The hidden matching-structure of the composition of strips: a polyhedral perspective

Stable set problems subsume matching problems since a matching is a stable set in a so- called line graph but stable set problems are hard in general while matching can be solved efficiently [11]. However, there are some classes of graphs where the stable set problem can be solved efficiently. A famous class is that of claw-free graphs; in fact, in 1980 Minty [19, 20] gave the first polynomial time algorithm for finding a maximum weighted stable set (mwss) in a claw-free graph. One of the reasons why stable set in claw-free graphs can be solved efficiently is because the so called augmenting path theorem [4] for matching generalizes to claw-free graphs [5] (this is what Minty is using). We believe that another core reason is structural and that there is a intrinsic matching structure in claw-free graphs. Indeed, recently Chudnovsky and Seymour [8] shed some light on this by proposing a decomposition theorem for claw-free graphs where they describe how to compose all claw-free graphs from building blocks. Interestingly the composition operation they defined seems to have nice consequences for the stable set problem that go much beyond claw-free graphs. Actually in a recent paper [21] Oriolo, Pietropaoli and Stauffer have revealed how one can use the structure of this composition to solve the stable set problem for composed graphs in polynomial time by reduction to matching. In this paper we are now going to reveal the nice polyhedral counterpart of this composition procedure, i.e. how one can use the structure of this composition to describe the stable set polytope from the matching one and, more importantly, how one can use it to separate over the stable set polytope in polynomial time. We will then apply those general results back to where they originated from: stable set in claw-free graphs, to show that the stable set polytope can be reduced to understanding the polytope in very basic structures (for most of which it is already known). In particular for a general claw-free graph G, we show two integral extended formulation for STAB(G) and a procedure to separate in polynomial time over STAB(G); moreover, we provide a complete characterization of STAB(G) when G is any claw-free graph with stability number at least 4 having neither homogeneous pairs nor 1-joins. We believe that the missing bricks towards the characterization of the stable set polytope of claw-free graphs are more technical than fundamentals; in particular, we have a characterization for most of the building bricks of the Chudnovsky-Seymour decomposition result and we are therefore very confident it is only a question of time before we solve the remaining case

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

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomass\'e, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O(k^5)} \cdot n^9$. In this article we improve this running time to $2^{O(k^2)} \cdot n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O(k^5)$ to $O(k^2)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p-1$ for $p \geq 3$, we speed up the running time to $2^{O(k \cdot k^{\frac{1}{p-1}})} \cdot n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p \geq 3$, Weighted Independent Set cannot be solved in time $2^{o(k)} \cdot n^{O(1)}$ in the class of $\{bull,C_4,\ldots,C_{2p-1}\}$-free graphs unless the ETH fails.Comment: 15 page