Fractional Matching Research Articles

Easy and hard separation of sparse and dense odd-set constraints in matching

We investigate polytopes intermediate between the fractional matching and the perfect matching polytopes, by imposing a strict subset of the odd-set (blossom) constraints. For sparse constraints, we give a polynomial time separation algorithm if only constraints on all odd sets bounded by a given size (e.g. ≤9+|V|/6) are present. Our algorithm also solves the more general problem of finding a T-cut subject to upper bounds on the cardinality of its defining node set and on its cost. In contrast, regarding dense constraints, we prove that for every 0<α≤12, it is NP-complete to separate over the class of constraints on odd sets of size 2⌊(1+α|V|)/2⌋−1 or ≥α|V|.

Conditional Fractional Matching Preclusion Number of Graphs

The conditional fractional matching preclusion number (CFMP number for short) [Formula: see text] of a graph [Formula: see text] is the minimum number of edges whose deletion results in a graph without isolated vertices and without fractional perfect matchings. In this paper, we study the CFMP number of complete graphs, complete bipartite graphs and twisted cubes. Also, we give Nordhaus–Gaddum-type results for the CFMP numbers of general graphs.

Class fairness in online matching

We initiate the study of fairness among classes of agents in online bipartite matching where there is a given set of offline vertices (aka agents) and another set of vertices (aka items) that arrive online and must be matched irrevocably upon arrival. In this setting, agents are partitioned into classes and the matching is required to be fair with respect to the classes. We adapt popular fairness notions (e.g. envy-freeness, proportionality, and maximin share) and their relaxations to this setting and study deterministic algorithms for matching indivisible items (leading to integral matchings) and for matching divisible items (leading to fractional matchings). For matching indivisible items, we propose an adaptive-priority-based algorithm, Match-and-Shift, prove that it achieves 12-approximation of both class envy-freeness up to one item and class maximin share fairness, and show that each guarantee is tight. For matching divisible items, we design a water-filling-based algorithm, Equal-Filling, that achieves (1−1e)-approximation of class envy-freeness and class proportionality; we prove 1−1e to be tight for class proportionality and establish a 34 upper bound on class envy-freeness. Finally, we discuss several challenges in designing randomized algorithms that achieve reasonable fairness approximation ratios. Nonetheless, we build upon Equal-Filling to design a randomized algorithm for matching indivisible items, Equal-Filling-OCS, which achieves 0.593-approximation of class proportionality.

Fractional matchings on regular graphs

Fractional matchings on regular graphs

Fractional Matchings in Graphs from the Spectral Radius

Fractional Matchings in Graphs from the Spectral Radius

Strong core and Pareto-optimality in the multiple partners matching problem under lexicographic preference domains

We study strong core and Pareto-optimal solutions for multiple partners matching problem under lexicographic preference domains from a computational point of view. The restriction to the two-sided case is called stable many-to-many matching problem and the general one-sided case is called stable fixtures problem. We provide an example to show that the strong core can be empty even for many-to-many problems, and that deciding the non-emptiness of the strong core is NP-hard. On the positive side, we give efficient algorithms for finding a near feasible strong core solution and for finding a fractional matching in the strong core of fractional matchings. In contrast with the NP-hardness result for the stable fixtures problem, we show that finding a maximum size matching that is Pareto-optimal can be done efficiently for many-to-many problems. Finally, we show that for reverse-lexicographic preferences the strong core is always non-empty in the many-to-many case.

Distance spectral radius and fractional matching in t-connected graphs

A fractional matching of a graph G is a function f assigning each edge a number in [ 0 , 1 ] so that ∑ e ∈ Γ ( v ) f ( e ) ≤ 1 for each v ∈ V ( G ) , where Γ ( v ) is the set of edges incident to v. The fractional matching number is the maximum of ∑ e ∈ Γ ( v ) f ( e ) over all fractional matchings. Motivated by progress in the study of relations between eigenvalues and matchings of graphs, in this paper, we characterize graphs with the minimum distance spectral radius among all t-connected graphs with n vertices and fractional matching number at most n − k 2 for 1 ≤ k ≤ n − 2 . Our characterization generalizes a result of Li, Miao, and Zhang [On the size, spectral radius, distance spectral radius and fractional matchings in graphs. Bull Aust Math Soc. 2023;187–199.], giving a distance spectral condition for the existence of a fractional perfect matching in a connected graph.

A note on the minimum size of matching-saturated graphs

Let s,m,n be positive integers and F be a graph. A graph G is called F-saturated if F is not a subgraph of G but G+e contain a copy of F for every edge e∈E(G¯), where G¯ is the complement graph of G. Let sat(n,F) be the minimum number of edges over all F-saturated graphs with order n and Sat(n,F) denotes the family of F-saturated graphs with sat(n,F) edges and n vertices. Let mK2 denote a matching of size m and let sK2f denote a fractional matching of size s. In this paper, we determine the exact value of sat(n,mK2), and characterize Sat(n,mK2) when n+2m<n<3m. Moreover, we also determine the exact value of sat(n,sK2f), and characterize Sat(n,sK2f) when n>2s. The main results provide partial answers to the open question posed by Faudree et al. (2011).

Distributed half-integral matching and beyond

By prior work, it is known that any distributed graph algorithm that finds a maximal matching requires Ω(log⁎⁡n) communication rounds, while it is possible to find a maximal fractional matching in O(1) rounds in bounded-degree graphs. However, all prior O(1)-round algorithms for maximal fractional matching use arbitrarily fine-grained fractional values. In particular, none of them is able to find a half-integral solution, using only values from {0,12,1}. We show that the use of fine-grained fractional values is necessary, and moreover we give a complete characterization on exactly how small values are needed: if we consider maximal fractional matching in graphs of maximum degree Δ=2d, and any distributed graph algorithm with round complexity T(Δ) that only depends on Δ and is independent of n, we show that the algorithm has to use fractional values with a denominator at least 2d. We give a new algorithm that shows that this is also sufficient.

Fractional matching preclusion numbers of Cartesian product graphs

The Cartesian product of two simple graphs G and H is the graph G□H whose vertex set is V(G)×V(H) and whose edge set is the set of all pairs (u1,v1)(u2,v2) such that either u1u2∈E(G) and v1=v2, or v1v2∈E(H) and u1=u2. The fractional matching preclusion number of a graph G, denoted by fmp(G), is the minimum number of edges whose deletion results in a graph with no fractional perfect matching. In this paper, we determine fmp(G□H) when H is a cycle or a path of even order; Moreover, given any integers a,b with a≥1 and 0≤b≤a+1, we construct a graph G such that δ(G)=a and fmp(G□H)=b when H is a path of odd order.

A note on maximum size of a graph without isolated vertices under the given matching number

In this paper, we determine the maximum size of a graph of fixed order without isolated vertices which contains (fractional) matching of given size, which is a variation of some known results involving the maximum size of all graphs of fixed order containing matching of given size by Erdős and Gallai. As corollaries of our results, Turán-type results for a connected graph on (fractional) matching are obtained.

Fractional matching, factors and spectral radius in graphs involving minimum degree

A fractional matching of a graph G is a function f:E(G)→[0,1] such that for any v∈V(G), ∑e∈EG(v)f(e)≤1, where EG(v)={e∈E(G):eis incident withvinG}. The fractional matching number of G is μf(G)=max{∑e∈E(G)f(e):f is a fractional matching of G}. Let k∈(0,n) is an integer. In this paper, we prove a tight lower bound of the spectral radius to guarantee μf(G)>n−k2 in a graph with minimum degree δ, which implies the result on the fractional perfect matching due to Fan et al. (2022) [6].For a set {A,B,C,…} of graphs, an {A,B,C,…}-factor of a graph G is defined to be a spanning subgraph of G each component of which is isomorphic to one of {A,B,C,…}. We present a tight sufficient condition in terms of the spectral radius for the existence of a {K2,{Ck}}-factor in a graph with minimum degree δ, where k≥3 is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a {K1,1,K1,2,…,K1,k}-factor with k≥2 in a graph with minimum degree δ, which generalizes the result of Miao et al. (2023) [10].

On sufficient conditions for spanning structures in dense graphs

AbstractWe study structural conditions in dense graphs that guarantee the existence of vertex‐spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of Kühn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Schülke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore‐type degree conditions, Pósa‐type degree conditions, deficiency‐type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.

Class Fairness in Online Matching

We initiate the study of fairness among classes of agents in online bipartite matching where there is a given set of offline vertices (aka agents) and another set of vertices (aka items) that arrive online and must be matched irrevocably upon arrival. In this setting, agents are partitioned into a set of classes and the matching is required to be fair with respect to the classes. We adopt popular fairness notions (e.g. envy-freeness, proportionality, and maximin share) and their relaxations to this setting and study deterministic and randomized algorithms for matching indivisible items (leading to integral matchings) and for matching divisible items (leading to fractional matchings). For matching indivisible items, we propose an adaptive-priority-based algorithm, MATCH-AND-SHIFT, prove that it achieves (1/2)-approximation of both class envy-freeness up to one item and class maximin share fairness, and show that each guarantee is tight. For matching divisible items, we design a water-filling-based algorithm, EQUAL-FILLING, that achieves (1-1/e)-approximation of class envy-freeness and class proportionality; we prove (1-1/e) to be tight for class proportionality and establish a 3/4 upper bound on class envy-freeness.

Spectral Radius and Fractional Perfect Matchings in Graphs

Spectral Radius and Fractional Perfect Matchings in Graphs

A Factor Matching of Optimal Tail Between Poisson Processes

Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension d at least 3. We construct a perfect matching between the two point sets that is a factor (i.e., a measurable function of the point configurations that commutes with translations), and with the property that the distance between two matched configuration points has a tail distribution that decays as fast as possible in magnitude, namely, as bexp (-cr^d) with suitable constants b,c>0. This settles the most difficult version of such matching problems: bicolored (versus unicolored) and deterministic (versus randomized). Our proof relies on two earlier results: an allocation (“land-division”) rule of similar tail for a Poisson point process by Markó and the author, and a recent breakthrough result of Bowen, Kun and Sabok that enables one to obtain perfect matchings from fractional perfect matchings under suitable conditions.

ON THE SIZE, SPECTRAL RADIUS, DISTANCE SPECTRAL RADIUS AND FRACTIONAL MATCHINGS IN GRAPHS

AbstractWe first establish a lower bound on the size and spectral radius of a graph G to guarantee that G contains a fractional perfect matching. Then, we determine an upper bound on the distance spectral radius of a graph G to ensure that G has a fractional perfect matching. Furthermore, we construct some extremal graphs to show all the bounds are best possible.

Maximum Size of a Graph with Given Fractional Matching Number

For three integers $n,k,d$, we determine the maximum size of a graph on $n$ vertices with fractional matching number $k$ and maximum degree at most $d$. As a consequence, we obtain the maximum size of a graph with given number of vertices and fractional matching number. This partially confirms a conjecture proposed by Alon et al. on the maximum size of $r$-uniform hypergraph with a fractional matching number for the special case when $r=2$.

Conditional fractional matching preclusion for burnt pancake graphs and pancake-like graphs

ABSTRACT The conditional fractional strong matching preclusion number of a graph G is the minimum size of F such that and G−F has neither a fractional perfect matching nor an isolated vertex. In this paper, we obtain the conditional fractional strong matching preclusion number for burnt pancake graphs and a subset of the class of pancake-like graphs.

File Placements, Fractional Matchings, and Normal Ordering

File Placements, Fractional Matchings, and Normal Ordering