On uncrossing games for skew-supermodular functions

TL;DR

This paper develops an improved polynomial-time strategy for a two-player uncrossing game related to skew-supermodular functions, leading to efficient solutions for associated linear programs and insights into laminar optimality.

Contribution

It extends previous work by providing a strongly polynomial uncrossing procedure and strategy for skew-supermodular functions, enhancing understanding of LP dual solutions.

Findings

Red has a polynomial-time winning strategy.

The uncrossing procedure is strongly polynomial.

Implications for laminar solution optimality.

Abstract

In this note, we consider the uncrossing game for a skew-supermodular function $f$, which is a two-player game with players, Red and Blue, and abstracts the uncrossing procedure in the cut-covering linear program associated with $f$. Extending the earlier results by Karzanov for $\{0,1\}$-valued skew-supermodular functions, we present an improved polynomial time strategy for Red to win, and give a strongly polynomial time uncrossing procedure for dual solutions of the cut-covering LP as its consequence. We also mention its implication on the optimality of laminar solutions.