We study graph ordering problems with a min-max objective. A classical problem of this type is cutwidth, where given a graph we want to order its vertices such that the number of edges crossing any point is minimized. We give a$\log^{1+o(1)}(n)$approximation for the problem, substantially improving upon the previous poly-logarithmic guarantees based on the standard recursive balanced partitioning approach of Leighton and Rao (FOCS'88). Our key idea is a new metric decomposition procedure that is suitable for handling min-max objectives, which could be of independent interest. We also use this to show other results, including an improved$\log^{1+o(1)}(n)$approximation for computing the pathwidth of a graph.

We consider the Max--Section problem, where we are given an undirected graph G=(V,E)equipped with non-negative edge weights w: E → R_+ and the goal is to find a partition of V into three equisized parts while maximizing the total weight of edges crossing between different parts. Max-3-Section is closely related to other well-studied graph partitioning problems, e.g., Max-Cut, Max-3-Cut, and Max-Bisection. We present a polynomial time algorithm achieving an approximation of 0.795, that improves upon the previous best known approximation of 0.673. The requirement of multiple parts that have equal sizes renders Max-3-Section much harder to cope with compared to, e.g., Max-Bisection. We show a new algorithm that combines the existing approach of Lassere hierarchy along with a random cut strategy that suffices to give our result.

Correlation Clustering is an elegant model where given a graph with edges labeled + or -, the goal is to produce a clustering that agrees the most with the labels: + edges should reside within clusters and - edges should cross between clusters. In this work we study the MaxCorr objective, aiming to find a clustering that maximizes the difference between edges classified correctly and incorrectly. We focus on the case of bipartite graphs and present an improved approximation of 0.254, improving upon the known approximation of 0.219 given by Charikar and Wirth [FOCS`2004] and going beyond the 0.2296 barrier imposed by their technique. Our algorithm is inspired by Krivine’s method for bounding Grothendieck’s constant, and we extend this method to allow for more than two clusters in the output. Moreover, our algorithm leads to two additional results: (1) the first known approximation guarantees for MaxCorr where the output is constrained to have a bounded number of clusters; and (2) a natural extension of Grothendieck’s inequality to large domains.