Vertex cover number of valued constraints is a structural parameter for efficient local search

Abstract

Many local search methods for problems in artificial intelligence can be viewed as an uphill climb on a corresponding discrete fitness landscapes. Finding even local peaks in these fitness landscapes is computationally intractable in theory, but often works in practice. So what features of fitness landscapes allow for efficient local search? Since any fitness landscapes can be represented by a (hyper)graph of valued constraints, I re-frame this as a question of parameterized complexity: what structural parameter of a valued constraint graph guarantees that a strict local search will find a local peak in the corresponding fitness landscape efficiently? Given a valued constrain graph of vertex cover number k, I prove that greedy local search will find the a local fitness peak in at most $2^{2k}\cdot(n - k + 1)$ steps and random uphill local search will find a local fitness peak in an expected number of at most $2^k\cdot n(n - k)$ steps. I also show that these results are asymptotically good for strict local search because there are valued constraint graphs of vertex cover number $k$ where every ascent from some initial assignment has a length of $\frac{9}{128} \cdot 2^k \cdot (n - k)$ or greater. This suggests vertex cover number as a good structural parameter for the complexity of local search.