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A Study of Width Bounded Arithmetic Circuits and the Complexity of Matroid Isomorphism

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A Study of Width Bounded Arithmetic Circuits and the Complexity of Matroid Isomorphism A STUDY OF WIDTH BOUNDED ARITHMETIC CIRCUITS AND THE COMPLEXITY OF MATROID ISOMORPHISM by Raghavendra Rao B. V. The Institute of Mathematical Sciences, Chennai. A thesis submitted to the Board of Studies in Mathematical Sciences In partial fulfillment of the requirements For the Degree of DOCTOR OF PHILOSOPHY of HOMI BHABHA NATIONAL INSTITUTE September 2009 Homi Bhabha National Institute Recommendations of the Viva Voce Board As members of the Viva Voce Board, we recommend that the dissertation prepared by Raghavendra Rao B. V. entitled “A study of width bounded arithmetic cir- cuits and the complexity of matroid isomorphism” may be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date : Chairman : V. Arvind (IMSc) Date : Convener : Meena Mahajan (IMSc) Date : Member : Venkatesh Raman (IMSc) Date : Member : K. V. Subrahmanyam (CMI) Date : Member : Sundar Vishwanathan (IIT-B) Final approval and acceptance of this dissertation is contingent upon the can- didate’s submission of the final copies of the dissertation to HBNI. I hereby certify that I have read this dissertation prepared under my direction and recommend that it may be accepted as fulfilling the dissertation requirement. Date : Guide : Meena Mahajan DECLARATION I hereby declare that the investigation presented in the thesis has been carried out by me. The work is original and the work has not been submitted earlier as a whole or in part for a degree/diploma at this or any other Institution or University. Raghavendra Rao B. V. ACKNOWLEDGEMENTS First of all I thank my supervisor Meena Mahajan for her kind support and for providing all the motiviation, inspiration and knowledge which led to this thesis. Also, I would like to thank Appa, Amma, Akka, Lathakka, Shubha, bhava, Meghana and Mrinal for their uncodintional kindness and support, without which I could not have pursued research as a career. I thank all the faculty members of the TCS group at imsc for offering excellent courses, which certainly has motivated me to pursue research as a career. I also thank G. Sajith, for motivating me into theoretical computer science. I would like thank my collaborators Jayalal Sarma and Maurice Jansen. Some part of the thesis is an outcome of discussion with them. I thank all my friends Aravind, Gaurav, Jayalal, Narayanan, Pradeep, Pushkar, Soumya, Srikanth, Thakur and Ved (I am sure I have missed many) for all the moral support they have provided. Finally I would like to thank all the administrative staff for the excellent facilities provided. Abstract The subject of computational complexity theory deals with characterization of computational problems with respect to various resource measures. This thesis can broadly be divided into two parts: (1) Study of width bounded arithmetic circuits and (2) Computational complexity of matroid isomorphism problems. In the first part of the thesis we further the study of various arithmetiza- tions of boolean complexity class NC1. In particular, we propose constant width arithmetic circuits of polynomial degree as a new possible arithmetization of NC1. Among the candidate arithmetizations of NC1, constant width arithmetic circuits seem to be the larger ones. In particular, it is not known if constant width arithmetic circuits of polynomial size and degree can be efficiently simulated by arithmetic formulas of polynomial size. However, the situation changes when we restrict ourselves to syntactic mul- tilinear circuits. We show that constant width syntactic multilinear circuits are equivalent to constant width syntactic multilinear branching programs at poly- nomial size and hence contained inside syntactic multilinear polynomial size formulas. This makes syntactic multilinear formulas as the stronger one among the competent classes in the syntactic multilinear world. Moving a step further, we show that the only known technique of simulat- ing arithmetic formulas by constant width algebraic branching programs due to Ben-Or and Cleve does not preserve syntactic multilinearity. Moreover, for a generalized version of Ben-Or and Cleve’s technique, we show a linear lower bound on the number of registers if it is to preserve syntactic multilinearity. Also, we propose width of the arithmetic circuit as a possible measure of space for arithmetic computations. This definition gives a reasonable definition of lower space complexity classes compared to the existing notions of space. We also define and propose read-once certificates as a natural model of non- determinism for space bounded arithmetic computations. In the second part we study the complexity of isomorphism problem on ma- troids. We classify the problems according to how the input matroids are repre- sented. In the most general case, when the matroids are given as independent p set oracles, the problem is in Σ2, the second level of the polynomial hierarchy. p For linear matroids, we show that isomorphism testing is in Σ2, and is unlikely p to be Σ2 complete. However when the rank of the given input matroid is a con- stant, the problem is polynomial time many-one equivalent to the graph isomor- phism problem. For the case of matroids represented by graphs, we show that the isomorphism testing problem is polynomial time equivalent to the graph isomorphism problem. Along the way, we develop colouring techniques for handling coloured instances of matroid isomorphism problems. We also prove polynomial time equivalence of isomorphism testing problem and the problem of computing automorphism groups for the case of linear and graphic matroids. 6 Contents 1 Introduction 1 1.1 Counting classes and Arithmetic Circuits . ... 1 1.1.1 Motivation............................ 3 1.1.2 Contributions of the thesis . 4 1.2 MatroidIsomorphism.......................... 7 1.2.1 Motivation............................ 7 1.2.2 Contribution of the thesis . 8 1.3 Organisationofthethesis . 10 I Arithmetic circuits around NC1 11 2 Preliminaries 12 2.1 BooleanCircuits ............................ 12 2.2 BranchingPrograms .......................... 14 2.2.1 BPsandskewcircuits . 15 2.2.2 Series Parallel Construction . 16 2.3 Uniformityofcircuits. 18 2.4 Circuitbasedcomplexityclasses . 19 2.5 The Chinese Remaindering Technique . 23 3 Width and degree bounded circuits 25 3.1 Introduction............................... 25 3.2 The sSC hierarchy............................ 26 3.3 Closureproperties............................ 27 3.4 Relations with other hierarchies . 29 i 3.5 Conclusion................................ 30 4 Counting and arithmetic variants of NC1 31 4.1 Introduction............................... 31 4.2 Counting in a log-widthformula ................... 35 4.3 Counting version of sSCo ....................... 36 4.3.1 Definition ............................ 36 4.3.2 ClosureProperties . 37 4.4 HigherWidth .............................. 39 4.5 Variousrestrictions . 40 4.5.1 Modular Counting . 40 4.5.2 Arithmetic-Booleanversions . 40 4.6 Counting paths in restricted grid graphs . 42 4.7 Translating into the Valiant’s model . 47 4.7.1 Valiants’ Algebraic Model . 48 4.7.2 Valiant’sclasses. 49 4.8 Conclusion and open questions . 51 4.9 Appendix ................................ 54 4.9.1 Closure Properties of #NC1 and #BWBP ........... 54 5 The syntactic multilinear world 57 5.1 Introduction............................... 57 5.2 Syntactic Multilinear Circuits . 59 5.3 Depth reduction in small width sm-circuits . 59 5.4 Makingacircuitskew ......................... 67 5.4.1 Multiplicatively disjointness and Weakly skewness . ..... 69 5.4.2 Weaklyskewtoskew . 70 5.4.3 Multiplicatively disjoint to skew . 74 5.5 Bigpictureofthesm-world . 77 5.6 Conclusion and open questions . 80 6 Limitations 81 6.1 Introduction............................... 81 6.2 Limitations of Ben-Or and Cleve’s simulation . 83 6.2.1 Generalized B-C simulation . 83 ii 6.2.2 Combinatorial Designs . 85 6.2.3 Arandomizedconstruction. 86 6.3 Skewformula .............................. 91 6.3.1 A characterization of VSkewF ................. 92 6.3.2 An upper bound for .VSkewF ................ 93 6.3.3 Multilinear VersionsP . 94 6.4 Conclusion and open questions . 95 7 Small space analogues 96 7.1 Introduction............................... 96 7.2 Notion of space for arithmetic computations? . 97 7.2.1 Previously studied notions . 98 7.2.2 Defining VPSPACE intermsofcircuitwidth . 99 7.2.3 Comparing VPSPACE and VWIDTH(poly)...........101 7.3 Read-Oncecertificates . 103 7.4 Read-Once exponential sums of some restricted circuits . 109 7.5 Appendix ................................116 7.5.1 Blum Shub Smale (BSS) model of computation . 116 II Complexity of Matroid Isomorphism Problems 117 8 Matroid Isomorphism Problems 118 8.1 Introduction...............................118 8.2 Preliminaries ..............................120 8.2.1 Matroids.............................120 8.2.2 Input Representations of Matroids . 121 8.2.3 2-isomorphism. 123 8.2.4 Some complexity notions . 126 8.3 LinearMatroidIsomorphism. 126 8.3.1 General Complexity Bounds . 126 8.3.2 The bounded rank case . 128 8.4 Graphic Matroid Isomorphism . 131 8.5 Improved upper bounds for special cases of GMI ..........144 8.5.1 PlanarMatroids. 144 iii 8.5.2 Matroids of bounded genus and bounded degree graphs . 146 8.6 Conclusion and open problems . 146 9 Structural Complexity of Matroid Isomorphism Problems 148 9.1 Introduction...............................148 9.2 Colouring Techniques . 149 9.3 Complexity of computing automorphism groups . 153 9.3.1 Relationship with isomorphism
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