Flow-Partitionable Signed Graphs

TL;DR

This paper introduces flow-partitionable signed graphs, a class where correlation clustering can be solved efficiently using linear programming, characterized by forbidden minors.

Contribution

It defines flow-partitionable signed graphs, characterizes them via forbidden minors, and connects the problem to ideal clutter theory.

Findings

Flow-partitionable graphs allow exact LP relaxation for correlation clustering.

Two infinite classes of forbidden minors identified for specific graph structures.

A new forbidden minor proposed for the general case, linking to open problems in ideal clutters.

Abstract

The NP-hard problem of correlation clustering is to partition a signed graph such that the number of conflicts between the partition and the signature of the graph is minimized. This paper studies graph signatures that allow the optimal partition to be found efficiently. We define the class of flow-partitionable signed graphs, which have the property that the standard linear programming relaxation based on so-called cycle inequalities is tight. In other words, flow-partitionable signed graphs satisfy an exact max-multiflow-min-multicut relation in the associated instances of minimum multicut. In this work we propose to characterize flow-partitionable signed graphs in terms of forbidden minors. Our initial results include two infinite classes of forbidden minors, which are sufficient if the positive subgraph is a circuit or a tree. For the general case we present another forbidden minor and point out a connection to open problems in the theory of ideal clutters.