Evolutionary algorithms (EAs) are heuristic algorithms inspired by natural evolution. They are often used to obtain satisficing solutions in practice. In this paper, we investigate a largely underexplored issue: the approximation performance of EAs in terms of how close the solution obtained is to an optimal solution. We study an EA framework named simple EA with isolated population (SEIP) that can be implemented as a single- or multi-objective EA. We analyze the approximation performance of SEIP using the partial ratio, which characterizes the approximation ratio that can be guaranteed. Specifically, we analyze SEIP using a set cover problem that is NP-hard. We find that in a simple configuration, SEIP efficiently achieves an $H_n$-approximation ratio, the asymptotic lower bound, for the unbounded set cover problem. We also find that SEIP efficiently achieves an $(H_k-\frac{k-1}/{8k^9})$-approximation ratio, the currently best-achievable result, for the k-set cover problem. Moreover, for an instance class of the k-set cover problem, we disclose how SEIP, using either one-bit or bit-wise mutation, can overcome the difficulty that limits the greedy algorithm.
Evolutionary algorithms (EAs) [3] have been successfully applied to many fields and can achieve extraordinary performance in addressing some real-world hard problems, particularly NP-hard problems [16,18,17,4]. To gain an understanding of the behavior of EAs, many theoretical studies have focused on the running time required to achieve exact optimal solutions [14,33,26,2]. In practice, EAs are most commonly used to obtain satisficing solutions, yet theoretical studies of the approximation ability of EAs have only emerged recently.
He and Yao [15] first studied conditions under which the wide-gap far distance and the narrow-gap long distance problems are hard to approximate using EAs. Giel and Wegener [12] investigated a (1+1)-EA for a maximum matching problem and found that the time taken time to find exact optimal solutions is exponential but is only O(n 2⌈1/ǫ⌉ ) for (1 + ǫ)-approximate solutions, which demonstrates the value of EAs as approximation algorithms.
Subsequently, further results on the approximation ability of EAs were reported. For the (1+1)-EA, the simplest type of EA, two classes of results have been obtained. On one hand, it was found that the (1+1)-EA has an arbitrarily poor approximation ratio for the minimum vertex cover problem and thus also for the minimum set cover problem [11,28]. On the other hand, it was also found that (1+1)-EA provides a polynomial-time randomized approximation scheme for a subclass of the partition problem [32]. Furthermore, for some subclasses of the minimum vertex cover problem for which the (1+1)-EA gets stuck, a multiple restart strategy allows the EA to recover an optimal solution with high probability [28]. Another result for the (1+1)-EA is that it improves a 2-approximation algorithm to a (2 -2/n)-approximation on the minimum vertex cover problem [10]. This implies that it might sometimes be useful as a post-optimizer.
Recent advances in multi-objective (usually bi-objective) EAs have shed light on the power of EAs as approximation optimizers. For a single-objective problem, multi-objective reformulation introduces an auxiliary objective function for which a multi-objective EA is used as the optimizer. Scharnow et al. [30] first suggested that multi-objective reformulation could be superior to use of a single-objective EA. This was confirmed for various problems [24,25,11,27] by showing that while a single-objective EA could get stuck, multi-objective reformulation helps to solve the problems efficiently.
Regarding approximations, it has been shown that multi-objective EAs are effective for some NP-hard problems. Friedrich et al. [11] proved that a multi-objective EA achieves a (ln n)-approximation ratio for the minimum set cover problem, and reaches the asymptotic lower bound in polynomial time.
Neumann and Reichel [23] showed that multi-objective EAs achieve a k-approximation ratio for the minimum multicuts problem in polynomial time.
In the present study, we investigate the approximation ability of EAs by introducing a framework called simple evolutionary algorithm with isolated population (SEIP), which uses an isolation function to manage competition among solutions. By specifying the isolation function, SEIP can be implemented as a single-or multi-objective EA. Multi-objective EAs previously analyzed [22,11,23] can be viewed as special cases of SEIP in term of the solutions maintained in the population. By analyzing the SEIP framework, we obtain a general characterization of EAs that guarantee approximation quality.
We then study the minimum set cover problem (MSCP), which is an NP-hard problem [9]. We prove that for the unbounded MSCP, a simple configuration of SEIP efficiently obtains an H k -approximation ratio (where H k is the harmonic number of the cardinality of the largest set), the asymptotic lower bound [9]. For the minimum k-set cover problem, this approach efficiently yields an (H k -k-1 8k 9 )approximation ratio, the currently best-achievable quality [13]. Moreover, for a subclass of the minimum k-set cover problem, we demonstrate how SEIP, with either one-bit or bit-wise mutation, can overcome the difficulty that limits the greedy algorithm.
The remainder of the paper is organized as follows. After introducing preliminaries in Section 2, we describe SEIP in Section 3 and characterize its behavior for approximation in Section 4. We then analyze the approximation ratio achieved by SEIP for the MSCP in Section 5. In Section 6 we conclude the paper with a discussion of the advantages and limitations of the SEIP framework.
We use bold small letters such as w, x, y, z to represent vectors. We denote [m] as the set {1, 2, . . . , m} and 2 S as the power set of S, which consists of all subsets of S. We denote
In this paper, we consider minimization problems as follows.
Given an evaluation function f and a set of feasibility constraints C, find a solution x ∈ {0, 1} n that minimizes f (x) while satisfying constraints in C. A problem
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