Exponential time complexity of the permanent and the Tutte polynomial

  • Authors:

  • Holger Dell;Thore Husfeldt;Martin Wahlén

  • Affiliations:

  • Humboldt University of Berlin, Germany;IT University of Copenhagen, Denmark and Lund University, Sweden;Lund University, Sweden and Uppsala University, Sweden

  • Venue:
  • ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
  • Year:
  • 2010

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Abstract

The Exponential Time Hypothesis (ETH) says that deciding the satisfiability of n-variable 3-CNF formulas requires time exp(Ω(n)). We relax this hypothesis by introducing its counting version #;ETH, namely that every algorithm that counts the satisfying assignments requires time exp(Ω(n)). We transfer the sparsification lemma for d-CNF formulas to the counting setting, which makes #ETH robust. Under this hypothesis, we show lower bounds for well-studied #P-hard problems: Computing the permanent of an n×n matrix with m nonzero entries requires time exp(Ω(m)). Restricted to 01-matrices, the bound is exp(Ω(m/logm)). Computing the Tutte polynomial of a multigraph with n vertices and m edges requires time exp(Ω(n)) at points (x, y) with (x - 1)(y - 1) ≠ = 1 and y ∉ {0,±1}. At points (x, 0) with x ∉ {0,±1} it requires time exp(Ω(n)), and if x = -2,-3, ..., it requires time exp(Ω(m)). For simple graphs, the bound is exp(Ω(m/log3 m)).