that maximizes the size of the subgraph covered by P. A k-core of graph G is a partial path ...

In this paper we describe a Sequential Importance Sampling (SIS) procedure for counting the number of vertex covers in general graphs. The optimal SIS proposal distribution is the uniform over a suitably restricted set, but is not implementable. We will consider two proposal distributions as approximations to the optimal. Both proposals are based on randomization techniques. The first randomiza...

We apply a Bethe-Peierls approach to statistical-mechanics models defined on random networks of arbitrary degree distribution and arbitrary correlations between the degrees of neighboring vertices. Using the nondeterministic polynomial time hard optimization problem of finding minimal vertex covers on these graphs, we show that such correlations may lead to a qualitatively different solution st...