Learning Lines with Ordinal Constraints

04/27/2020

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by   Diego Ihara Centurion, et al.

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We study the problem of finding a mapping f from a set of points into the real line, under ordinal triple constraints. An ordinal constraint for a triple of points (u,v,w) asserts that |f(u)-f(v)|<|f(u)-f(w)|. We present an approximation algorithm for the dense case of this problem. Given an instance that admits a solution that satisfies (1-ε)-fraction of all constraints, our algorithm computes a solution that satisfies (1-O(ε^1/8))-fraction of all constraints, in time O(n^7) + (1/ε)^O(1/ε^1/8) n.