A Multilevel Tabu Search with Backtracking for Exploring Weak Schur Numbers
In the field of Ramsey theory, the weak Schur number WS(k) is the largest integer n for which their exists a partition into k subsets of the integers [1,n] such that there is no x < y < z all in the same subset with x + y = z. Although studied since 1941, only the weak Schur numbers WS(1) through WS(4) are precisely known, for k ≥ 5 the WS(k) are only bracketed within rather loose bounds. We tackle this problem with a tabu search scheme, enhanced by a multilevel and backtracking mechanism. While heuristic approaches cannot definitely settle the value of weak Schur numbers, they can improve the lower bounds by finding suitable partitions, which in turn can provide ideas on the structure of the problem. In particular we exhibit a suitable 6-partition of [1,574] obtained by tabu search, improving on the current best lower bound for WS(6).
Keywordstabu search weak Schur numbers optimization
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