A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees

TL;DR

This paper introduces the first strongly polynomial-time algorithm for a specific class of weighted general factor problems with real weights, expanding the scope of efficiently solvable graph problems beyond weighted matchings.

Contribution

It provides a novel strongly polynomial-time algorithm for weighted general factors with feasible degrees, not reducible to weighted matchings, for the first time.

Findings

Algorithm runs in strongly polynomial time

Handles real-valued edge weights

Applicable to problems beyond weighted matchings

Abstract

General factors are a generalization of matchings. Given a graph $G$ with a set $\pi(v)$ of feasible degrees, called a degree constraint, for each vertex $v$ of $G$, the general factor problem is to find a (spanning) subgraph $F$ of $G$ such that $\text{deg}_F(x) \in \pi(v)$ for every $v$ of $G$. When all degree constraints are symmetric $\Delta$-matroids, the problem is solvable in polynomial time. The weighted general factor problem is to find a general factor of the maximum total weight in an edge-weighted graph. In this paper, we present the first strongly polynomial-time algorithm for a type of weighted general factor problems with real-valued edge weights that is provably not reducible to the weighted matching problem by gadget constructions.