Structures and Duality in Combinatorial Programming

Transcription

1 Structures and Duality in Combinatorial Programming HDR Denis Cornaz Mourad Baiou Université Blaise Pascal examinateur Gérard Cornuéjols Carnegie Mellon University rapporteur Jérôme Lang Université Paris-Dauphine président Ridha Mahjoub Université Paris-Dauphine coordinateur Gianpaolo Oriolo Università di Roma Tor Vergata rapporteur Christophe Picouleau Conservatoire national des arts et métiers examinateur András Sebő Institut national polytechnique Grenoble rapporteur LAMSADE, Paris-Dauphine HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 1 / 24

2 Introduction (P) = {Ax b} is TDI: min{y b : y A w } = min{y b : y A w, y integer} ( w integer) P = {x : Ax b} is integer: max{w x : x P} = max{w x : x P, x integer} ( w) A is 0-1: a clutter H = (E, C) α w (H) := max{w x : Ax 1, x 0, x integer} (packing) τ w (H) := min{w x : Ax 1, x 0, x integer} (covering) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 2 / 24

3 Contents 1 Minimal forbidden structures 2 Graph coloring 3 Min-max relations 4 Election problems HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 3 / 24

4 Contents 1 Minimal forbidden structures HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 3 / 24

5 The minimum biclique cover problem (with Fonlupt) (a) (b) (c) (d) (e) (f) H = (E, C) is the minimal non-biclique clutter of G = (V, E) ω(g) 1 max C ω(g) C C (with equality at the right if ω(g) is odd) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 4 / 24

6 The minimum biclique cover problem (with Fonlupt) (a) (b) (c) (d) (e) (f) H = (E, C) is the minimal non-biclique clutter of G = (V, E) ω(g) 1 max C ω(g) C C (with equality at the right if ω(g) is odd) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 4 / 24

7 A general method (with Fonlupt, Kerivin, and Mahjoub) B is a set of edge subsets of G, C(B) := {C E : C B but C B, C C} Finding a minimum cost C C(B) is P when B is: (a) edge sets of complete bipartite subgraphs of G, (b) edge sets of complete multipartite subgraphs of G, (c) edge sets of vertex-induced bipartite subgraphs of G, (d) arc sets of vertex-induced acyclic subdigraphs of G. HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 5 / 24

8 The maximum complete multipartite subgraph problem H = (E, C) is the minimal non-multiclique clutter of G C = 2 ( C C) G is fan- and prism-free HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 6 / 24

9 1 2 Graph coloring 3 4 HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 7 / 24

10 Chromatic Gallai identities (with Jost and Meurdesoif) (a) (b) (c) (d) (e) (f) G is an orientation of G, without 3-dicycle χ(g) + α(s( G)) = V (G) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 8 / 24

11 Chromatic Gallai identities (with Jost and Meurdesoif) (a) (b) (c) (d) (e) (f) G is an orientation of G, without 3-dicycle χ(g) + α(s( G)) = V (G) = α(g) + χ(s( G)) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 8 / 24

12 Appl. in mathematical programming (with Meurdesoif) χ f (G) = = max 1 x s.t. x(k) 1 ( clique K) x v 0 ( v V ) min s.t. 1 y K v y K = 1 ( v) y K 0 ( clique K) y optimal for G x feasible for S( G) x uv = clique-stars S K uv y K HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 9 / 24

13 Appl. in mathematical programming (with Meurdesoif) ϑ(g) = max v x v 2 s.t. x o 2 = 1 xo x v = x v 2 ( v) xu x v = 0 ( uv E) = min y o 2 s.t. y v 2 = 1 ( v) yo y v = 1 ( v) yu y v = 0 ( uv / E) x R d m+1 optimal for S( G) y R nd n+1 feasible for G y o v = x o y o v if u = v x uv and y v u = x uv if uv E( G) 0 otherwise u:uv E( G) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 10 / 24

14 Appl. in mathematical programming (with Meurdesoif) If α(g) β(g) χ(g), G, then α(g) Φ β (G) := V (G) β(s( G)) χ(g) ( G) Improving Lovász s ϑ bound for coloring α(g) Φ χf (G) χ f (G) and ϑ(g) Φ ϑ (G) χ(g) ( G) V E min ρ := Φ ϑ ϑ ϑ mean ρ max ρ M % M % 27.3% 28.7% M % 27.6% 29.5% M % 27.8% 29.5% HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 11 / 24

15 Coloring clustered graphs (with Bonomo, Ekim and Ries) (a) (b) (c) M(G, V) = (d) and M(G/V) = ω(g/v) χ(g/v) χ sel (G, V) (e) ( ) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 12 / 24

16 Coloring clustered graphs (with Bonomo, Ekim and Ries) G is selective-perfect if M(G, V) is perfect for all V uv E G is i-threshold if R(t(u) + t(v)) I(t(u))I(t(v)) > 0 Minimal clustered graphs such that M(G, V) M(G/V) are (a)-(c) G is selective-perfect G is i-threshold HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 13 / 24

17 The clique-connecting forest polytope 0 x e 1 ( e E) (CCFO) { 1 if U is a clique of G x(e(u)) U ( U V ) 2 otherwise (a) (b) Facets of the clique-connecting forest polytope Inequalities induced by complete sets or by the clique polytope are facets HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 14 / 24

18 1 2 3 Min-max relations 4 HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 15 / 24

19 The star polytope (with Nguyen) G 1... G k = G with G i G Cost = i (G i) (a) (b) Figure : If max (G i ) = 2, then Cost > (G) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 16 / 24

20 The star polytope (with Nguyen) ocm set (C, M): M is a matching C = C 1... C k (C i odd-circuit) C 1,..., C k, M are pairwise vertex-disjoint Generalizing Kőnig s min-max relation to a best-posible one { (P star ) 1 2 is minimally TDI x e 0 for all e E x(c) + x(m) 1 for all maximal ocm set (C, M) of G, HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 17 / 24

21 Max-multiflow vs. min-multicut (a) (b) (c) A multicut disconnects the demands HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 18 / 24

22 Max-multiflow vs. min-multicut The multicut polytope: conv.hull{χ δ(v 1,...,V p) } { (MCUT) 0 x e 1 ( e) x(c \ {e}) x e ( circuit C e) ( p) Weights: (+) demands and ( ) links (MCUT) is TDI the graph is series-parallel G + H is series-parallel = max-multiflow=min-multicut HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 19 / 24

23 Election problems HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 20 / 24

24 Structure and algorithm in elections (with Galand and Spanjaard) P = a c b b b a c a c is a subprofile of P = a c b b a b d a a c c b c d b d a d c d χ c v := position of candidate c C in voter v V representative k-set C C minimizes v V min c C χc v P single-peaked if path P with vertex-set C: {c C : χ c v i} induces a connected subgraph of P ( v V, i {1,..., C }) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 21 / 24

25 Structure and algorithm in elections (with Galand and Spanjaard) Kemeny voter u minimizes v V d(u, v), where d(u, v) = ab pair of C (u may be outside V ) (1 u ab 1v ab )2, 1 u ab = { 1 if χ a u < χ b u 0 otherwise P single-crossing if path P with vertex-set V : {v V : χ a v < χ b v } induces a connected subgraph of P ( ab pair of C) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 22 / 24

26 Structure and algorithm in elections (with Galand and Spanjaard) C = partition of C into intervals P = d d x a y v c b b c a x v y = d d x a y v c b b c a x v y P/C = subprofile with one candidate per interval of C P/(abcv, d, x, y) = d d x a y x a y and P/(abv, d, cxy) = d d c a a c HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 23 / 24

27 Structure and algorithm in elections (with Galand and Spanjaard) single-peaked width:= maximum size of an interval in C such that P/C is single-peaked finding the single-peaked width is P bounded single-peaked width = representative k-set becomes P single-crossing width:= maximum size of an interval in C such that P/C is single-crossing finding the single-crossing width is P bounded single-crossing width = Kemeny voter becomes P HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 24 / 24

28 Question: Max-multiflow vs. min-multicut (a) (b) (c) (G, R) signed graph: R is the set of demands (red) C odd-circuit: C R odd C flow: C R = 1 T = R D (D cut or multicut) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 24 / 24

29 Question: The star polytope (a) (b) Figure : If max (G i ) = 4, then Cost > (G) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 24 / 24

30 Publications Denis Cornaz, Hervé Kerivin, Ali Ridha Mahjoub: Structure of minimal arc-sets vertex-inducing dicycles. (submitted) Denis Cornaz, Philippe Meurdesoif: Chromatic Gallai identities operating on Lovász number. Mathematical Programming, Series A (to appear) (online springer.com) Flavia Bonomo, Denis Cornaz, Tinaz Ekim, Bernard Ries: Perfectness of clustered graphs. Discrete Optimization 10(4) : (2013) Denis Cornaz, Viet Hung Nguyen: Kőnig s edge-colouring theorem for all graphs. Operations Research Letters 41: (2013) Denis Cornaz, Lucie Galand, Olivier Spanjaard: Kemeny elections with bounded single-peaked or single-crossing width. In Proceedings of 23rd. IJCAI : (2013) Cédric Bentz, Denis Cornaz, Bernard Ries: Packing and covering with linear programming: A survey. European Journal of Operational Research 227(3): (2013) Denis Cornaz, Lucie Galand, Olivier Spanjaard: Bounded single-peaked width and proportional representation. In Proceedings of 20th. ECAI : (2012) Denis Cornaz: Max-multiflow/min-multicut for G+H series-parallel. Discrete Mathematics 311(17): (2011) Denis Cornaz, Philippe Meurdesoif: The sandwich line graph. Electronic Notes in Discrete Mathematics 36: (2010) Denis Cornaz: Clique-connecting forest and stable set polytopes. RAIRO - Operations Research 44(1): (2010) Denis Cornaz, Vincent Jost: A one-to-one correspondence between colorings and stable sets. Operations Research Letters 36(6): (2008) Denis Cornaz: On co-bicliques. RAIRO - Operations Research 41(3): (2007) Denis Cornaz, Ali Ridha Mahjoub: The maximum induced bipartite subgraph problem with edge weights. SIAM Journal on Discrete Mathematics 21(3): (2007) Denis Cornaz: A linear programming formulation for the maximum complete multipartite subgraph problem. Mathematical Programming, Series B 105(2-3): (2006) Denis Cornaz, Jean Fonlupt: Chromatic characterization of biclique covers. Discrete Mathematics 306(5): (2006) HDR Denis Cornaz Structures and Duality in Comb. Prog. LAMSADE, Paris-Dauphine 24 / 24