Greed is slow on sparse graphs of oriented valued constraints

TL;DR

This paper constructs a simple, sparse VCSP with an oriented structure where greedy local search is provably slow, highlighting the limitations of greedy methods for solving certain constraint satisfaction problems.

Contribution

It introduces the first known sparse, oriented VCSP where greedy local search exhibits exponential slowdowns, demonstrating the inherent limitations of greedy approaches.

Findings

Greedy local search takes exponential time on the constructed VCSP.

The VCSP is sparse with pathwidth 2 and maximum degree 3.

Many non-greedy local search methods can find the peak efficiently.

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.