On the Average-case Complexity of Parameterized Clique

TL;DR

This paper investigates the average-case complexity of the parameterized k-Clique problem, introducing two analogs of efficient algorithms and analyzing their applicability on random graphs and specific distributions.

Contribution

It defines natural parameterized average-case algorithms and demonstrates their feasibility on Erdős-Rényi graphs, while also identifying limitations for certain distributions.

Findings

Both analogs are feasible for Erdős-Rényi graphs of any density.

The problem likely does not admit these analogs for some specific distributions.

Advances understanding of average-case complexity in parameterized problems.

Abstract

The k-Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most well-known NP-complete and W[1]-complete problems. Moreover, its average-case complexity analysis has created a long thread of research already since the 1970s. Here, we continue this line of research by studying the dependence of the average-case complexity of the k-Clique problem on the parameter k. To this end, we define two natural parameterized analogs of efficient average-case algorithms. We then show that k-Clique admits both analogues for Erd\H{o}s-R\'{e}nyi random graphs of arbitrary density. We also show that k-Clique is unlikely to admit neither of these analogs for some specific computable input distribution.