77 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
Extended formulations from communication protocols in output-efficient time
Deterministic protocols are well-known tools to obtain extended formulations,
with many applications to polytopes arising in combinatorial optimization.
Although constructive, those tools are not output-efficient, since the time
needed to produce the extended formulation also depends on the number of rows
of the slack matrix (hence, on the exact description in the original space). We
give general sufficient conditions under which those tools can be implemented
as to be output-efficient, showing applications to e.g.~Yannakakis' extended
formulation for the stable set polytope of perfect graphs, for which, to the
best of our knowledge, an efficient construction was previously not known. For
specific classes of polytopes, we give also a direct, efficient construction of
extended formulations arising from protocols. Finally, we deal with extended
formulations coming from unambiguous non-deterministic protocols
Clique-circulants and the stable set polytope of fuzzy circular interval graphs
In this paper, we give a complete and explicit description of the rank facets of the stable set polytope for a class of claw-free graphs, recently introduced by Chudnovsky and Seymour (Proceedings of the Bristish Combinatorial Conference, 2005), called fuzzy circular interval graphs. The result builds upon the characterization of minimal rank facets for claw-free graphs by Galluccio and Sassano (J. Combinatorial Theory 69:1-38, 2005) and upon the introduction of a superclass of circulant graphs that are called clique-circulants. The new class of graphs is invetigated in depth. We characterize which clique-circulants C are facet producing, i.e. are such that Sigma upsilon epsilon V(C) chi(upsilon) <= alpha(C) is a facet of STAB(C), thus extending a result of Trotter (Discrete Math. 12:373-388, 1975) for circulants. We show that a simple clique family inequality (Oriolo, Discrete Appl. Math. 132(2):185-201, 2004) may be associated with each clique-circulant C subset of G, when G is fuzzy circular interval. We show that these inequalities provide all the rank facets of STAB(G), if G is a fuzzy circular interval graph. Moreover we conjecture that, in this case, they also provide all the non-rank facets of STAB(G) and offer evidences for this conjecture
Clique-Stable Set separation in perfect graphs with no balanced skew-partitions
Inspired by a question of Yannakakis on the Vertex Packing polytope of
perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary
subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates
a clique K and a stable set S if and . A
Clique-Stable Set Separator is a family of cuts such that for every clique K,
and for every stable set S disjoint from K, there exists a cut in the family
that separates K and S. Given a class of graphs, the question is to know
whether every graph of the class admits a Clique-Stable Set Separator
containing only polynomially many cuts. It is open for the class of all graphs,
and also for perfect graphs, which was Yannakakis' original question. Here we
investigate on perfect graphs with no balanced skew-partition; the balanced
skew-partition was introduced in the proof of the Strong Perfect Graph Theorem.
Recently, Chudnovsky, Trotignon, Trunck and Vuskovic proved that forbidding
this unfriendly decomposition permits to recursively decompose Berge graphs
using 2-join and complement 2-join until reaching a basic graph, and they found
an efficient combinatorial algorithm to color those graphs. We apply their
decomposition result to prove that perfect graphs with no balanced
skew-partition admit a quadratic-size Clique-Stable Set Separator, by taking
advantage of the good behavior of 2-join with respect to this property. We then
generalize this result and prove that the Strong Erdos-Hajnal property holds in
this class, which means that every such graph has a linear-size biclique or
complement biclique. This property does not hold for all perfect graphs (Fox
2006), and moreover when the Strong Erdos-Hajnal property holds in a hereditary
class of graphs, then both the Erdos-Hajnal property and the polynomial
Clique-Stable Set Separation hold.Comment: arXiv admin note: text overlap with arXiv:1308.644
Characterizing Structurally Cohesive Clusters in Networks: Theory and Algorithms
This dissertation aims at developing generalized network models and solution approaches for studying cluster detection problems that typically arise in networks. More specifically, we consider graph theoretic relaxations of clique as models for characterizing structurally cohesive and robust subgroups, developing strong upper bounds for the maximum clique problem, and present a new relaxation that is useful in clustering applications.
We consider the clique relaxation models of k-block, and k-robust 2-club for describing cohesive clusters that are reliable and robust to disruptions, and introduce a new relaxation called s-stable cluster, for modeling stable clusters. First, we identify the structural properties associated with the models, and investigate the computational complexity of these problems. Next, we develop mathematical programming techniques for the optimization problems introduced, and apply them in presenting effective solution approaches to the problems.
We present integer programming formulations for the optimization problems of interest, and provide a detailed study of the associated polytopes. Particularly, we develop valid inequalities and identify different classes of facets for the polytopes. Exact solution approaches developed for solving the problems include simple branch and bound, branch and cut, and combinatorial branch and bound algorithms. In addition, we introduce many preprocessing techniques and heuristics to enhance their performance. The presented algorithms are tested computationally on a number of graph instances, that include social networks and random graphs, to study the capability of the proposed solution methods.
As a fitting conclusion to this work, we propose new techniques to get easily computable and strong upper bounds for the maximum clique problem. We investigate k-core and its stronger variant k-core/2-club in this light, and present minimization problems to get an upper bound on the maximization problems. Simple linear programming relaxations are developed and strengthened by valid inequalities, which are then compared with some standard relaxations from the literature. We present a detailed study of our computational results on a number of benchmark instances to test the effectiveness of our technique for getting good upper bounds
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