Exponential Time Complexity of Weighted Counting of Independent Sets

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


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 x k . 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 2 cn ), 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.


Graph Transformation Satisfying Assignment Instance Size Tutte Polynomial Weighted Counting 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christian Hoffmann
    • 1
  1. 1.Donghua UniversityShanghaiChina

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