Binary constraints on one additional variable can create exponential ascents for local search

Abstract

Local search in combinatorial optimisation can be viewed as an uphill climb on a corresponding fitness landscape, where the assignments visited by a strict local search follow an ascent. This hill-climbing is sometimes surprisingly efficient, but not always. Since fitness landscapes can be succinctly represented by Boolean valued constraint satisfaction problems (VCSPs), we ask: what properties of VCSPs ensure that all ascents are polynomial? Or alternatively, what are the "simplest" VCSPs with exponential ascents? Prior examples of VCSPs with exponential ascents were built up as a chain of gadgets of constraints. Here we investigate what happens for simpler starlike graphs. To establish lower bounds on the \emph{longest} ascent, we construct: (1) a star of binary constraints with a quadratic longest ascent, (2) a binary VCSP of treedepth 3 on 4n + 1 Boolean variables (made by gluing 2n triangles of constraints at a common centre variable) with an exponential longest ascent of length $10\cdot2^n - 9$, and (3) starlike VCSP of logarithmic vertex cover number with an exponential longest ascent. For steepest ascent, we prove (4) a tight bound of 2(n - 1) for VCSPs with star constraint graphs, and construct (5) a starlike tree VCSP with treedepth 3 with a quadratic steepest ascent. Finally, we prove upper bounds on the shortest ascent from any initial assignment: (6) linear for VCSPs of treedepth 3; and (7) $O(2^k (n - k))$ for VCSPs of vertex cover number k. Together (2) and (6) establish treedepth 3, and (3) and (7) establish logarithmic vertex cover number, as the first structural graph parameters for which the longest and shortest ascents can be exponentially separated. We discuss the consequences of our results for the parameterized complexity of local search.