Focused Jump-and-Repair Constraint Handling for Fixed-Parameter Tractable Graph Problems Closed Under Induced Subgraphs

TL;DR

This paper introduces a focused jump-and-repair operator for the (1+1) EA to efficiently solve fixed-parameter tractable graph problems, achieving probabilistic guarantees and exponential improvements over previous methods.

Contribution

It develops a novel jump-and-repair approach enabling the (1+1) EA to efficiently handle fixed-parameter graph problems with probabilistic performance guarantees.

Findings

Proves the (1+1) EA finds a k-vertex cover in expected O(2^k n^2 log n) time.

Establishes the first parameterized results for evolutionary algorithms on FeedbackVertexSet and OddCycleTransversal.

Demonstrates exponential improvements over existing bounds for VertexCover.

Abstract

Repair operators are often used for constraint handling in constrained combinatorial optimization. We investigate the (1+1)~EA equipped with a tailored jump-and-repair operation that can be used to probabilistically repair infeasible offspring in graph problems. Instead of evolving candidate solutions to the entire graph, we expand the genotype to allow the (1+1)~EA to develop in parallel a feasible solution together with a growing subset of the instance (an induced subgraph). With this approach, we prove that the EA is able to probabilistically simulate an iterative compression process used in classical fixed-parameter algorithmics to obtain a randomized FPT performance guarantee on three NP-hard graph problems. For $k$-VertexCover, we prove that the (1+1) EA using focused jump-and-repair can find a $k$-vertex cover (if one exists) in $O(2^k n^2\log n)$ iterations in expectation. This leads to an exponential (in $k$) improvement over the best-known parameterized bound for evolutionary algorithms on VertexCover. For the $k$-FeedbackVertexSet problem in tournaments, we prove that the EA finds a feasible feedback set in $O(2^kk!n^2\log n)$ iterations in expectation, and for OddCycleTransversal, we prove the optimization time for the EA is $O(3^k k m n^2\log n)$. For the latter two problems, this constitutes the first parameterized result for any evolutionary algorithm. We discuss how to generalize the framework to other parameterized graph problems closed under induced subgraphs and report experimental results that illustrate the behavior of the algorithm on a concrete instance class.