Binary constraints on one additional variable can create exponential ascents

TL;DR

This paper constructs a simpler VCSP example demonstrating exponential ascent lengths in local search landscapes, highlighting how a central variable can intertwine sublandscapes and affect complexity.

Contribution

It introduces a new, simpler star-of-gadgets construction for VCSPs with exponential ascents, reducing complexity parameters compared to prior examples.

Findings

Constructed a VCSP with exponential ascent length of 10*2^n - 9.

Reduced treedepth complexity from (log n) to 3.

Lowered feedback vertex set number from (n) to 1.

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 in the landscape. This hill-climbing is sometimes surprisingly efficient, but not always. Since fitness landscapes can be succinctly represented by valued constraint satisfaction problems (VCSPs), it is natural to 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 long ascents were built up as a chain of gadgets of constraints. Here we give a simpler star of gadgets construction by gluing 2n triangles of constraints at a common centre variable. We obtain a binary VCSP on 4n + 1 Boolean variables with an exponential ascent of length 10*2^n - 9. The variable at the centre of our construction intertwines two sublandscapes with only linear ascents into one with exponential ascents. The VCSP that we construct is significantly simpler than prior constructions in terms of treedepth (reducing \Omega(log n) to 3) and feedback vertex set number (reducing \Omega(n) to 1). We discuss the consequences of this simplicity for the parameterized complexity of local search.