On Enumerating Monomials and Other Combinatorial Structures by Polynomial Interpolation

We study the problem of generating the monomials of a black box polynomial in the context of enumeration complexity. We present three new randomized algorithms for restricted classes of polynomials with a polynomial or incremental delay, and the same global running time as the classical ones. We introduce TotalBPP, IncBPP and DelayBPP, which are probabilistic counterparts of the most common classes for enumeration problems. Our interpolation algorithms are applied to algebraic representations of several combinatorial enumeration problems, which are so proved to belong to the introduced complexity classes. In particular, the spanning hypertrees of a 3-uniform hypergraph can be enumerated with a polynomial delay. Finally, we study polynomials given by circuits and prove that we can derandomize the interpolation algorithms on classes of bounded-depth circuits. We also prove the hardness of some problems on polynomials of low degree and small circuit complexity, which suggests that our good interpolation algorithm for multilinear polynomials cannot be generalized to degree 2 polynomials. This article is an improved and extended version of Strozecki (Mathematical Foundations of Computer Science, pp. 629–640, 2010) and of the third chapter of the author’s Ph.D. Thesis (Strozecki, Ph.D. Thesis, 2010).

[1]  Arnaud Durand,et al.  First-order queries on structures of bounded degree are computable with constant delay , 2005, TOCL.

[2]  Nitin Saxena,et al.  An Almost Optimal Rank Bound for Depth-3 Identities , 2011, SIAM J. Comput..

[3]  Stefan Mengel,et al.  Monomials in arithmetic circuits: Complete problems in the counting hierarchy , 2012, STACS.

[4]  Seinosuke Toda,et al.  Classes of Arithmetic Circuits Capturing the Complexity of Computing the Determinant , 1992 .

[5]  E. W. Ng.  Symbolic and Algebraic Computation , 1979, Lecture Notes in Computer Science.

[6]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[7]  Éric Schost,et al.  Interpolation of polynomials given by straight-line programs , 2009, Theor. Comput. Sci..

[8]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[9]  Vikraman Arvind,et al.  Arithmetic Circuit Size, Identity Testing, and Finite Automata , 2009, Electron. Colloquium Comput. Complex..

[10]  Ilya Volkovich,et al.  Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[11]  Yann Strozecki.  Enumeration of the Monomials of a Polynomial and Related Complexity Classes , 2010, MFCS.

[12]  Richard J. Lipton,et al.  A Probabilistic Remark on Algebraic Program Testing , 1978, Inf. Process. Lett..

[13]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[14]  Ján Plesník,et al.  The NP-Completeness of the Hamiltonian Cycle Problem in Planar Digraphs with Degree Bound Two , 1979, Inf. Process. Lett..

[15]  Martin Aigner,et al.  A Course in Enumeration , 2007 .

[16]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[17]  Guillaume Bagan.  Algorithmes et complexité des problèmes d'énumération pour l'évaluation de requêtes logiques. (Algorithms and complexity of enumeration problems for the evaluation of logical queries) , 2009 .

[18]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[19]  Leslie Ann Goldberg.  Listing Graphs That Satisfy First-Order Sentences , 1994, J. Comput. Syst. Sci..

[20]  Guillaume Malod,et al.  Characterizing Valiant's algebraic complexity classes , 2006, J. Complex..

[21]  Michael Ben-Or,et al.  A deterministic algorithm for sparse multivariate polynomial interpolation , 1988, STOC '88.

[22]  Ilya Volkovich,et al.  Black-Box Identity Testing of Depth-4 Multilinear Circuits , 2011, Combinatorica.

[23]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[24]  Erich Kaltofen,et al.  Expressing a fraction of two determinants as a determinant , 2008, ISSAC '08.

[25]  Takeaki Uno,et al.  Algorithms for Enumerating All Perfect, Maximum and Maximal Matchings in Bipartite Graphs , 1997, ISAAC.

[26]  A new matrix-tree theorem , 2001, math/0109104.

[27]  Peter Bürgisser,et al.  Completeness and Reduction in Algebraic Complexity Theory , 2000, Algorithms and computation in mathematics.

[28]  Meena Mahajan,et al.  Identity Testing, Multilinearity Testing, and Monomials in Read-Once/Twice Formulas and Branching Programs , 2012, MFCS.

[29]  Neeraj Kayal,et al.  On the Sum of Square Roots of Polynomials and Related Problems , 2011, Computational Complexity Conference.

[30]  Joël Ouaknine,et al.  Language Equivalence for Probabilistic Automata , 2011, CAV.

[31]  Bruno Courcelle,et al.  Linear delay enumeration and monadic second-order logic , 2009, Discret. Appl. Math..

[32]  Kousha Etessami,et al.  Polynomial time algorithms for multi-type branching processesand stochastic context-free grammars , 2012, STOC '12.

[33]  Oscar H. Ibarra,et al.  Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs , 1983, JACM.

[34]  Mihalis Yannakakis,et al.  On Generating All Maximal Independent Sets , 1988, Inf. Process. Lett..

[35]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[36]  Nitin Saxena,et al.  Jacobian hits circuits: hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits , 2011, STOC '12.

[37]  Yann Strozecki,et al.  The Complexity of Acyclic Subhypergraph Problems , 2011, WALCOM.

[38]  Yann Strozecki,et al.  Enumeration Complexity of Logical Query Problems with Second-order Variables , 2011, CSL.

[39]  Yann Strozecki,et al.  Enumeration complexity and matroid decomposition , 2010 .

[40]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[41]  Ilya Volkovich,et al.  Deterministic Identity Testing of Depth-4 Multilinear Circuits with Bounded Top Fan-in , 2013, SIAM J. Comput..

[42]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[43]  László Lovász,et al.  Matroid matching and some applications , 1980, J. Comb. Theory, Ser. B.

[44]  Frank Ruskey,et al.  Generating Linear Extensions Fast , 1994, SIAM J. Comput..

[45]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[46]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[47]  Daniel A. Spielman,et al.  Randomness efficient identity testing of multivariate polynomials , 2001, STOC '01.

[48]  M. Jerrum.  Counting, Sampling and Integrating: Algorithms and Complexity , 2003 .

[49]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[50]  Joachim von zur Gathen,et al.  Feasible Arithmetic Computations: Valiant's Hypothesis , 1987, J. Symb. Comput..

[51]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[52]  Richard Zippel,et al.  Interpolating Polynomials from Their Values , 1990, J. Symb. Comput..

[53]  Erich Kaltofen,et al.  Early termination in Ben-Or/Tiwari sparse interpolation and a hybrid of Zippel's algorithm , 2000, ISSAC.