The Parameterized Complexity of Counting Problems

Abstract

We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W[t], for t ≥ 1, that corresponds to Downey and Fellows's W-hierarchy [13] and show that a few central W-completeness results for decision problems translate to #W-completeness results for the corresponding counting problems. Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, is #W[1]-complete. This makes it highly unlikely that any of these problems is fixed-parameter tractable, even though their decision versions are fixed-parameter tractable. More explicitly, our result shows that most likely there is no f (k) • n c-algorithm for counting cycles or paths of length k in a graph of size n for any computable function f : N → N and constant c, even though there is a 2 O(k) • n 2.376 algorithm for finding a cycle or path of length k [2].

Exponential Time Complexity of the Permanent and the Tutte Polynomial

“…Flum and Grohe [FG04] introduce the class #W[1] of parameterized counting problems. This class is characterized by complete problems such as computing the number of cliques of size k or computing the number of simple paths of length k in an n-vertex graph.…”

“…This class is characterized by complete problems such as computing the number of cliques of size k or computing the number of simple paths of length k in an n-vertex graph. Implicitly, Flum and Grohe [FG04] show that these problems are not fixed-parameter tractable under #ETH. The latter is only an implication and, as in the case of decision problems, we do not know whether the two claims are equivalent.…”

The parameterized complexity of probability amplification

“…There are few connections between this paper and previous work on parameterized Counting Complexity. Parameterized Counting Complexity is not yet a mature theory, there are few works on the topic [6,8], and no structural theorems like the one of Toda or the one of Stockmeyer. It is important to remark that one of the main open problems in parameterized Counting Complexity is the proof of a parameterized analogue of Valiant's Theorem on the complexity of counting matchings, however Flum and Grohe in [6] were able to prove that the counting of cycles and paths is hard from the parameterized point of view although the corresponding decision problems are fix parameter tractable.…”

On Counting 3-D Matchings of Size k

“…The p-3D MATCHING problem is a generalization of the problem of counting parameterized matchings in bipartite graphs, and the latter has been conjectured to be parameterized infeasible (i.e., W [1]-hard) [6]. Therefore, p-3D MATCHING has no known fixed-parameter tractable algorithms, and is probably W [1]-hard.…”