ICALP 2010: Automata, Languages and Programming pp 426-437

# Exponential Time Complexity of the Permanent and the Tutte Polynomial

(Extended Abstract)
• Holger Dell
• Thore Husfeldt
• Martin Wahlén
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

## Abstract

The Exponential Time Hypothesis (ETH) says that deciding the satisfiability of n-variable 3-CNF formulas requires time $$\exp(\Omega(n))$$. We relax this hypothesis by introducing its counting version #ETH, namely that every algorithm that counts the satisfying assignments requires time $$\exp(\Omega(n))$$. 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 $$\exp(\Omega(m))$$. Restricted to 01-matrices, the bound is $$\exp(\Omega(m/\log m))$$. Computing the Tutte polynomial of a multigraph with n vertices and m edges requires time $$\exp(\Omega(n))$$ at points (x,y) with (x − 1)(y − 1) ≠ 1 and y ∉ {0,±1}. At points (x,0) with $$x \not \in \{0,\pm 1\}$$ it requires time $$\exp(\Omega(n))$$, and if x = − 2, − 3,..., it requires time $$\exp(\Omega(m))$$. For simple graphs, the bound is $$\exp(\Omega(m/\log^3 m))$$.

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## Authors and Affiliations

• Holger Dell
• 1
• Thore Husfeldt
• 2
• Martin Wahlén
• 3
1. 1.Humboldt University of BerlinGermany
2. 2.IT University of Copenhagen, Denmark and Lund UniversitySweden
3. 3.Lund University, Sweden and Uppsala UniversitySweden