Efficient algorithms for clique problems

Abstract

The k-clique problem is a cornerstone of NP-completeness and parametrized complexity. When k is a fixed constant, the asymptotically fastest known algorithm for finding a k-clique in an n-node graph runs in O ( n 0.792 k ) time (given by Nešetřil and Poljak). However, this algorithm is infamously inapplicable, as it relies on Coppersmith and Winograd's fast matrix multiplication. We present good combinatorial algorithms for solving k-clique problems. These algorithms do not require large constants in their runtime, they can be readily implemented in any reasonable random access model, and are very space-efficient compared to their algebraic counterparts. Our results are the following: • We give an algorithm for k-clique that runs in O ( n k / ( ε log n ) k − 1 ) time and O ( n ε ) space, for all ε > 0 , on graphs with n nodes. This is the first algorithm to take o ( n k ) time and O ( n c ) space for c independent of k. • Let k be even. Define a k-semiclique to be a k-node graph G that can be divided into two disjoint subgraphs U = { u 1 , … , u k / 2 } and V = { v 1 , … , v k / 2 } such that U and V are cliques, and for all i ⩽ j , the graph G contains the edge { u i , v j } . We give an O ˜ ( k 2 k n k / 2 + 1 ) time algorithm for determining if a graph has a k-semiclique. This yields an approximation algorithm for k-clique, in the following sense: if a given graph contains a k-clique, then our algorithm returns a subgraph with at least 3/4 of the edges in a k-clique.