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We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a long-standing conjecture of Bang-Jensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments.

In the Discrepancy problem, we are given M sets {S 1 , . . . , S M } on N elements. Our goal is to find an assignment χ of {−1, +1} values to elements, so as to minimize the maximum discrepancy max j | i∈Sj χ(i)|. Recently, Bansal gave an efficient algorithm for achieving O( √ N ) discrepancy for any set system where M = O(N ) [Ban10], giving a constructive version of Spencer's proof that the discrepancy of any set system is at mostWe show that from the perspective of computational efficiency, these results are tight for general set systems where M = O(N ). Specifically, we show that it is NP-hard to distinguish between such set systems with discrepancy zero and those with discrepancy Ω( √ N ). This means that even if the optimal solution has discrepancy zero, we cannot hope to efficiently find a coloring with discrepancy o( √ N ). We also consider the hardness of the Discrepancy problem on sets with bounded shatter function, and show that the upper bounds due to Matoušek [Mat95] are tight for these sets systems as well.The hardness results in both settings are obtained from a common framework: we compose a family of high discrepancy set systems with set systems for which it is NP-hard to distinguish instances with discrepancy zero from instances in which a large number of the sets (i.e. constant fraction of the sets) have non-zero discrepancy. Our composition amplifies this zero versus non-zero gap.

In the traveling salesman path problem, we are given a set of cities, traveling costs between city pairs and fixed source and destination cities. The objective is to find a minimum cost path from the source to destination visiting all cities exactly once. In this paper, we study polyhedral and combinatorial properties of a variant we call the traveling salesman walk problem, in which the objective is to find a minimum cost walk from the source to destination visiting all cities at least once. We first characterize traveling salesman walk perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities. We show these graphs have a description by way of forbidden minors and also characterize them constructively. We also address the asymmetric traveling salesman path problem (ATSPP) and give a factor O( √ n)-approximation algorithm for this problem. Mathematics Subject Classification (2000)68Q25 · 68R10 · 90C05 · 90C27