A heuristic search based on diversity for solving combinatorial problems

Abstract

In this paper we propose a novel heuristic search for solving combinatorial optimization problems which we call Diverse Search (DS). Like beam search, this constructive approach expands only a selected subset of the solutions in each level of the search tree. However, instead of selecting the solutions with the best values, we use an efficient method to select a diverse subset, after filtering out uninteresting solutions. DS also distinguishes solutions that do not produce better offspring, and applies a local search process to them. The intuition is that the combination of these strategies allows to reach more—and more diverse—local optima, increasing the chances of finding the global optima. We test DS on several instances of the Köerkel–Ghosh (KG) and K-median benchmarks for the Simple Plant Location Problem. We compare it with a state-of-the-art heuristic for the KG benchmark and the relatively old POPSTAR solver, which also relies on the idea of maintaining a diverse set of solutions and, surprisingly, reached a comparable performance. With the use of a Path Relinking post-optimization step, DS can achieve results of the same quality that the state-of-the-art in similar CPU times. Furthermore, DS proved to be slightly better on average for large scale problems with small solution sizes, proving to be an efficient algorithm that delivers a set of good and diverse solutions.

Appendices

A Alternative distance for SPLP solutions

In this section we propose an alternative distance to compare SPLP solutions when the mean geometric error described in Sect. 4.5.2 is inconvenient, which we call per-client delta.

This distance does not require a distance between facilities $d_{\text {facs}}(\cdot ,\cdot )$. Instead, it compares the costs of connecting each client in both solutions as if they were a vector of size M:

$$\begin{aligned} d(A,B)&= \sum _{j \in J} \left| g_{Aj} - g_{Bj} \right| \, . \end{aligned}$$

where $g_{Sj}$ corresponds to the cost of the most convenient facility assignment within solution S for client j, as defined in Eq. 3.

It satisfies the triangle inequality since it corresponds to the $L^1$ distance between these two vectors.

Under the assumption that each solution S stores its $g_{Sj}$ values, the computational cost of computing this distance is $\varTheta (M)$, which may be more convenient than the mean geometric error if solutions are large.

B Implementation details

To test both DS and other search strategies we developed a SPLP and p-median solver called dc2 whose source code is available online¹.

dc2 can perform DS as described in Algorithm 1, but it also supports restarts, different selection operators and combinations of them, as well as specifying different branching factors b. These configurations allow using dc2 as a GRASP, sample greedy, beam search, or stochastic beam search method.

Additionally, dc2 has support for Whitaker’s local search (Whitaker 1983), Path Relinking, and different solution distances $d(\cdot ,\cdot )$. In order to solve big problem instances faster, this solver is also parallelized, but we only reported total CPU times in this work.