Introduction Previous polyhedral study New IP formulation Preliminary computational results Concluding remarks On the facial structure of the Common Edge Subgraph polytope Gordana Mani´ c, Laura Bahiense and Cid de Souza Universidade Estadual de Campinas, SP, Brazil and Universidade Federal do Rio de Janeiro, RJ, Brazil CTW 2008 Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks Summary Common Edge Subgraph problem – Definition – Applications Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks Summary Common Edge Subgraph problem – Definition – Applications Previous polyhedral study Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks Summary Common Edge Subgraph problem – Definition – Applications Previous polyhedral study Our contribution – New integer programming formulation – Valid inequalities and facets of the polytope Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks Summary Common Edge Subgraph problem – Definition – Applications Previous polyhedral study Our contribution – New integer programming formulation – Valid inequalities and facets of the polytope Preliminary computational results Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks Maximum Common Edge Subgraph Problem Definition ( Bokhari 81 ): Given: two graphs with | V G | = | V H | Find: a common subgraph of G and H , (not necessary induced) with the maximum number of EDGES. Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks Maximum Common Edge Subgraph Problem Definition ( Bokhari 81 ): Given: two graphs with | V G | = | V H | Find: a common subgraph of G and H , (not necessary induced) with the maximum number of EDGES. We denote this problem by MSEC (Maximum Common Edge Subgraph). Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks Maximum Common Edge Subgraph Problem Definition ( Bokhari 81 ): Given: two graphs with | V G | = | V H | Find: a common subgraph of G and H , (not necessary induced) with the maximum number of EDGES. We denote this problem by MSEC (Maximum Common Edge Subgraph). Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks MCES-Example Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks MCES-Application Application 1: Parallel programming environments G : task interaction graph (edges join pairs of tasks with communication demands) H : processors graph (pair of processors being joined by an edge when they are directly connected). Problem: Find mapping of tasks to processors s.t. number of neighboring tasks assigned onto connected processors is maximized. Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks MCES-Application Application 1: Parallel programming environments G : task interaction graph (edges join pairs of tasks with communication demands) H : processors graph (pair of processors being joined by an edge when they are directly connected). Problem: Find mapping of tasks to processors s.t. number of neighboring tasks assigned onto connected processors is maximized. Application 2: Graph isomorphism problem When | E G | = | E H | , there exists a common subgraph with | E G | edges, iff, G and H are isomorphic. Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks MCES-More applications and complexity Application 3: Chemistry and biology Matching 2 D and 3 D chemical structures Raymond 02 Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks MCES-More applications and complexity Application 3: Chemistry and biology Matching 2 D and 3 D chemical structures Raymond 02 Complexity MCES is NP -hard. Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study Summary New IP formulation MCES Preliminary computational results Concluding remarks MCES-More applications and complexity Application 3: Chemistry and biology Matching 2 D and 3 D chemical structures Raymond 02 Complexity MCES is NP -hard. Goal: Find exact/optimal solution of MCES instances using integer programming (IP) techniques and polyhedral combinatorics. Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study New IP formulation IP formulation Preliminary computational results Concluding remarks Previous polyhedral study Master’s thesis Marenco 99 presented: IP formulation for MCES some valid inequalities and facets for corresponding polytope computational results. Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study New IP formulation IP formulation Preliminary computational results Concluding remarks Previous polyhedral study Master’s thesis Marenco 99 presented: IP formulation for MCES some valid inequalities and facets for corresponding polytope computational results. Subsequent works by Marenco Marenco 06 present new classes of valid inequalities for MCES , but no new computational experiments. Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study New IP formulation IP formulation Preliminary computational results Concluding remarks IP formulation for MCES � 1 if vertex i is mapped to vertex k y ik := 0 otherwise . � 1 if exists kl ∈ E H such that i is mapped to k and j to l x ij := 0 otherwise . IP formulation presented by Marenco: max � ij ∈ E G x ij � k ∈ V H y ik = 1 , ∀ i ∈ V G � i ∈ V G y ik = 1 , ∀ k ∈ V H x ij + y ik ≤ 1 + � ∀ ij ∈ E G , ∀ k ∈ V H l ∈ N ( k ) y jl , y ik ∈ { 0 , 1 } , ∀ i ∈ V G , ∀ k ∈ V H ; x ij ∈ { 0 , 1 } , ∀ ij ∈ E G Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study New IP formulation IP formulation Preliminary computational results Concluding remarks IP formulation for MCES Note: Consider inequality x ij + y ik ≤ 1 + � l ∈ N ( k ) y jl , ∀ ij ∈ E G , ∀ k ∈ V H . Let ij be a fixed edge in G , and k a fixed vertex from H . Then x ij = 1 iff j is mapped to a neighbour of k . Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study New IP formulation IP formulation Preliminary computational results Concluding remarks IP formulation for MCES Note: Consider inequality x ij + y ik ≤ 1 + � l ∈ N ( k ) y jl , ∀ ij ∈ E G , ∀ k ∈ V H . Let ij be a fixed edge in G , and k a fixed vertex from H . Then x ij = 1 iff j is mapped to a neighbour of k . Theorem ( Marenco 99 ): dim( conv ( S )) = ( | V G | − 1) 2 + | E G | , where S is the set of feasible integer solutions of the problem, and conv ( S ) its convex hull. Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Introduction Previous polyhedral study New IP formulation New IP formulation Valid inequalities and facets Preliminary computational results Concluding remarks New IP formulation � 1 if ij is mapped to kl c ijkl := 0 otherwise . New IP formulation: max � � kl ∈ E H c ijkl ij ∈ E G � k ∈ V H y ik ≤ 1 , ∀ i ∈ V G � i ∈ V G y ik ≤ 1 , ∀ k ∈ V H � kl ∈ E H c ijkl ≤ � k ∈ V H y ik , ∀ ij ∈ E G � ij ∈ E G c ijkl ≤ � ∀ kl ∈ E H i ∈ V G y ik , � j ∈ N ( i ) c ijkl ≤ y ik + y il , ∀ i ∈ V G , ∀ kl ∈ E H � l ∈ N ( k ) c ijkl ≤ y ik + y jk , ∀ ij ∈ E G , ∀ k ∈ V H c ijkl ∈ { 0 , 1 } , ∀ ij ∈ E G , ∀ kl ∈ E H Mani´ c, Bahiense and Souza Common Edge Subgraph polytope y ik ∈ { 0 , 1 } , ∀ i ∈ V G , ∀ k ∈ V H
Introduction Previous polyhedral study New IP formulation New IP formulation Valid inequalities and facets Preliminary computational results Concluding remarks New IP formulation We decided to work with the monotonous model since the proofs of facet-defining inequalities are easier than in the model given in Marenco 99 . Mani´ c, Bahiense and Souza Common Edge Subgraph polytope
Recommend
More recommend
