Randomized greedy multi-start algorithm for the minimum common integer partition problem

In this paper, we propose a randomized greedy multi-start algorithm for the minimum common integer partition problem. Given k multisets S1,…,Sk of positive integers (Si={si1,…,sij,…,simi}), the goal is to find the common integer partition T with minimal cardinality, i.e., a unique and reduced multiset T that, for each Si, it can be partitioned into mi multisets Tj so that the elements in Tj sum to sij. This mathematical problem is reported to appear in computational molecular biology, when assigning orthologs on a genome scale or assembling DNA fingerprints in particular. Our proposed metaheuristic approach constitutes the construction of multiple solutions by a new greedy method that embeds a diversification agent to allow diverse and promising solutions to be reached. Furthermore, we formulate an integer programming model for this problem and show that the CPLEX solver can only solve small instances of the problem. However, computational results for problem instances involving up to 1000 multisets (each one with up to 1000 elements) show that our innovative metaheuristic produces very good feasible solutions in reasonable computing times, arising as a very attractive alternative to the existing approaches.