At Play with Combinatorial Optimization, Integer Programming and Polyhedra Gautier Stauffer To cite this version: Gautier Stauffer. At Play with Combinatorial Optimization, Integer Programming and Polyhedra. Optimization and Control [math.OC]. Université Sciences et Technologies - Bordeaux I, 2011. tel- 00653059 HAL Id: tel-00653059 https://tel.archives-ouvertes.fr/tel-00653059 Submitted on 17 Dec 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universite´ Bordeaux 1 Institut de Math´ematiques de Bordeaux Habilitation a` diriger des recherches Ecole doctorale de Math´ematiques et Informatique de Bordeaux At play with Combinatorial Optimization, Integer Programming and Polyhedra ou Excursions en Optimisation Combinatoire, Programmation Enti`ere et Polyh`edres. Soutenue et pr´esent´ee publiquement par Gautier Stauffer le 28 novembre 2011 devant la commission d’examen form´ee de : Michele Conforti Universit`adi Padova rapporteur Volker Kaibel Otto-von-Guericke Universit¨at rapporteur Ridha Majhoub Universit´ede Paris-Dauphine rapporteur Arnaud Pˆecher Universit´ede Bordeaux 1 examinateur Fran¸cois Vanderbeck Universit´ede Bordeaux 1 examinateur 2 `aMadame Patate 3 4 Contents 1 Introduction 9 2 Dynamic programming and shortest path 13 2.1 Inventorycontrol ................................ 13 2.2 Stable sets in graphs with bounded width stability number ............ 14 3 Building upon network flows 17 3.1 Set packing problem associated with circular one matrices............. 17 3.2 Separating over the set packing polytope associated with circular one matrices . 18 3.3 Inventory control in one-warehouse multi-retailer systems as a fixed charge net- workflow ........................................ 19 4 Matching and beyond 21 4.1 Linegraphsandcompositionofstrips . ........ 21 4.2 Stablesetsinclaw-freegraphs. ........ 24 4.3 p-medianinY-freegraphs . ..... 25 5 Structural graph theory 29 5.1 Algorithmic decomposition theorem for claw-free graphs.............. 29 5.2 Recognizing fuzzy circular interval graphs : a simple induction .......... 32 6 Polyhedra 35 6.1 Some properties of the matching polytope . ......... 35 6.2 p-medianpolytopeofY-freegraphs. ........ 36 6.3 Stable set polytope of quasi-line graphs . .......... 37 6.4 Rank-facets for SSP of quasi-line graphs by sequential lifting ........... 39 7 Extended formulations 43 7.1 Extended formulation for distance claw-free graphs . .............. 43 7.2 Combinatorial union of polytopes . ........ 43 7.3 Extended formulation for the stable set polytope of claw-freegraphs . 45 8 Cutting-plane and Column generation 47 8.1 Chv`atal-Gomory cuts, split cuts and the SSP of quasi-linegraphs. 47 8.2 A lower bound on the Chv`atal-Gomory rank for polytope in the 0/1 cube . 48 8.3 An industrial cutting stock problem by Column Generation ............ 50 9 Approximation algorithms 53 9.1 OWMRCD : Performance analysis through convex optimization.......... 53 9.2 OWMR : a 2-approximation through extended formulation . ........... 55 10 Conclusion and perspectives 59 Glossary α(G) Stability number of G i.e. size of a maximum cardinality stable set. We sometimes abuse notation and given a subset of vertices U of a graph G, use α(U) for α(G[V ]). CIG Circular interval graphs. Circular interval graphs A circular interval graph G(V, E) is defined by the following con- struction: Take a circle , a set of vertices V and a mapping Φ : V of the vertices on C 7→ C the circle. Take a subset of closed and proper intervals of , with no interval including J C another, and say that u, v V are adjacent (i.e. (u, v) E) if Φ(u), Φ(v) is a subset of ∈ ∈ { } one of the intervals. (V, Φ, ) is a interval model of G. J Circular one matrix Let A be a m n matrix with zeros and ones. A is a circular one matrix × if in each row, the ones appear in consecutive columns (where column 1 is considered consecutive to column n). Claw A claw (u; v,w,z) is a graph with vertex set u,v,w,z and edge set (u, v), (u, w), (u, z) . { } { } Clique A clique is a complete subgraph. Clique family inequality Let G be a graph and let = K ,...,K be a set of cliques o G, F { 1 n} 1 p n be integral and r = n mod p. Let V V (G) be the set of vertices covered ≤ ≤ p−1 ⊆ by exactly (p 1) cliques of and V V (G) the set of vertices covered by p or more − F ≥p ⊆ cliques of . The inequality F n (p r 1) x(v) + (p r) x(v) (p r) − − − ≤ − p v∈XVp−1 v∈XV≥p is valid for STAB(G) and is called the clique family inequality associated with and p. F n Combinatorial union of polytopes Let Pi R , i = 1, ..., k be k non empty polyhedra. Let 1 l k ⊆ λ , ..., λ 0, 1 . The combinatorial union of the polyhedron Pi w.r.t. λ1, ..., λl is the ∈ { } l j { } set P = conv.hull( j=1 λ (i)Pi) where the sum is the Minkowsky sum. Composition of stripsSLet P = (Gi, i), i = 1, ..., k be a family of vertex disjoint strips. H { A } Let ( ) denote the multi-family of the extremities of those strips, i.e., ( )= i, A H A H i=1..k A and let = P , P , ..., P be the classes of a partition of ( ). We associate to the pair P 1 2 q A H S ( , ) the graph G that is made of the disjoint union of the graphs G1,...Gk, with H P additional edges E := (u, v) : u = v and u and v belong to different extremities in a same { 6 class P , for some 1 i q . G is called the composition of the strips with respect to i ≤ ≤ } H partition . P Distance simplicial strip A strip (G, ) is distance simplicial if for all A , α(N (A)) 1 A ∈A i ≤ for all i 0. ≥ 5 6 Glossary Edmonds’ inequality Let G be a graph. Given a set of n cliques and a set of vertices Q covered by at least two of those cliques, one can write an inequality x(v) Q /2 v∈Q ≤ ⌊| | ⌋ which is valid for STAB(G) and such an inequality is called an Edmonds’ inequality. P FCIG Fuzzy circular interval graphs. Fuzzy circular interval graphs A graph G = (V, E) is a fuzzy circular interval graph if the following conditions hold: (i) There is a map Φ from V to a circle ; (ii) There is a set C of closed and proper intervals of , none including another, such that no point of is J C C the end of more than one interval and (iia) If two vertices u and v are adjacent, then Φ(u) and Φ(v) belong to a common interval ; (iib) If two vertices u and v belong to the same interval, which is not an interval with endpoints Φ(u) and Φ(v), then they are adjacent. Γ(S) is the strong neighborhood of S i.e. given a graph G(V, E) and a set S V , Γ(S)= v ⊆ { ∈ V S : (u, v) E for all u S . \ ∈ ∈ } Intersection property a polyhedra P has the intersection property if P x : x = k ∩ { e∈E e } is integral for every integer k. P JRP Joint Replenishment Problem, see Section 3.3. Krausz family Given a graph G, we call a family of cliques of G a Krausz family (of G) if F it satisfies that every edge of G is covered by some clique of and every vertex of G is F covered by exactly two cliques of . F Line graph The line graph L(G) of a graph G is the intersection graph of the edges of G i.e. there is a vertex for each edge in G and two vertices are adjacent if and only if their corresponding edges in G are incident. Line strip a strip (S, ) is line if it admits a Krausz family with . A K A ⊆ K MATCH(G) The matching polytope of a graph G i.e. the convex hull of all incidence vectors of matchings in a graph G. Matching A matching in a (multi-) graph G(V, E) is a subset of pairwise non incident edges. Minkowsky sum Given two polyhedra P1, P2, the Minkowsky sum of P1 and P2 is denoted by P + P and is defined by P + P = x + x : x P ,x P . 1 2 1 2 { 1 2 1 ∈ 1 2 ∈ 2} MWSS Maximum Weighted Stable Set. N (S) the neighborhood of the set S i.e. given an undirected graph G(V, E), and S V , ⊆ N(S)= v V S : (u, v) E for some u S . { ∈ \ ∈ ∈ } N(v) the neighborhood of v i.e N( v ). { } N[S] the closed neighborhood of S i.e. N(S) S. ∪ N[v] the closed neighborhood of v i.e N[ v ]. { } N (v) the i-th neighborhood of v defined by the following recursion : N (v) = v , N (v) = i 0 { } i+1 N(N (v)) i N (v). i \ l=1 l S Glossary 7 Net A net (u , u , u ; v , v , v ) is a graph with vertex set u , u , u , v , v , v and edge set 1 2 3 1 2 3 { 1 2 3 1 2 3} (u , u ), (u , u ), (u , u ), (u , v ), (u , v ), (u , v ) .