Optimal General Factor Problem and Jump System Intersection

TL;DR

This paper reformulates the optimal general factor problem as a jump system intersection, providing a simpler correctness proof for a polynomial-time algorithm and extending the results to a valuated case.

Contribution

It introduces a new jump system intersection framework for the problem and simplifies the proof of the existing algorithm's correctness.

Findings

The problem can be formulated as a jump system intersection.

The algorithm's correctness is validated through this new abstraction.

The results are extended to a valuated case.

Abstract

In the optimal general factor problem, given a graph $G=(V, E)$ and a set $B(v) \subseteq \mathbb Z$ of integers for each $v \in V$, we seek for an edge subset $F$ of maximum cardinality subject to $d_F(v) \in B(v)$ for $v \in V$, where $d_F(v)$ denotes the number of edges in $F$ incident to $v$. A recent crucial work by Dudycz and Paluch shows that this problem can be solved in polynomial time if each $B(v)$ has no gap of length more than one. While their algorithm is very simple, its correctness proof is quite complicated. In this paper, we formulate the optimal general factor problem as the jump system intersection, and reveal when the algorithm by Dudycz and Paluch can be applied to this abstract form of the problem. By using this abstraction, we give another correctness proof of the algorithm, which is simpler than the original one. We also extend our result to the valuated case.