Approximation and Parameterized Runtime Analysis of Evolutionary Algorithms for the Maximum Cut Problem

Abstract

The maximum cut (MAX-CUT) problem is to find a bipartition of the vertices in a given graph such that the number of edges with ends in different sets reaches the largest. Though, several experimental investigations have shown that evolutionary algorithms (EAs) are efficient for this NP-complete problem, there is little theoretical work about EAs on the problem. In this paper, we theoretically investigate the performance of EAs on the MAX-CUT problem. We find that both the (1+1) EA and the (1+1) EA*, two simple EAs, efficiently achieve approximation solutions of (m/2)+(1/4)s(G) and (m/2)+(1/2)(√{8m+1}-1), where m and s(G) are respectively the number of edges and the number of odd degree vertices in the input graph. We also reveal that for a given integer k the (1+1) EA* finds a cut of size at least k in expected runtime O(nm+1/δ(4k)) and a cut of size at least (m/2)+k in expected runtime O(n(2)m+1/δ((64/3)k(2))), where δ is a constant mutation probability and n is the number of vertices in the input graph. Finally, we show that the (1+1) EA and the (1+1) EA* are better than some local search algorithms in one instance, and we also show that these two simple EAs may not be efficient in another instance.

Error analysis of elitist randomized search heuristics

“…To take the second best, it is expected that an RSH can get approximate optima in polynomial FHT/RT. In this way, approximate FHT/RT analyses are performed for given approximation ratios [17,18,19,20,21,22,23,24,25]. However, approximate FHT/RT analysis is not flexible enough to address some cases.…”

“…Although popular in theoretical study, FHT/RT analysis does not work well when RSHs are not anticipated to locate the global optimal solutions in polynomial FHT/RHT. For this case, a remedy is to investigate the approximate FHT/RT by taking an approximation set as the hitting destination of consecutive iterations [17,18,19,20,21,22,23,24,25].…”

Estimating Approximation Errors of Elitist Evolutionary Algorithms

“…For this case, it is much more interesting to investigate the quality of approximate solutions obtained in polynomial FHT/RT. In this way, researchers have estimated approximation ratios of approximate solutions that EAs can obtain for various NPC combinatorial optimization problems in polynomial expected FHT/RT [12,13,14,15,16,17].…”

Stochastic runtime analysis of a Cross-Entropy algorithm for traveling salesman problems

“…The first steps towards this type of analysis were made for the so-called (1+1) EA [11] on some test problems that use pseudo boolean functions as cost functions, e.g., OneMax [15], LeadingOnes [27] and BinVar [11]. Recent research addresses problems of practical importance, such as the computing a minimum spanning trees (MST) [28], matroid optimization [29], traveling salesman problem [30], the shortest path problem [23], the maximum satisfiability problem [31] and the max-cut problem [32].…”