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 ! 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 ( G i , A i , B i ) and a partition P of all extremities. The graph G obtained from the union of the G i by adding complete adjacencies between extremities in the same set of the partition is called the composition of strips ( G i , A i , B i ) 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" ! " 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 ( 3 k ) + 4 k . T ( n )) providing one can solve the stable set problem on each strip of 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 ∈ R V , 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 . ! 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 t i for which it can be expressed as a convex combination of stable sets with t i picking both extremities of strip i . This is an interval [ t i , ¯ t i ] . Lemma Membership only depends on the intervals [ t i , ¯ t i ] ’s and on x ( A i ) , x ( B i ) 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 [ t i , ¯ t i ] and x ( A i ) , x ( B i ) ’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. x ( A ) ! x ( B ) − t + ¯ t 2 1 + t − x ( A ) − x ( B ) x ( A ) − t + ¯ t 2 " x ( B ) 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 ? t i is feasible iff the extended point on the left is in the stable set polytope of the gadgetized strip ! Thus t i , ¯ t i are thus constrained by the facets of this polytope x ( B ) − t and thus their value can be expressed as affine functions of 1 + t − x ( A ) − x ( B ) x . x ( A ) − t " 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
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