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Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Perspective

Yuri Faenza 1 Gianpaolo Oriolo 1 Gautier Stauffer 2

1Università di Roma “Tor Vergata” 2Institut de Mathematiques de Bordeaux, Université de Bordeaux I

Aussois, January 2010

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

A strip

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A strip (G,A,B) is a graph G with 2 designated cliques A and B (possibly intersecting) We call A and B the two extremities of the strip

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

The graph composition procedure of C.S.

Definition Given a set of disjoint strips (Gi, Ai, Bi) and a partition P of all extremities. The graph G obtained from the union of the Gi by adding complete adjacencies between extremities in the same set of the partition is called the composition of strips (Gi, Ai, Bi) with respect to partition P. Composition of strips 2-join

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Overview

Everything you can do with matching, you can do with stable sets in composed graphs provided that the strips are "purpose-friendly"

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Polytime optimization = ⇒ Polytime algo for the stable set problem (using a matching algorithm) Polytime separation = ⇒ Polytime separation (using matching separation) SSP Characterization = ⇒ Polyhedral characterization (extended matching inequalities + strip inequalities)

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Solving the stable set problem: an observation

Compatibility of stable sets from each strip depends only on intersection with extremities

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

A simple algorithm by substituting all strips (Oriolo, Pietropaoli, S. 08)

Original graph G Auxiliary graph G′ Auxiliary graph is line graph thus use matching algorithm

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

A simple algorithm by substituting all strips (Oriolo, Pietropaoli, S. 08)

Original graph G Auxiliary graph G′ Auxiliary graph is line graph thus use matching algorithm

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

A simple algorithm by substituting all strips (Oriolo, Pietropaoli, S. 08)

Original graph G Auxiliary graph G′ Auxiliary graph is line graph thus use matching algorithm

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

A simple algorithm by substituting all strips (Oriolo, Pietropaoli, S. 08)

Original graph G Auxiliary graph G′ Auxiliary graph is line graph thus use matching algorithm

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

A simple algorithm by substituting all strips (Oriolo, Pietropaoli, S. 08)

Original graph G Auxiliary graph G′ Auxiliary graph is line graph thus use matching algorithm

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

From optimization to polyhedra

Theorem One can solve the weighted stable set problem of a composed graph with k strips in O(match(3k) + 4k.T(n)) providing one can solve the stable set problem on each strip

• f size n in O(T(n)). In particular if the problem on the strips is polynomial, the overall

problem is polynomial too. Grotschel, Lovasz, Schrijver 86 implies that we can separate in polytime using ellipsoid method. Questions:

Can we solve the separation problem more efficiently ? Can we derive a complete characterization of the stable set polytope ? What about extended formulation(s) ?

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Testing Membership

Given a composed graph G (and its decomposition) and given a point x ∈ RV , we want to understand if x ∈ STAB(G). Recall that a point lies in STAB(G) iff it can be expressed as a convex combination of stable sets of G.

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Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Testing Membership

For the composition of strips, the restriction on each strip has to be a convex combination of stable sets of the strip. In particular, the restriction of x on each strip i has a non empty set of feasible ti for which it can be expressed as a convex combination of stable sets with ti picking both extremities of strip i. This is an interval [ti, ¯ ti]. Lemma Membership only depends on the intervals [ti, ¯ ti]’s and on x(Ai), x(Bi) for all i.

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Compatibility of convex combinations

Since the membership only depends on [ti, ¯ ti] and x(Ai), x(Bi)’s, to assert membership, we can replace each strip with a simpler gadget (and a point) having the same property. The original point x is in STAB(G) iff the new point y is in STAB(H). Observe that H is a line graph so STAB(H) is a matching polytope.

1 + t − x(A) − x(B)

x(A) x(B)

x(B) − t + ¯ t 2 x(A) − t + ¯ t 2

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Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Computing the intervals ?

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1 + t − x(A) − x(B) x(A) − t x(B) − t

ti is feasible iff the extended point on the left is in the stable set polytope of the gadgetized strip Thus ti, ¯ ti are thus constrained by the facets of this polytope and thus their value can be expressed as affine functions of x. We can get those values algorithmically using a separation algorithm for the SSP of the gadgetized strip

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

From membership to extended formulation

Lemma Let G be a composition of strips Gi and H be the composition with the gadgets. Let G+

i

the gadgetized strip. A point x lies in STAB(G) iff there exist 0 ≤ ti ≤ 1 for all i such that

zi = aff(x|Gi , ti) ∈ STAB(G+

i ) for all i

y = aff(x(Ai ), x(Bi), ti, i = 1, .., k) ∈ STAB(H)

We can easily write this as a linear system of inequalities in x, ti if we know the description of the stable set polytope of gadgetized strip (similarly if we have an extended formulation). This yields an linear extended formulation for the problem.

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Projection and separation

Projection

Relatively easy to project this extended formulation on the x space using Fourier-Motzin because the extra variables ti do not interfere too much. The resulting inequalities are matching inequalities that are "blown up". This gives a complete characterization (in the original space). NB: We have other easier extended formulations stemming directly from the algorithm but are not easy to project.

Separation

While testing membership, if the different intervals are not compatible, we can get a violated matching inequality from a separating procedure for matching. This can be converted into a violated inequality in the original space by substituting each ti , ¯ ti using their linear expression in term of x. This inequality is valid because it is exactly the way we derive the projection.

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Recap

Everything you can do with matching, you can do with stable sets in composed graphs provided that the strips are "purpose-friendly"

! "

Polytime optimization = ⇒ Polytime algo for the stable set problem (using a matching algorithm) Polytime separation = ⇒ Polytime separation (using matching separation) SSP Characterization = ⇒ Polyhedral characterization (extended matching inequalities + strip inequalities)

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Stable set in claw-free graphs

A graph is claw-free if it does not contain an induced subgraph which is isomorphic to a claw. The characterization of the stable set polytope of claw-free graphs is opened for 30 years : Arbitrary many coefficients and arbitrary large coefficients

a X v∈◦ xv + (a + 1) X v∈• xv ≤ 2a(a + 1)

2 X v∈✷ xv + 3 X v∈⋄ xv + 4 X v∈◦ xv + 5 X v∈△ xv + 6 X v∈• xv ≤ 8

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

Stable set polytope of claw-free graphs

Polytime separation procedure for the stable set polytope of ALL claw-free graphs using only simple compact linear programs + separation algorithm for matching. Characterize the stable set polytope of claw-free graphs with stability number at least 4 and that do not have homogeneous pair of cliques nor 1-join.

Indeed those graphs are either fuzzy circular interval graphs or composition of "friendly" strips [Chudnovsky, Seymour 08] Characterization for fuzzy circular interval graphs was done in Eisenbrand, Oriolo, S. Ventura 08.

Currently extending our result to the stable set polytope of claw-free graphs with stability number at least 4. It will then only remain to characterize the facets with roots of size 2 and 3 of the stable set polytope of claw-free graphs with stability number 3 to close the characterization of claw-free graphs. Preliminary results of Pecher and Wagler 08 might help to solve this case.

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp

Definitions and known results Testing membership for SSP of composition of strips Extended formulation, projection and separation Consequences for claw-free graphs

THANK YOU !

Yuri Faenza , Gianpaolo Oriolo , Gautier Stauffer The Hidden Matching-Structure of the Composition of Strips: a Polyhedral Persp