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arXiv · 1206.1775

Exponential Time Complexity of the Permanent and the Tutte Polynomial

Holger Dell·Thore Husfeldt·Dániel Marx·Nina Taslaman·Martin Wáhlen
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Abstract

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting.

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BibTeXRIS

Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, Martin Wáhlen. 2012-06-08. Exponential Time Complexity of the Permanent and the Tutte Polynomial. https://doi.org/10.1145/2635812

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