Degree-constrained Subgraph Reconfiguration is in P

The degree-constrained subgraph problem asks for a subgraph of a given graph such that the degree of each vertex is within some specified bounds. We study the following reconfiguration variant of this problem: Given two solutions to a degree-constrained subgraph instance, can we transform one solution into the other by adding and removing individual edges, such that each intermediate subgraph satisfies the degree constraints and contains at least a certain minimum number of edges? This problem is a generalization of the matching reconfiguration problem, which is known to be in P. We show that even in the more general setting the reconfiguration problem is in P.Comment: Full version of the paper published at Mathematical Foundations of Computer Science (MFCS) 201

Stabilization of Capacitated Matching Games

An edge-weighted, vertex-capacitated graph G is called stable if the value of a maximum-weight capacity-matching equals the value of a maximum-weight fractional capacity-matching. Stable graphs play a key role in characterizing the existence of stable solutions for popular combinatorial games that involve the structure of matchings in graphs, such as network bargaining games and cooperative matching games. The vertex-stabilizer problem asks to compute a minimum number of players to block (i.e., vertices of G to remove) in order to ensure stability for such games. The problem has been shown to be solvable in polynomial-time, for unit-capacity graphs. This stays true also if we impose the restriction that the set of players to block must not intersect with a given specified maximum matching of G. In this work, we investigate these algorithmic problems in the more general setting of arbitrary capacities. We show that the vertex-stabilizer problem with the additional restriction of avoiding a given maximum matching remains polynomial-time solvable. Differently, without this restriction, the vertex-stabilizer problem becomes NP-hard and even hard to approximate, in contrast to the unit-capacity case. Finally, in unit-capacity graphs there is an equivalence between the stability of a graph, existence of a stable solution for network bargaining games, and existence of a stable solution for cooperative matching games. We show that this equivalence does not extend to the capacitated case.Comment: 14 pages, 3 figure

Submodular Maximization Meets Streaming: Matchings, Matroids, and More

We study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order, where the quantity to be maximized is given by a monotone submodular function on subsets of edges. This problem, which we call maximum submodular-function matching (MSM), is a natural generalization of maximum weight matching (MWM), which is in turn a generalization of maximum cardinality matching (MCM). We give two incomparable algorithms for this problem with space usage falling in the semi-streaming range---they store only $O(n)$ edges, using $O(n\log n)$ working memory---that achieve approximation ratios of $7.75$ in a single pass and $(3+\epsilon)$ in $O(\epsilon^{-3})$ passes respectively. The operations of these algorithms mimic those of Zelke's and McGregor's respective algorithms for MWM; the novelty lies in the analysis for the MSM setting. In fact we identify a general framework for MWM algorithms that allows this kind of adaptation to the broader setting of MSM. In the sequel, we give generalizations of these results where the maximization is over "independent sets" in a very general sense. This generalization captures hypermatchings in hypergraphs as well as independence in the intersection of multiple matroids.Comment: 18 page

A study on supereulerian digraphs and spanning trails in digraphs

A strong digraph D is eulerian if for any v ∈ V (D), d+D (v) = d−D (v). A digraph D is supereulerian if D contains a spanning eulerian subdigraph, or equivalently, a spanning closed directed trail. A digraph D is trailable if D has a spanning directed trail. This dissertation focuses on a study of trailable digraphs and supereulerian digraphs from the following aspects. 1. Strong Trail-Connected, Supereulerian and Trailable Digraphs. For a digraph D, D is trailable digraph if D has a spanning trail. A digraph D is strongly trail- connected if for any two vertices u and v of D, D posses both a spanning (u, v)-trail and a spanning (v,u)-trail. As the case when u = v is possible, every strongly trail-connected digraph is also su- pereulerian. Let D be a digraph. Let S(D) = {e ∈ A(D) : e is symmetric in D}. A digraph D is symmetric if A(D) = S(D). The symmetric core of D, denoted by J(D), has vertex set V (D) and arc set S(D). We have found a well-characterized digraph family D each of whose members does not have a spanning trail with its underlying graph spanned by a K2,n−2 such that for any strong digraph D with its matching number α′(D) and arc-strong-connectivity λ(D), if n = |V (D)| ≥ 3 and λ(D) ≥ α′(D) − 1, then each of the following holds. (i) There exists a family D of well-characterized digraphs such that for any digraph D with α′(D) ≤ 2, D has a spanning trial if and only if D is not a member in D. (ii) If α′(D) ≥ 3, then D has a spanning trail. (iii) If α′(D) ≥ 3 and n ≥ 2α′(D) + 3, then D is supereulerian. (iv) If λ(D) ≥ α′(D) ≥ 4 and n ≥ 2α′(D) + 3, then for any pair of vertices u and v of D, D contains a spanning (u, v)-trail. 2. Supereulerian Digraph Strong Products. A cycle vertex cover of a digraph D is a collection of directed cycles in D such that every vertex in D lies in at least one dicycle in this collection, and such that the union of the arc sets of these directed cycles induce a connected subdigraph of D. A subdigraph F of a digraph D is a circulation if for every vertex v in F, the indegree of v equals its outdegree, and a spanning circulation if F is a cycle factor. Define f(D) to be the smallest cardinality of a cycle vertex cover of the digraph D/F obtained from D by contracting all arcs in F , among all circulations F of D. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if D1 and D2 are nontrivial strong digraphs such that D1 is supereulerian and D2 has a cycle vertex cover C′ with |C′| ≤ |V (D1)|, then the Cartesian product D1 and D2 is also supereulerian. We prove that for strong digraphs D1 and D2, if for some cycle factor F1 of D1, the digraph formed from D1 by contracting arcs in F1 is hamiltonian with f(D2) not bigger than |V (D1)|, then the strong product D1 and D2 is supereulerian

A Note on Extreme Sets

In decomposition theory, extreme sets have been studied extensively due to its connection to perfect matchings in a graph. In this paper, we first define extreme sets with respect to degree-matchings and next investigate some of their properties. In particular, we prove the generalized Decomposition Theorem and give a characterization for the set of all extreme vertices in a graph

A PTAS for Triangle-Free 2-Matching

In the Triangle-Free (Simple) 2-Matching problem we are given an undirected graph $G=(V,E)$. Our goal is to compute a maximum-cardinality $M\subseteq E$ satisfying the following properties: (1) at most two edges of $M$ are incident on each node (i.e., $M$ is a 2-matching) and (2) $M$ does not induce any triangle. In his Ph.D. thesis from 1984, Harvitgsen presents a complex polynomial-time algorithm for this problem, with a very complex analysis. This result was never published in a journal nor reproved in a different way, to the best of our knowledge. In this paper we have a fresh look at this problem and present a simple PTAS for it based on local search. Our PTAS exploits the fact that, as long as the current solution is far enough from the optimum, there exists a short augmenting trail (similar to the maximum matching case).Comment: 27 pages, 18 figure