When Gravity Fails: Local Search Topology

TL;DR

This paper analyzes the structure of plateaus in local search for SAT problems, revealing features that influence algorithm performance and suggesting directions for improved local search strategies.

Contribution

It characterizes plateau features in SAT problems, examines their impact on local search, and discusses implications for designing more effective algorithms.

Findings

Local minima are usually small but can be very large.

Escaping local minima without many clause violations is possible but costly for large minima.

Benches are larger plateau regions with few exit points, affecting search efficiency.

Abstract

Local search algorithms for combinatorial search problems frequently encounter a sequence of states in which it is impossible to improve the value of the objective function; moves through these regions, called plateau moves, dominate the time spent in local search. We analyze and characterize plateaus for three different classes of randomly generated Boolean Satisfiability problems. We identify several interesting features of plateaus that impact the performance of local search algorithms. We show that local minima tend to be small but occasionally may be very large. We also show that local minima can be escaped without unsatisfying a large number of clauses, but that systematically searching for an escape route may be computationally expensive if the local minimum is large. We show that plateaus with exits, called benches, tend to be much larger than minima, and that some benches have very few exit states which local search can use to escape. We show that the solutions (i.e., global minima) of randomly generated problem instances form clusters, which behave similarly to local minima. We revisit several enhancements of local search algorithms and explain their performance in light of our results. Finally we discuss strategies for creating the next generation of local search algorithms.