Iterated local search with Trellis-neighborhood for the partial Latin square extension problem

• Published in: Journal of heuristics Vol. 22; no. 5; pp. 727 - 757
• Main Author: Haraguchi, Kazuya
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2016
• Subjects: Artificial Intelligence; Calculus of Variations and Optimal Control; Optimization; Management Science; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Partial Latin square extension problem; Local search; Maximum independent set problem; Metaheuristics
• ISSN: 1381-1231; 1572-9397
• DOI: 10.1007/s10732-016-9317-6

A partial Latin square ( PLS ) is a partial assignment of n symbols to an n × n grid such that, in each row and in each column, each symbol appears at most once. The PLS extension problem is an NP-hard problem that asks for a largest extension of a given PLS. We consider the local search such that the neighborhood is defined by ( p ,  q ) -swap , i.e., the operation of dropping exactly p symbols and then assigning symbols to at most q empty cells. As a fundamental result, we provide an efficient ( p , ∞ ) -neighborhood search algorithm that finds an improved solution or concludes that no such solution exists for p ∈ { 1 , 2 , 3 } . The running time of the algorithm is O ( n p + 1 ) . We then propose a novel swap operation, Trellis-swap, which is a generalization of ( p ,  q )-swap with p ≤ 2 . The proposed Trellis-neighborhood search algorithm runs in O ( n 3.5 ) time. The iterated local search (ILS) algorithm with Trellis-neighborhood is more likely to deliver a high-quality solution than not only ILSs with ( p , ∞ ) -neighborhood but also state-of-the-art optimization solvers such as IBM ILOG CPLEX and LocalSolver .