Exponential Time Complexity of Weighted Counting of Independent Sets

  • Christian Hoffmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6478)

Abstract

We consider weighted counting of independent sets using a rational weight x: Given a graph with n vertices, count its independent sets such that each set of size k contributes xk. This is equivalent to computation of the partition function of the lattice gas with hard-core self-repulsion and hard-core pair interaction. We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time 2Ω(n) (i.e. there is a c > 0 such that no algorithm can solve #3SAT in time 2cn), counting the independent sets of size n/3 of an n-vertex graph needs time 2Ω(n) and weighted counting of independent sets needs time \(2^{\Omega(n/\log^3 n)}\) for all rational weights x ≠ 0.

We have two technical ingredients: The first is a reduction from 3SAT to independent sets that preserves the number of solutions and increases the instance size only by a constant factor. Second, we devise a combination of vertex cloning and path addition. This graph transformation allows us to adapt a recent technique by Dell, Husfeldt, and Wahlén which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christian Hoffmann
    • 1
  1. 1.Donghua UniversityShanghaiChina

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