A Space–Time Tradeoff for Permutation Problems

Many combinatorial problems—such as the traveling salesman, feedback arcset, cutwidth, and treewidth problem—can be formulated as finding a feasible permutation of n elements. Typically, such problems can be solved by dynamic programming in time and space O*(2ⁿ), by divide and conquer in time O*(4ⁿ) and polynomial space, or by a combination of the two in time O*(4ⁿ2^−s) and space O*(2^s) for s = n, n/2, n/4, …. Here, we show that one can improve the tradeoff to time O*(Tⁿ) and space O*(Sⁿ) with TS < 4 at any . The idea is to find a small family of “thin” partial orders on the n elements such that every linear order is an extension of one member of the family. Our construction is optimal within a natural class of partial order families.