Algorithms for Weighted Matching Generalizations II: f-factors and the Special Case of Shortest Paths

Harold N Gabow,Piotr Sankowski

doi:10.1137/16m1106225

Abstract

For an undirected graph or multigraph $G=(V,E)$ and a function $f:V\to \mathbb{Z_+}$, an $f$-factor is a subgraph whose degree function is $f$. For integral edge weights of maximum magnitude $W$ our algorithm finds a maximum weight $f$-factor in time $\tilde{O}(Wf(V)^{\omega})$, where $f(V)=\sum_{v\in V} f(v)$ and $\omega$ is the exponent of matrix multiplication. The algorithm is randomized and has two versions. For worst-case time the algorithm is correct with high probability. For expected time the algorithm is Las Vegas. The algorithm is based on a detailed analysis of the structure of the optimum blossoms. A special case gives a representation for single-source shortest-paths in conservative undirected graphs, generalizing the standard shortest-path tree to a “tree of cycles”. The representation can be constructed by a randomized algorithm with the same time bound as above, or deterministically by an algorithm for maximum weight matching, achieving time $O(n(m + n \log n))$ or $O(\sqrt{n }\ m \log (nW))$.

Highlights

This paper (Part II) studies the problem of finding f -factors in general graphs. f -factors are important combinatorial notions as they generalize both nonbipartite matching and network flow, as well as shortest paths and min-cost flow
The second part of this paper presents an analogue of the shortest-paths tree, for conservative undirected graphs
[10] Gabow (1990) [12] this paper root’s subgraph may be a tree or a cycle. (In the absence of negative edges there are no cycle nodes and the root’s subgraph is the usual shortest-paths tree.) We prove the existence of this shortest-paths structure and give an algebraic algorithm to construct it in time O(W nω)

Summary

Introduction

This paper (Part II) studies the problem of finding f -factors in general graphs. f -factors are important combinatorial notions as they generalize both nonbipartite matching and network flow, as well as shortest paths and min-cost flow. The edges with maximum ζ-value span a set of vertices B where all the factors Fv, v ∈ B, are identical outside of γ(B) (Lemma 3.8 gives the stronger statement of this property; recall the functions γ, δ from Part I, section 2).

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Conclusion