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

TL;DR

This work addresses the weighted general factor problem ($WGFP$) on graphs with degree constraints that have gaps of length at most $1$, extending tractability beyond constraints reducible to weighted matchings. The authors introduce a recursive, parity-splitting algorithm for $WGFP(\\mathscr{G}\\cup\\mathscr{T})$ that leverages local-to-global optimality via basic augmenting subgraphs, enabling a strongly polynomial-time solution for interval, parity-interval, type-1, and type-2 constraints on subcubic graphs, with a running time bound of $O(n^6)$. Two central results underpin the algorithm: (i) a local-global optimality theorem guaranteeing that a globally optimal factor can be obtained from a near-optimal local factor, and (ii) a robust existence lemma for basic factors in key instances, proven through extensive case analysis of five basic-factor types. The approach also yields a complexity dichotomy for the WGFP on subcubic graphs and connects to Δ-matroids and edge CSPs, highlighting practical implications for network design problems such as terminal backup.

Abstract

General factors are a generalization of matchings. Given a graph $G$ with a set $π(v)$ 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 $\text{deg}_F(x) \in π(v)$ for every $v$ of $G$. When all degree constraints are symmetric $Δ$-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.

Figures (2)

• Figure 1: An example that violates Theorem \ref{['Thm:local-global-optimal']} when $F$ is optimal in $\Omega_u^1$ instead of $\Omega_u^0$
• Figure 13: A matching gadget realizing $D=\{p, p+1, \ldots, p+r\}$ of arity $n$

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2: 2-vertex-connectivity
• Theorem 2.3: Menger's Theorem
• Definition 2.4: Bridge and 2-edge-connectivity
• Theorem 2.5
• Lemma 2.6
• Lemma 2.7