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 $$\exp(\Omega(m/\log^3 m))$$.

## Keywords

Perfect Matchings Simple Graph Graph Class Satisfying Assignment Tutte Polynomial
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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## 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