In this paper we are interested in algorithms for finding 2-factors that cover certain prescribed edge-cuts in bridgeless cubic graphs. Since a Hamilton cycle is a 2-factor covering all edge-cuts, imposing the constraint of covering those edge-cuts makes the obtained 2-factor closer to a Hamilton cycle. We present an algorithm for finding a minimum-weight 2-factor covering all the 3edge cuts in weighted bridgeless cubic graphs, together with a polyhedral description of such 2-factors and that of perfect matchings intersecting all the 3-edge cuts in exactly one edge. We further give an algorithm for finding a 2-factor covering all the 3-and 4-edge cuts in bridgeless cubic graphs. Both of these algorithms run in O(n 3) time, where n is the number of vertices. As an application of the latter algorithm, we design a 6/5-approximation algorithm for finding a minimum 2-edge-connected spanning subgraph in 3-edge-connected cubic graphs, which improves upon the previous best ratio of 5/4. The algorithm begins with finding a 2-factor covering all 3-and 4-edge cuts, which is the bottleneck in terms of complexity, and thus it has running time O(n 3). We then improve this time complexity to O(n 2 log 4 n) by relaxing the condition of the initial 2-factor and elaborating on the subsequent processes. Key words. bridgeless cubic graphs, minimum-weight 2-factor covering 3-edge cuts, polyhedral description of 2-factors covering 3-edge cuts, 2-factor covering 3-and 4-edge cuts, minimum 2-edgeconnected spanning subgraphs

In this paper, we introduce the concept of b-branchings in digraphs, which is a generalization of branchings serving as a counterpart of b-matchings. Here b is a positive integer vector on the vertex set of a digraph, and a b-branching is defined as a common independent set of two matroids defined by b: an arc set is a b-branching if it has at most b(v) arcs sharing the terminal vertex v, and it is an independent set of a certain sparsity matroid defined by b. We demonstrate that b-branchings yield an appropriate generalization of branchings by extending several classical results on branchings. We first present a multi-phase greedy algorithm for finding a maximum-weight b-branching. We then prove a packing theorem extending Edmonds' disjoint branchings theorem, and provide a strongly polynomial algorithm for finding optimal disjoint b-branchings. As a consequence of the packing theorem, we prove the integer decomposition property of the b-branching polytope. Finally, we deal with a further generalization in which a matroid constraint is imposed on the b(v) arcs sharing the terminal vertex v. IntroductionSince the pioneering work of Edmonds [12,14], the importance of matroid intersection has been well appreciated. A special class of matroid intersection is branchings (or arborescences) in digraphs. Branchings have several good properties which do not hold for general matroid intersection. The objective of this paper is to propose a class of matroid intersection which generalizes branchings and inherits those good properties of branchings.One of the good properties of branchings is that a maximum-weight branching can be found by a simple combinatorial algorithm [4,6,11,23]. This algorithm is much simpler than general weighted matroid intersection algorithms, and is referred to as a "multi-phase greedy algorithm" in the textbook by Kleinberg and Tardos [34].

The matching forest problem in mixed graphs is a common generalization of the matching problem in undirected graphs and the branching problem in directed graphs. Giles presented an O(n 2 m)-time algorithm for finding a maximum-weight matching forest, where n is the number of vertices and m is that of edges, and a linear system describing the matching forest polytope. Later, Schrijver proved total dual integrality of the linear system.In the present paper, we reveal another nice property of matching forests: the degree sequences of the matching forests in any mixed graph form a delta-matroid and the weighted matching forests induce a valuated delta-matroid. We remark that the delta-matroid is not necessarily even, and the valuated delta-matroid induced by weighted matching forests slightly generalizes the well-known notion of Dress and Wenzel's valuated delta-matroids. By focusing on the delta-matroid structure and reviewing Giles' algorithm, we design a simpler O(n 2 m)-time algorithm for the weighted matching forest problem. We also present a faster O(n 3 )-time algorithm by using Gabow's method for the weighted matching problem.

For a digraph D = (V, A) and a partition {S, T } of V , an arc set B ⊆ A is called an S-T bibranching if each vertex in T is reachable from S and each vertex in S reaches T in the subgraph (V, B). Bibranchings commonly generalize bipartite edge covers and arborescences. A totally dual integral linear system determining the S-T bibranching polytope is provided by Schrijver, and the shortest S-T bibranching problem, whose objective is to find an S-T bibranching of minimum total arc weight, can be solved in polynomial time by the ellipsoid method or a faster combinatorial algorithm due to Keijsper and Pendavingh. The valuated matroid intersection problem, introduced by Murota, is a weighted generalization of the independent matching problem, including the independent assignment problem and the weighted matroid intersection problem. The valuated matroid intersection problem can be solved efficiently with polynomially many value oracles by extending classical combinatorial algorithms for the weighted matroid intersection problem. In this paper, we show that the shortest S-T bibranching problem is polynomially reducible to the valuated matroid intersection problem. This reduction suggests one answer to why the shortest S-T bibranching problem is tractable, and implies new combinatorial algorithms for the shortest S-T bibranching problem based on the valuated matroid intersection algorithm, where a value oracle corresponds to computing a minimum-weight arborescence.

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