164 research outputs found
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
On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs
We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Giles and Trotter [7] by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a+1)-valued coefficients (rank facets have only coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already observed by Cook [3] and Shepherd [18
Small Chvatal rank
We propose a variant of the Chvatal-Gomory procedure that will produce a
sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax
<= b} as b varies. The number of steps needed is called the small Chvatal rank
(SCR) of A. We characterize matrices for which SCR is zero via the notion of
supernormality which generalizes unimodularity. SCR is studied in the context
of the stable set problem in a graph, and we show that many of the well-known
facet normals of the stable set polytope appear in at most two rounds of our
procedure. Our results reveal a uniform hypercyclic structure behind the
normals of many complicated facet inequalities in the literature for the stable
set polytope. Lower bounds for SCR are derived both in general and for
polytopes in the unit cube.Comment: 24 pages, 3 figures, v3. Major revision: additional author, new
application to stable-set polytopes, reorganization of sections. Accepted for
publication in Mathematical Programmin
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
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