Claw-free t-perfect graphs can be recognised in polynomial time

A graph is called t-perfect if its stable set polytope is defined by non-negativity, edge and odd-cycle inequalities. We show that it can be decided in polynomial time whether a given claw-free graph is t-perfect

Hybrid tractability of soft constraint problems

The constraint satisfaction problem (CSP) is a central generic problem in computer science and artificial intelligence: it provides a common framework for many theoretical problems as well as for many real-life applications. Soft constraint problems are a generalisation of the CSP which allow the user to model optimisation problems. Considerable effort has been made in identifying properties which ensure tractability in such problems. In this work, we initiate the study of hybrid tractability of soft constraint problems; that is, properties which guarantee tractability of the given soft constraint problem, but which do not depend only on the underlying structure of the instance (such as being tree-structured) or only on the types of soft constraints in the instance (such as submodularity). We present several novel hybrid classes of soft constraint problems, which include a machine scheduling problem, constraint problems of arbitrary arities with no overlapping nogoods, and the SoftAllDiff constraint with arbitrary unary soft constraints. An important tool in our investigation will be the notion of forbidden substructures.Comment: A full version of a CP'10 paper, 26 page

Thick Forests

We consider classes of graphs, which we call thick graphs, that have their vertices replaced by cliques and their edges replaced by bipartite graphs. In particular, we consider the case of thick forests, which are a subclass of perfect graphs. We show that this class can be recognised in polynomial time, and examine the complexity of counting independent sets and colourings for graphs in the class. We consider some extensions of our results to thick graphs beyond thick forests.Comment: 40 pages, 19 figure

On the recognition and characterization of M-partitionable proper interval graphs

For a symmetric {0, 1, ⋆ }-matrix M of size m, a graph G is said to be M-partitionable, if its vertices can be partitioned into sets V1, V2, . . . , Vm, such that two parts Vi, Vj are completely adjacent if Mi,j = 1, and completely non-adjacent if Mi,j = 0 (Vi is considered completely adjacent to itself if it induces a clique, and completely non-adjacent if it induces an independent set). The complexity problem (or the recognition problem) for a matrix M asks whether the M-partition problem is polynomial-time solvable or NP-complete. The characterization problem for a matrix M asks if all M-partitionable graphs can be characterized by the absence of a finite set of forbidden induced subgraphs. These forbidden induced subgraphs are called obstructions to M. In the literature, many results were obtained by restricting the input graphs. In this thesis, we survey these results when the questions are restricted to the class of perfect graphs. We then study the recognition problem and the characterization problem when the inputs are restricted to proper interval graphs. The recognition problem can be solved by an existing algorithm, but we simplify its proof of correctness. As our main result, we prove that all the matrices of size 3 and size 4 with constant diagonal, have finitely many minimal proper interval obstructions. We also obtain partial results about matrices of arbitrary size if they have a zero diagonal

Counting Weighted Independent Sets beyond the Permanent

Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671--697] showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair, and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chvátal and Sbihi [J. Combin. Theory Ser. B, 44 (1988)], Maffray and Reed [J. Combin. Theory Ser. B, 75 (1999)], and Lozin and Milanič [J. Discrete Algorithms, 6 (2008), pp. 595--604]

Cliques, colouring and satisfiability : from structure to algorithms

We examine the implications of various structural restrictions on the computational complexity of three central problems of theoretical computer science (colourability, independent set and satisfiability), and their relatives. All problems we study are generally NP-hard and they remain NP-hard under various restrictions. Finding the greatest possible restrictions under which a problem is computationally difficult is important for a number of reasons. Firstly, this can make it easier to establish the NP-hardness of new problems by allowing easier transformations. Secondly, this can help clarify the boundary between tractable and intractable instances of the problem. Typically an NP-hard graph problem admits an infinite sequence of narrowing families of graphs for which the problem remains NP-hard. We obtain a number of such results; each of these implies necessary conditions for polynomial-time solvability of the respective problem in restricted graph classes. We also identify a number of classes for which these conditions are sufficient and describe explicit algorithms that solve the problem in polynomial time in those classes. For the satisfiability problem we use the language of graph theory to discover the very first boundary property, i.e. a property that separates tractable and intractable instances of the problem. Whether this property is unique remains a big open problem