DOCSLIB.ORG

The Power of Algorithmic Approaches to the Graph Isomorphism Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Power of Algorithmic Approaches to the Graph Isomorphism Problem The Power of Algorithmic Approaches to the Graph Isomorphism Problem Von der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Daniel Neuen, Master of Science aus Dormagen Berichter: Universit¨atsprofessorDr. Martin Grohe Universit¨atsprofessorDr. Pascal Schweitzer Universit¨atsprofessorDr. L´aszl´oBabai Tag der m¨undlichen Pr¨ufung:17. Dezember 2019 Diese Dissertation ist auf den Internetseiten der Universit¨atsbibliothek online verf¨ugbar. ii Abstract The Graph Isomorphism Problem asks, given two input graphs, whether they are structurally the same, that is, whether there is a renaming of the vertices of the first graph in order to transform it to the second graph. By a recent breakthrough result of Babai (STOC 2016), this problem can be solved in quasipolynomial time. However, despite extensive research efforts, it remains one of only few natural problems in NP that are neither known to be solvable in polynomial time nor known to be NP- complete. Over the past five decades several powerful techniques tackling the Graph Isomor- phism Problem have been investigated uncovering various surprising links between different approaches. Also, the situation has led to a number of algorithms solving the isomorphism problem on restricted classes of input graphs. In this thesis, we continue the investigation of various standard approaches to the Graph Isomorphism Problem to further broaden our understanding on the power and limits of such approaches. In particular, this leads to several improved algorithms solving the isomorphism problem for important restricted classes of graphs. One of the most fundamental methods in the context of graph isomorphism test- ing is the Weisfeiler-Leman algorithm, which iteratively computes an isomorphism- invariant coloring of vertex-tuples. While the algorithm is unable to decide the isomorphism problem itself, it is commonly used as a subroutine and, for various restricted graph classes, it already serves as a complete isomorphism test. In the latter direction we prove for example that the Weisfeiler-Leman dimension of graph classes of bounded rank-width is bounded. While it was already known that the isomorphism problem for graphs of rank-width at most k is polynomial-time solvable (Grohe and Schweitzer, FOCS 2015), the previous best algorithm is complicated and the exponent of the running time depends non-elementary on k. In contrast, our analysis of the Weisfeiler-Leman algorithm yields a simple isomorphism test running in time nO(k). A framework closely related to the Weisfeiler-Leman algorithm and which works particularly well in practice is the Individualization-Refinement paradigm. Extending our understanding on the limits of combinatorial approaches we provide the first exponential lower bounds on the worst case complexity of a large and natural class of algorithms within this framework. In particular, this includes all practical state- of-the-art isomorphism tools answering an open question from Babai (STOC 2016) on the worst-case complexity of such solvers. A second crucial approach to the Graph Isomorphism Problem is based on group- theoretic techniques. In this direction, one of the algorithmic cornerstones is Luks's polynomial time algorithm for testing isomorphism of bounded degree graphs (JCSS 1982). Adapting the novel group-theoretic methods by Babai developed for his quasipolynomial time isomorphism test (STOC 2016) we give an isomorphism test for graphs of maximum degree d running in time npolylog(d). This significantly improves over the previous best isomorphism test for graphs of maximum degree d running in time nO(d= log d) (Babai, Kantor and Luks, FOCS 1983). With Luks's algorithm being used as a subroutine in a number of other algo- rithms it is natural to ask for the consequences of this improvement. Besides simple applications regarding structures of small degree, we present an isomorphism test for graphs of tree-width k running in time 2k polylog(k) poly(n) improving the fixed- parameter tractable algorithm of Lokshtanov et al. (FOCS 2014) running in time 5 2O(k log k) poly(n). iii Zusammenfassung Das Graphisomorphieproblem fragt, gegeben zwei Graphen, ob diese strukturell iden- tisch sind, d.h. ob man durch Umbenennung der Knoten des ersten Graphen diesen in den zweiten Graphen transformieren kann. Mit dem j¨ungstenDurchbruch von Babai (STOC 2016) kann dieses Problem in quasipolynomieller Zeit gel¨ostwerden. Jedoch bleibt es, trotz intensiver Forschung, eines von nur wenigen nat¨urlichen Problemen in NP, f¨urdas weder ein Polynomialzeitalgorithmus noch die NP-Schwere bekannt ist. In den letzten f¨unfJahrzehnten wurden vielf¨altige algorithmische Techniken f¨ur das Graphisomorphieproblem erforscht, was ¨uberraschende Verbindungen zwischen verschiedenen Ans¨atzenhervorgebracht hat. Außerdem f¨uhrtedies zu zahlreichen Algorithmen, die das Isomorphieproblem auf eingeschr¨anktenKlassen von Eingabe- graphen l¨osen. Diese Dissertation verfolgt das Ziel unser Verst¨andnis¨uber St¨arken und Grenzen verschiedener wichtiger Ans¨atzezum Graphisomorphieproblem zu er- weitern. Dies resultiert insbesondere in mehreren verbesserten Algorithmen die das Isomorphieproblem f¨urwichtige Klassen von Graphen l¨osen. Eine der grundlegendsten Methoden f¨urdas Testen von Isomorphie ist der Weisfei- ler-Leman Algorithmus, der iterativ eine isomorphieinvariante F¨arbungder Knotentu- pel berechnet. Obwohl dieser Algorithmus das Isomorphieproblem nicht alleine l¨osen kann, wird er regelm¨aßigals Unterprogramm verwendet und dient als vollst¨andiger Isomorphietest f¨urmehrere eingeschr¨ankteKlassen von Graphen. In diesen Zusam- menhang zeigen wir, dass die Weisfeiler-Leman Dimension von Graphen beschr¨ankter Rangweite beschr¨anktist. W¨ahrendein Polynomialzeitalgorithmus f¨urdas Isomor- phieproblem von Graphen mit Rangweite h¨ochstens k bereits vorher bekannt war (Grohe und Schweitzer, FOCS 2015), so sind bisherige Algorithmen kompliziert und der Exponent der Laufzeit h¨angtnicht-elementar von k ab. Im Gegensatz dazu ergibt unsere Analyse des Weisfeiler-Leman Algorithmus einen einfachen Isomorphietest mit Laufzeit nO(k). Ein mit dem Weisfeiler-Leman Algorithmus eng verwandter Ansatz, der besonders gut in der Praxis funktioniert, ist das Paradigma des Individualisierens und Verfei- nerns. Um unser Verst¨andnisder Grenzen kombinatorischer Ans¨atzezu erweitern, geben wir die ersten exponentiellen unteren Schranken f¨urdie Laufzeit einer großen und nat¨urlichen Klasse von Algorithmen innerhalb dieses Paradigmas. Insbeson- dere erhalten wir exponentielle untere Schranken f¨urdie Laufzeit s¨amtlicher moder- ner praktischer Isomorphieprogramme und beantworten damit eine offene Frage von Babai (STOC 2016). Ein zweiter fundamentaler Ansatz f¨urdas Graphisomorphieproblem ist die Ver- wendung gruppentheoretischer Methoden. In dieser Hinsicht ist Luks Algorithmus f¨ur das Testen von Isomorphie von Graphen beschr¨anktenGrades einer der Grundpfeiler der algorithmischen Theorie des Graphisomorphieproblems. Indem wir die grup- pentheoretischen Methoden, die Babai f¨urseinen Quasipolynomialzeitalgorithmus entwickelt hat, anpassen, erhalten wir einen Isomorphietest f¨urGraphen mit Ma- ximalgrad h¨ochstens d mit einer Laufzeit von npolylog(d). Dies stellt eine deutliche Verbesserung im Vergleich zum bisher schnellsten Algorithmus f¨urdieses Problem dar, welcher nO(d= log d) Schritte ben¨otigt(Babai, Kantor und Luks, FOCS 1983). Da Luks Algorithmus als Unterprogramm f¨ureine Reihe weiterer Algorithmen dient, stellt sich die Frage welche Konsequenzen sich aus obiger Verbesserung ergeben. Neben einfachen Anwendungen f¨urdas Isomorphieproblem von relationalen Struk- turen mit kleinem Grad zeigen wir, dass man Isomorphie von Graphen mit Baumweite h¨ochstens k in Zeit 2k polylog(k) poly(n) testen kann und verbessern damit den FPT- 5 Algorithmus von Lokshtanov et al. (FOCS 2014) mit Laufzeit 2O(k log k) poly(n). iv v Acknowledgments First and foremost, I am grateful to my supervisor Martin Grohe for his guidance and support during the time I spent in his group researching for this thesis. Also, I want to thank Pascal Schweitzer for his continued support and for always having an open door when still being at RWTH Aachen University. Moreover, I would like to thank my colleagues Sandra Kiefer and Daniel Wiebking for joint collaborations and for proof-reading parts of this thesis. I am particularly thankful to Sandra Kiefer for sharing an office all these years and making the time at our group more joyful. Finally, I am grateful to my family, and in particular my parents, for their endless support throughout my time as an undergraduate and graduate student. vi Contents 1 Introduction 1 2 Isomorphism and Combinatorial Algorithms 11 2.1 Graphs and Isomorphism . 11 2.1.1 Graphs and Notation . 11 2.1.2 Tree Decompositions and Tree-Width . 12 2.1.3 Isomorphisms . 13 2.1.4 Isomorphism Invariance and Canonization . 14 2.2 The Weisfeiler-Leman Algorithm . 16 2.2.1 The Algorithm . 16 2.2.2 Pebble Games . 19 2.2.3 Connection to Logic . 20 2.3 Individualization-Refinement . 21 2.3.1 The Basic Paradigm . 21 2.3.2 Pruning with Invariants . 23 2.3.3 Pruning with Automorphisms . 23 3 Upper Bounds on the WL Dimension 25 3.1 Tree-Width . 25 3.2 Rank-Width . 30 3.2.1 Definition and Properties . 30 3.2.2 Split Pairs and Flip Functions . 32 3.2.3 A Recursive Strategy for Spoiler . 36 3.3 Further Results . 41 4 Lower Bounds 43 4.1 Weisfeiler-Leman Algorithm . 43 4.2 The I/R-Method in Theory . 48 4.2.1 A Framework for a Lower Bound . 49 4.2.2 The Multipede Construction . 50 4.2.3 The Weisfeiler-Leman Refinement and Closure Operators . 53 4.2.4 Meager Graphs . 56 4.2.5 Lower Bounds for I/R-Algorithms . 58 4.3 The I/R-Method in Practice . 61 vii viii CONTENTS 5 Group Theory 65 5.1 Permutation Groups . 65 5.1.1 Basics . 65 5.1.2 Algorithms for Permutation Groups . 67 5.1.3 Groups with Restricted Composition Factors . 68 5.2 String Isomorphism . 69 5.2.1 Graphs of Bounded Color Class Size . 69 5.2.2 String Isomorphism Problem . 70 5.2.3 Recursion Mechanisms . 71 5.3 Bounded Degree Graphs and Group Theory .
Load more

Recommended publications
  • Nondeterministic Direct Product Reducations and the Success
    Nondeterministic Direct Product Reductions and the Success Probability of SAT Solvers [Preliminary version] Andrew Drucker∗ May 6, 2013 Last updated: May 16, 2013 Abstract We give two nondeterministic reductions which yield new direct product theorems (DPTs) for Boolean circuits. In these theorems one assumes that a target function f is mildly hard against nondeterministic circuits, and concludes that the direct product f ⊗t is extremely hard against (only polynomially smaller) probabilistic circuits. The main advantage of these results compared with previous DPTs is the strength of the size bound in our conclusion. As an application, we show that if NP is not in coNP=poly then, for every PPT algorithm attempting to produce satisfying assigments to Boolean formulas, there are infinitely many instances where the algorithm's success probability is nearly-exponentially small. This furthers a project of Paturi and Pudl´ak[STOC'10]. Contents 1 Introduction 2 1.1 Hardness amplification and direct product theorems..................2 1.2 Our new direct product theorems.............................4 1.3 Application to the success probability of PPT SAT solvers...............6 1.4 Our techniques.......................................7 2 Preliminaries 9 2.1 Deterministic and probabilistic Boolean circuits..................... 10 2.2 Nondeterministic circuits and nondeterministic mappings............... 10 2.3 Direct-product solvers................................... 12 2.4 Hash families and their properties............................ 13 2.5 A general nondeterministic circuit construction..................... 14 ∗School of Mathematics, IAS, Princeton, NJ. Email: [email protected]. This material is based upon work supported by the National Science Foundation under agreements Princeton University Prime Award No. CCF- 0832797 and Sub-contract No. 00001583.
    [Show full text]
  • On the Probabilistic Closure of the Loose Unambiguous Hierarchy
    Electronic Colloquium on Computational Complexity, Report No. 50 (2014) On the probabilistic closure of the loose unambiguous hierarchy Edward A. Hirsch∗ Dmitry Sokolovy March 21, 2014 Abstract Unambiguous hierarchies [NR93, LR94, NR98] are defined similarly to the polyno- mial hierarchy; however, all witnesses must be unique. These hierarchies have subtle differences in the mode of using oracles. We consider a \loose" unambiguous hierarchy prUH• with relaxed definition of oracle access to promise problems. Namely, we allow to make queries that miss the promise set; however, the oracle answer in this case can be arbitrary (a similar definition of oracle access has been used in [CR08]). We prove that the first part of Toda's theorem PH ⊆ BP · ⊕P ⊆ PPP can be rectified to PH = BP · prUH•, that is, the closure of our hierarchy under Sch¨oning's BP operator equals the polynomial hierarchy. It is easily seen that BP · prUH• ⊆ BP · ⊕P. The proof follows the same lines as Toda's proof, so the main contribution of the present note is a new definition. 1 Introduction Around 1990, there was a burst of results about interactive protocols [GMR89, Bab85, GS86, BOGKW88, BM88, LFKN92, BFL91, Sha90]. In the same time, Seinosuke Toda proved that PH ⊆ BP · ⊕P ⊆ PPP [Tod91]. The first part of his result can be viewed as an Arthur-Merlin game (recall that AM = BP · NP); however, Merlin must have an odd number of correct proofs. One can describe its proof as follows. We depart from a relativized version of Valiant-Vazirani's lemma and turn the polynomial hierarchy, level by level, into a multi-round Arthur-Merlin game where Merlin always has a unique witness.
    [Show full text]
  • The Complexity of Verifying Loop-Free Programs As
    The Complexity of Verifying Loop-Free Programs as Differentially Private Marco Gaboardi Boston University, USA Kobbi Nissim Georgetown University, USA David Purser University of Warwick, UK MPI-SWS, Germany Abstract We study the problem of verifying differential privacy for loop-free programs with probabilistic choice. Programs in this class can be seen as randomized Boolean circuits, which we will use as a formal model to answer two different questions: first, deciding whether a program satisfies a prescribed level of privacy; second, approximating the privacy parameters a program realizes. We show that the problem of deciding whether a program satisfies ε-differential privacy is coNP#P-complete. In fact, this is the case when either the input domain or the output range of the program is large. Further, we show that deciding whether a program is (ε, δ)-differentially private is coNP#P-hard, #P and in coNP#P for small output domains, but always in coNP#P . Finally, we show that the problem of approximating the level of differential privacy is both NP-hard and coNP-hard. These results complement previous results by Murtagh and Vadhan [35] showing that deciding the optimal composition of differentially private components is #P-complete, and that approximating the optimal composition of differentially private components is in P. 2012 ACM Subject Classification Security and privacy; Theory of computation Probabilistic → computation Keywords and phrases differential privacy, program verification, probabilistic programs Digital Object Identifier 10.4230/LIPIcs.ICALP.2020.129 Category Track B: Automata, Logic, Semantics, and Theory of Programming Related Version A full version of the paper is available at https://arxiv.org/abs/1911.03272 Funding M.
    [Show full text]
  • Download This PDF File
    T G¨ P 2012 C N Deadline: December 31, 2011 The Gödel Prize for outstanding papers in the area of theoretical computer sci- ence is sponsored jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery, Special Inter- est Group on Algorithms and Computation Theory (ACM-SIGACT). The award is presented annually, with the presentation taking place alternately at the Inter- national Colloquium on Automata, Languages, and Programming (ICALP) and the ACM Symposium on Theory of Computing (STOC). The 20th prize will be awarded at the 39th International Colloquium on Automata, Languages, and Pro- gramming to be held at the University of Warwick, UK, in July 2012. The Prize is named in honor of Kurt Gödel in recognition of his major contribu- tions to mathematical logic and of his interest, discovered in a letter he wrote to John von Neumann shortly before von Neumann’s death, in what has become the famous P versus NP question. The Prize includes an award of USD 5000. AWARD COMMITTEE: The winner of the Prize is selected by a committee of six members. The EATCS President and the SIGACT Chair each appoint three members to the committee, to serve staggered three-year terms. The committee is chaired alternately by representatives of EATCS and SIGACT. The 2012 Award Committee consists of Sanjeev Arora (Princeton University), Josep Díaz (Uni- versitat Politècnica de Catalunya), Giuseppe Italiano (Università a˘ di Roma Tor Vergata), Mogens Nielsen (University of Aarhus), Daniel Spielman (Yale Univer- sity), and Eli Upfal (Brown University). ELIGIBILITY: The rule for the 2011 Prize is given below and supersedes any di fferent interpretation of the parametric rule to be found on websites on both SIGACT and EATCS.
    [Show full text]
  • The Complexity of Verifying Loop-Free Programs As
    The Complexity of Verifying Loop-Free Programs as Differentially Private Marco Gaboardi Boston University, MA, USA Kobbi Nissim Georgetown University, Washington, DC, USA David Purser University of Warwick, Coventry, UK Max Planck Institute for Software Systems, Saarbrücken, Germany Abstract We study the problem of verifying differential privacy for loop-free programs with probabilistic choice. Programs in this class can be seen as randomized Boolean circuits, which we will use as a formal model to answer two different questions: first, deciding whether a program satisfies a prescribed level of privacy; second, approximating the privacy parameters a program realizes. We show that the problem of deciding whether a program satisfies ε-differential privacy is coNP#P-complete. In fact, this is the case when either the input domain or the output range of the program is large. Further, we show that deciding whether a program is (ε, δ)-differentially private is coNP#P-hard, #P and in coNP#P for small output domains, but always in coNP#P . Finally, we show that the problem of approximating the level of differential privacy is both NP-hard and coNP-hard. These results complement previous results by Murtagh and Vadhan [35] showing that deciding the optimal composition of differentially private components is #P-complete, and that approximating the optimal composition of differentially private components is in P. 2012 ACM Subject Classification Security and privacy; Theory of computation → Probabilistic computation Keywords and phrases differential privacy, program verification, probabilistic programs Digital Object Identifier 10.4230/LIPIcs.ICALP.2020.129 Category Track B: Automata, Logic, Semantics, and Theory of Programming Related Version A full version of the paper is available at https://arxiv.org/abs/1911.03272.
    [Show full text]
  • On TC Lower Bounds for the Permanent
    On TC0 Lower Bounds for the Permanent Jeff Kinne Indiana State University [email protected] September 5, 2012 Abstract In this paper we consider the problem of proving lower bounds for the permanent. An ongoing line of research has shown super-polynomial lower bounds for slightly-non-uniform small-depth threshold and arithmetic circuits [All99, KP09, JS11, JS12]. We prove a new parameterized lower bound that includes each of the previous results as sub-cases. Our main result implies that the permanent does not have Boolean threshold circuits of the following kinds. 1. Depth O(1), poly-log(n) bits of non-uniformity, and size s(n) such that for all constants c, s(c)(n) < 2n. The size s must satisfy another technical condition that is true of functions normally dealt with (such as compositions of polynomials, logarithms, and exponentials). 2. Depth o(log log n), poly-log(n) bits of non-uniformity, and size nO(1). 3. Depth O(1), no(1) bits of non-uniformity, and size nO(1). Our proof yields a new “either or” hardness result. One instantiation is that either NP does not have polynomial-size constant-depth threshold circuits that use no(1) bits of non-uniformity, or the permanent does not have polynomial-size general circuits. 1 Introduction The Search for Hard Problems Computational complexity aims to determine the compu- tational costs of solving important problems, requiring both upper bounds and lower bounds. Though many types of lower bounds have been difficult to prove, conjectured lower bounds have become central across complexity theory.
    [Show full text]
  • The Complexity of Equivalence Relations
    The Complexity of Equivalence Relations Joshua A. Grochow November 20, 2008 Abstract To determine if two given lists of numbers are the same set, we would sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms for equivalences arise in graph isomorphism and its variants, and the equality of permutation groups given by generators. To determine if two given graphs are cospectral, however, we compute their characteristic polynomials and see if they are the same; the characteristic polynomial is a complete invariant for the equivalence relation of cospectrality. This is weaker than a canonical form, and it is not known whether a canonical form for cospectrality exists. Note that it is a priori possible for an equivalence relation to be decidable in polynomial time without either a complete invariant or canonical form. Blass and Gurevich (“Equivalence relations, invariants, and normal forms, I and II”, 1984) ask whether these conditions on equivalence relations – having an FP canonical form, having an FP complete invariant, and simply being in P – are in fact different. They showed that this question requires non-relativizing techniques to resolve. Here we extend their results using generic oracles, and give new connections to probabilistic and quantum computation. We denote the class of equivalence problems in P by PEq, the class of problems with com- plete FP invariants Ker, and the class with FP canonical forms CF; CF ⊆ Ker ⊆ PEq, and we ask whether these inclusions are proper. If x ∼ y implies |y| ≤ poly(|x|), we say that ∼ is polynomially bounded; we denote the corresponding classes of equivalence relation CFp, Kerp, and PEqp.
    [Show full text]
  • Stal Aanderaa Hao Wang Lars Aarvik Martin Abadi Zohar Manna James
    Don Heller J. von zur Gathen Rodney Howell Mark Buckingham Moshe VardiHagit Attiya Raymond Greenlaw Henry Foley Tak-Wah Lam Chul KimEitan Gurari Jerrold W. GrossmanM. Kifer J.F. Traub Brian Leininger Martin Golumbic Amotz Bar-Noy Volker Strassen Catriel Beeri Prabhakar Raghavan Louis E. Rosier Daniel M. Kan Danny Dolev Larry Ruzzo Bala Ravikumar Hsu-Chun Yen David Eppstein Herve Gallaire Clark Thomborson Rajeev Raman Miriam Balaban Arthur Werschulz Stuart Haber Amir Ben-Amram Hui Wang Oscar H. Ibarra Samuel Eilenberg Jim Gray Jik Chang Vardi Amdursky H.T. Kung Konrad Jacobs William Bultman Jacob Gonczarowski Tao Jiang Orli Waarts Richard ColePaul Dietz Zvi Galil Vivek Gore Arnaldo V. Moura Daniel Cohen Kunsoo Park Raffaele Giancarlo Qi Zheng Eli Shamir James Thatcher Cathy McGeoch Clark Thompson Sam Kim Karol Borsuk G.M. Baudet Steve Fortune Michael Harrison Julius Plucker NicholasMichael Tran Palis Daniel Lehmann Wilhelm MaakMartin Dietzfelbinger Arthur Banks Wolfgang Maass Kimberly King Dan Gordon Shafee Give'on Jean Musinski Eric Allender Pino Italiano Serge Plotkin Anil Kamath Jeanette Schmidt-Prozan Moti Yung Amiram Yehudai Felix Klein Joseph Naor John H. Holland Donald Stanat Jon Bentley Trudy Weibel Stefan Mazurkiewicz Daniela Rus Walter Kirchherr Harvey Garner Erich Hecke Martin Strauss Shalom Tsur Ivan Havel Marc Snir John Hopcroft E.F. Codd Chandrajit Bajaj Eli Upfal Guy Blelloch T.K. Dey Ferdinand Lindemann Matt GellerJohn Beatty Bernhard Zeigler James Wyllie Kurt Schutte Norman Scott Ogden Rood Karl Lieberherr Waclaw Sierpinski Carl V. Page Ronald Greenberg Erwin Engeler Robert Millikan Al Aho Richard Courant Fred Kruher W.R. Ham Jim Driscoll David Hilbert Lloyd Devore Shmuel Agmon Charles E.
    [Show full text]
  • Lipics-CCC-2021-16.Pdf (1
    On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors Joshua A. Grochow # Ñ Departments of Computer Science and Mathematics, University of Colorado Boulder, CO, USA Youming Qiao # Centre for Quantum Software and Information, University of Technology Sydney, Australia Abstract In this paper we study some classical complexity-theoretic questions regarding Group Isomorphism (GpI). We focus on p-groups (groups of prime power order) with odd p, which are believed to be a bottleneck case for GpI, and work in the model of matrix groups over finite fields. Our main results are as follows. Although search-to-decision and counting-to-decision reductions have been known for over four decades for Graph Isomorphism (GI), they had remained open for GpI, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from Tensor Isomorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p-groups of class 2 and exponent p. Despite the widely held belief that p-groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p-groups of “small” class and exponent p to those of class two and exponent p. For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI. Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets.
    [Show full text]
  • Bibliography Y
    UvA-DARE (Digital Academic Repository) On quantum computation theory van Dam, W.K. Publication date 2002 Link to publication Citation for published version (APA): van Dam, W. K. (2002). On quantum computation theory. Institute for Logic, Language and Computation. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:26 Sep 2021 Bibliography y [1]] Leonard M. Adleman, Jonathan Demarrais, and Ming-Deh A. Huang. Quantum computability.. SIAM Journal on Computing, 26(5): 1524-1540, October 1997. [2]] Manindra Agrawal and V. Arvind. Quasi-linear truth-table reductions to p- selectivee sets. Theoretical Computer Science, 158(l-2):361-370, May 1996. [3]] Dorit Aharonov, Alexei Kitaev, and Noam Nisan. Quantum circuits with mixed states.. In Proceedings of the 30th Annual ACM Symposium on Theory of Com- puting,puting, pages 20-30,1998.
    [Show full text]
  • 39 Non-Uniform ACC Circuit Lower Bounds
    39 Non-Uniform ACC Circuit Lower Bounds RYAN WILLIAMS, Stanford University The class ACC consists of circuit families with constant depth over unbounded fan-in AND, OR, NOT, and MODm gates, where m > 1 is an arbitrary constant. We prove: — NEXP, the class of languages accepted in nondeterministic exponential time, does not have non-uniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasi-polynomials and other less natural functions. — ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have non-uniform ACC o(1) circuits of 2n size. The lower bound gives an exponential size-depth tradeoff: for every d, m there is a NP nδ δ > 0 such that E doesn’t have depth-d ACC circuits of size 2 with MODm gates. NP Previously, it was not known whether EXP had depth-3 polynomial size circuits made out of only MOD6 gates. The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithms combine known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a strengthening of the author’s prior work. Categories and Subject Descriptors: F.1.3 [Theory of Computation]: Complexity Measures and Classes General Terms: Algorithms, Theory Additional Key Words and Phrases: circuit complexity, lower bounds, satisfiability, NEXP, ACC ACM Reference Format: Ryan Williams, 2013. Alternation-trading proofs, linear programming, and lower bounds. J. ACM 9, 4, Arti- cle 39 (March 2010), 31 pages.
    [Show full text]
  • Fixed-Polynomial Size Circuit Bounds
    Fixed-Polynomial Size Circuit Bounds Lance Fortnow Rahul Santhanam Ryan Williams Northwestern University University of Edinburgh Institute for Advanced Study Evanston, IL, USA Edinburgh, Scotland, UK Princeton, NJ, USA [email protected] [email protected] [email protected] P Abstract—In 1982, Kannan showed that Σ2 does not have this question remains open for NP – we do not even know k n -sized circuits for any k. Do smaller classes also admit of superlinear size lower bounds for NP. such circuit lower bounds? Despite several improvements of Kannan’s result, we still cannot prove that PNP does not have However, we do have such lower bounds for some classes linear size circuits. Work of Aaronson and Wigderson provides slightly above NP. In 1982, Kannan [Kan82] proved that strong evidence – the “algebrization” barrier – that current P k Σ2 does not have n -sized circuits for any k. This re- techniques have inherent limitations in this respect. sult was progressively improved using relativizing tech- We explore questions about fixed-polynomial size circuit niques ([BCG+96], [KW98], [Cai07]) culminating in the lower bounds around and beyond the algebrization barrier. analogous circuit lower bound for SP [Cai07]. We find several connections, including 2 Recently, non-relativizing techniques from the theory of • The following are equivalent: k k interactive proofs have been applied to this problem. In 2005, – NP is in SIZE(n ) (has O(n )-size circuit families) Vinodchandran [Vin05] showed that the class PP does not for some k nc k k – For each c, PNP[ ] is in SIZE(n ) for some k have n -sized circuits.
    [Show full text]