The Complexity of Testing Monomials in Multivariate Polynomials

The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its sum-product expansion. The complexity aspects of this problem and its variants are investigated with two objectives. One is to understand how this problem relates to critical problems in complexity, and if so to what extent. The other is to exploit possibilities of applying algebraic properties of polynomials to the study of those problems. A series of results about ΠΣΠ and ΠΣ polynomials are obtained in this paper, laying a basis for further study along this line.

[1]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[2]  Bin Fu,et al.  Algorithms for Testing Monomials in Multivariate Polynomials , 2010, COCOA.

[3]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[4]  Adam R. Klivans,et al.  Learning DNF in time 2 Õ(n 1/3 ) . , 2001, STOC 2001.

[5]  Richard Beigel,et al.  The polynomial method in circuit complexity , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[6]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[7]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[8]  Zhixiang Chen,et al.  On the learnability of Zn-DNF formulas (extended abstract) , 1995, COLT '95.

[9]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[10]  Rocco A. Servedio,et al.  Learning DNF in time , 2001, STOC '01.

[11]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[12]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[13]  Marvin Minsky,et al.  Perceptrons: expanded edition , 1988 .

[14]  John Michael Robson,et al.  Algorithms for Maximum Independent Sets , 1986, J. Algorithms.

[15]  B. M. Fulk.  MATH , 1992 .

[16]  Ioannis Koutis,et al.  Faster Algebraic Algorithms for Path and Packing Problems , 2008, ICALP.

[17]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[18]  Ryan Williams,et al.  Finding paths of length k in O*(2k) time , 2008, Inf. Process. Lett..

[19]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas , 1979, Inf. Process. Lett..

[20]  Zhi-Zhong Chen,et al.  Reducing randomness via irrational numbers , 1997, STOC '97.

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

[22]  Bin Fu,et al.  Approximating multilinear monomial coefficients and maximum multilinear monomials in multivariate polynomials , 2010, J. Comb. Optim..

[23]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[24]  Fu Bin.  SeparatingPH fromPP by relativization , 1992 .

[25]  Rocco A. Servedio,et al.  Learning DNF in time 2Õ(n1/3) , 2004, J. Comput. Syst. Sci..