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Second-Order Birkhoff Polytope and the Problem of Graph Isomorphism

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Second-Order Birkhoff Polytope and the Problem of Graph Isomorphism Second-order Birkhoff Polytope and the Problem of Graph Isomorphism Detection A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Pawan Kumar Aurora to the DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY KANPUR June, 2015 ii iii iv Synopsis Graph Isomorphism (GI) problem is a classic problem in the theory of computing. Given two graphs G; G0 on n vertices each, GI problem is to decide if G; G0 are isomorphic or not. The problem remains elusive for general graphs in the sense that neither a polynomial time algorithm is known for it nor is it established that this problem is NP-complete. Here it must be acknowledged that polynomial time algorithms are known for the GI problem on special classes of graphs. The best known algorithm for general graphs is moderately exponential. In this research work we attempt to devise a polynomial time algorithm for GI for general graphs. Most of the attempts for designing a polynomial time algorithm for this problem have been based on combinatorial, group theoretic or graph theoretic techniques or a combination of these. A small number of attempts have also been based on linear programming. In this work we revisit linear programming approach to design an algorithm for the problem. If we label the vertices of both the input graphs by 1; : : : ; n, then each isomor- phism is a permutation σ such that vertices i and j are adjacent in G if and only if the vertices σ(i) and σ(j) are adjacent in G0. The earlier LP based approaches use the commutation relation AP = PA0 satisfied by isomorphic pairs of graphs where A and A0 denote the adjacency matrices of the two graphs and P denotes the permutation matrix corresponding to an isomorphism between the two graphs. Replacing P by a doubly stochastic variable matrix Y gives a linear program whose zero-one solutions are the isomorphisms between the two graphs. The feasible region of this linear program is known as the standard GI polytope or Tinhofer polytope (let's say T). It has been shown that GI can be decided for certain classes of graphs in a constant number of rounds of the Sherali-Adams (SA) lift and project method starting with the polytope T. Consider the polytope P obtained by one SA lift step applied to the Birkhoff polytope, B. It can be shown that there is a one to one correspondence between the integer points of P and the integer points of B, with each integer point corresponding to a permutation. We refer to the convex hull of integer points in P as the second- order Birkhoff polytope and denote it by B[2]. In the literature this polytope is studied in the context of the Quadratic Assignment Problem and hence is also called the QAP-polytope. We show that B[2] is full-dimensional in the affine plane of polytope P, i.e., the two polytopes have the same dimension. Adding graph based conditions, also referred to as the edge/non-edge condi- tions, to the linear description of polytope P, we get our linear program, LP-GI. The feasible region of this linear program contains exactly those integer points (per- mutations) which correspond to the isomorphisms between the given pair of graphs. It can be shown that the two graphs are isomorphic if and only the feasible region intersects with B[2]. In the case of non-isomorphic pairs of graphs this fact implies that the feasible region is either empty or confined to P B[2]. We present an exact n algorithm to determine if the feasible region is contained in P B[2] (i.e., deciding n non-isomorphism). The most significant result of this thesis is that under a reason- able assumption this algorithm decides non-isomorphism in polynomial time. The said assumption is related to the facial structure of B[2] which is discussed next. The non-negativity constraints in the linear program LP-GI define all the facets of polytope P. Each of these constraints also define facets of polytope B[2]. We refer to these as the trivial facets of B[2]. There are other facets of the polytope which we will refer to as non-trivial facets. Exponentially many non-trivial facets of B[2] are known in the QAP literature and in this thesis we identify exponentially many additional facets. We give three general inequalities which define three classes of (exponentially many) supporting planes of B[2]. These supporting planes define faces of the polytope including all its known facets. We also define a partial ordering on these supporting planes/inequalities and show that no minimal inequality is ever violated by any solution of the LP, irrespective of whether the graphs are isomorphic or not. We analyze our exact algorithm for those cases when each point in the feasible region of LP-GI, for a pair of non-isomorphic graphs, violates one of the above men- tioned inequalities. In such case clearly each point violates at least one minimal inequality (an inequality that it violates but does not violate any lower inequality in the ordering). If there exists a single inequality which is a minimal violated inequal- ity for all points in the feasible region, then we show that the algorithm terminates in polynomial time. We perform several experiments with strongly regular graphs and CFI-graphs and report the results. In every non-isomorphic instance in these experiments we find that the feasible region is zero-one reducible, a property which ensures that the algorithm takes polynomial time to detect non-isomorphism. We also consider the general case when no single minimal violated inequality exists for every point in the feasible region. We modify our exact algorithm to handle the general case efficiently. If k is the minimal number of regions into which the feasible region can be divided such that each region has a single minimal violated inequality then the modified algorithm runs in time exponential in k. We believe that the value of k should be small. We also investigate if all the non-trivial facets of B[2] are discovered or more are yet to be found out. We develop a single generic inequality such that all the known facets are its instances and prove that there must be at least one facet of the polytope which is not an instance of this generic equation, implying that all facets of the polytope are not yet known. Finally, we restrict the feasible region of LP-GI to the cone of positive semidef- inite matrices and observe that the resulting semidefinite program is the Lov´asz Theta function of a product of the input graphs. We also perform experiments us- ing this formulation and find that the algorithm converges in fewer iterations than with the linear program, as should be expected. viii Acknowledgements I am forever indebted to my advisor Prof. Shashank K. Mehta for making this thesis possible. Most of the work in this thesis is the outcome of several hours of discussions that we have had over the past five years. I would like to thank Prof. Arunava Banerjee at UFL for teaching with such passion and enthusiasm that sparked my interest in algorithms and theoretical com- puter science. I would like to express my gratitude and thanks to my parents for being sup- portive of my decision to pursue doctoral studies and for sharing several of my responsibilites during the past several years so as to let me focus on my research. Last but not the least I would like to thank my wife Sonal for bearing with me all these years. Your support at difficult times kept me going. x To Mummy, Papa, Sonal, Kartikeya and Kritika xii Contents List of Figures xvii 1 Introduction1 1.1 The Graph Isomorphism Problem . .1 1.1.1 Combinatorial Approach to Graph Isomorphism . .2 1.1.2 Graph Theoretic Approach to Graph Isomorphism . .4 1.1.3 Group Theoretic Approach to Graph Isomorphism . .6 1.1.4 Polyhedral Approach to Graph Isomorphism . .7 1.2 Our Contributions . 10 1.3 Organization . 12 2 A Linear Programming Approach to Graph Isomorphism 13 2.1 Introduction . 13 2.2 Integer Linear Program for GI . 13 2.2.1 Affine Plane of B[2] ........................ 14 2.3 Linear Programming Relaxation . 16 2.4 Using the LP to Solve GI . 19 2.4.1 Zero-One Reducibility . 19 2.4.2 The Search Algorithm . 20 2.5 Conclusion . 22 3 Facial Structure of B[2] 23 3.1 Introduction . 23 3.2 Some Facets of B[2] ............................ 23 3.2.1 A Useful Identity . 25 xiv CONTENTS 3.2.2 Facets Due to the Non-negativity Constraint . 27 3.2.3 The Connection Lemma . 28 3.2.4 A Polynomial Sized Family of Facets . 29 3.2.5 An Exponential Sized Family of Facets . 31 3.3 Conclusion . 34 4 Non-Isomorphism Detection 35 4.1 Introduction . 35 4.2 Partial Ordering on Supporting Planes of B[2] ............. 36 4.3 Polynomiality of Algorithm1...................... 38 4.3.1 A Minimal Violated Inequality of Type (4.1).......... 38 4.3.2 A Minimal Violated inequality of Type (4.2).......... 38 4.3.2.1 Restriction to Facets . 38 4.3.2.2 General 1-box Inequality . 41 4.3.3 A Minimal Violated Inequality of Type (4.3).......... 41 4.4 The General Case . 44 4.4.1 A Generalized Algorithm for GI . 44 4.4.1.1 k-SearchVar() . 45 4.4.1.2 The Procedure .
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