A primogenitary linked quad tree approach for solution storage and retrieval in heuristic binary optimization

Abstract

A data structure, called the primogenitary linked quad tree (PLQT), is used to store and retrieve solutions in heuristic solution procedures for binary optimization problems. Two ways are proposed to use integer vectors to represent solutions represented by binary vectors. One way is to encode binary vectors into integer vectors in a much lower dimension and the other is to use the sorted indices of binary variables with values equal to 0 or equal to 1. The integer vectors are used as composite keys to store and retrieve solutions in the PLQT. An algorithm processing trial solutions for insertion into or retrieval from the PLQT is developed. Examples are provided to demonstrate the way the algorithm works. Another algorithm traversing the PLQT is also developed. Computational results show that the PLQT approach takes only a very tiny portion of the CPU time taken by a linear list approach for the same purpose for any reasonable application. The CPU time taken by the PLQT managing trial solutions is negligible as compared to that taken by a heuristic procedure for any reasonably hard to solve binary optimization problem, as shown in a tabu search heuristic procedure for the capacitated facility location problem. Compared to the hashing approach, the PLQT approach takes the same or less amount of CPU time but much less memory space while completely eliminating collision.