Greed is slow on sparse graphs of oriented valued constraints

Abstract: Greedy local search is especially popular for solving valued constraint satisfaction problems (VCSPs). Since any method will be slow for some VCSPs, we ask: what is the simplest VCSP on which greedy local search is slow? We construct a VCSP on 6n Boolean variables for which greedy local search takes 7(2^n - 1) steps to find the unique peak. Our VCSP is simple in two ways. First, it is very sparse: its constraint graph has pathwidth 2 and maximum degree 3. This is the simplest VCSP on which some local search could be slow. Second, it is "oriented" - there is an ordering on the variables such that later variables are conditionally-independent of earlier ones. Being oriented allows many non-greedy local search methods to find the unique peak in a quadratic number of steps. Thus, we conclude that - among local search methods - greed is particularly slow.