A quasi linear-time b-Matching algorithm on distance-hereditary graphs and bounded split-width graphs

We present a quasi linear-time algorithm for Maximum Matching on distance-hereditary graphs and some of their generalizations. This improves on [Dragan, WG'97], who proposed such an algorithm for the subclass of (tent,hexahedron)-free distance-hereditary graphs. Furthermore, our result is derived from a more general one that is obtained for b-Matching. In the (unit cost) b-Matching problem, we are given a graph G = (V, E) together with a nonnegative integer capacity b v for every vertex v $\in$ V. The objective is to assign nonnegative integer weights (x e) e$\in$E so that: for every v $\in$ V the sum of the weights of its incident edges does not exceed b v , and e$\in$E x e is maximized. We present the first algorithm for solving b-Matching on cographs, distance-hereditary graphs and some of their generalizations in quasi linear time. For that, we use a decomposition algorithm that outputs for any graph G a collection of subgraphs of G with no edge-cutsets inducing a complete bipartite subgraph (a.k.a., splits). The latter collection is sometimes called a split decomposition of G. Furthermore, there exists a generic method in order to design graph algorithms based on split decomposition [Rao, DAM'08]. However, this technique only applies to "localized" problems: for which a "best" partial solution for any given subgraph in a split decomposition can be computed almost independently from the remaining of the graph. Such framework does not apply to matching problems since an augmenting path may cross the subgraphs arbitrarily. We introduce a new technique that somehow captures all the partial solutions for a given union of subgraphs in a split decomposition, in a compact and amenable way for algorithms - assuming some piecewise linear assumption holds on the value of such solutions. The latter assumption is shown to hold for b-Matching. Doing so, we prove that solving b-Matching on any pair G, b can be reduced in quasi linear-time to solving this problem on a collection of smaller graphs: that are obtained from the subgraphs in any split decomposition of G by replacing every vertex with a constant-size module. In particular, if G has a split decomposition where all subgraphs have order at most a fixed k, then we can solve b-Matching for G, b in O((k log 2 k)$\times$(m+n)$\times$log ||b|| 1)-time. This answers an open question of [Coudert et al., SODA'18]

Concave Flow on Small Depth Directed Networks

Small depth networks arise in a variety of network related applications, often in the form of maximum flow and maximum weighted matching. Recent works have generalized such methods to include costs arising from concave functions. In this paper we give an algorithm that takes a depth $D$ network and strictly increasing concave weight functions of flows on the edges and computes a $(1 - \epsilon)$-approximation to the maximum weight flow in time $mD \epsilon^{-1}$ times an overhead that is logarithmic in the various numerical parameters related to the magnitudes of gradients and capacities. Our approach is based on extending the scaling algorithm for approximate maximum weighted matchings by [Duan-Pettie JACM`14] to the setting of small depth networks, and then generalizing it to concave functions. In this more restricted setting of linear weights in the range $[w_{\min}, w_{\max}]$, it produces a $(1 - \epsilon)$-approximation in time $O(mD \epsilon^{-1} \log( w_{\max} /w_{\min}))$. The algorithm combines a variety of tools and provides a unified approach towards several problems involving small depth networks.Comment: 25 page

A Proof of the MV Matching Algorithm

The Micali-Vazirani (MV) algorithm for maximum cardinality matching in general graphs, which was published in 1980 \cite{MV}, remains to this day the most efficient known algorithm for the problem. This paper gives the first complete and correct proof of this algorithm. Central to our proof are some purely graph-theoretic facts, capturing properties of minimum length alternating paths; these may be of independent interest. An attempt is made to render the algorithm easier to comprehend.Comment: 43 pages. arXiv admin note: text overlap with arXiv:1210.459

Scaling Algorithms for Weighted Matching in General Graphs

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in $O(m\sqrt{n}\log(nN))$ time, $O(m\sqrt{n})$ per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs. Here $m,n,$ and $N$ bound the number of edges, vertices, and magnitude of any edge weight. Our result improves on a 25-year old algorithm of Gabow and Tarjan, which runs in $O(m\sqrt{n\log n\alpha(m,n)} \log(nN))$ time.Comment: Extended abstract published in SODA'1

A New Community Definition For MultiLayer Networks And A Novel Approach For Its Efficient Computation

As the use of MultiLayer Networks (or MLNs) for modeling and analysis is gaining popularity, it is becoming increasingly important to propose a community definition that encompasses the multiple features represented by MLNs and develop algorithms for efficiently computing communities on MLNs. Currently, communities for MLNs, are based on aggregating the networks into single graphs using different techniques (type independent, projection-based, etc.) and applying single graph community detection algorithms, such as Louvain and Infomap on these graphs. This process results in different types of information loss (semantics and structure). To the best of our knowledge, in this paper we propose, for the first time, a definition of community for heterogeneous MLNs (or HeMLNs) which preserves semantics as well as the structure. Additionally, our basic definition can be extended to appropriately match the analysis objectives as needed. In this paper, we present a structure and semantics preserving community definition for HeMLNs that is compatible with and is an extension of the traditional definition for single graphs. We also present a framework for its efficient computation using a newly proposed decoupling approach. First, we define a k-community for connected k layers of a HeMLN. Then we propose a family of algorithms for its computation using the concept of bipartite graph pairings. Further, for a broader analysis, we introduce several pairing algorithms and weight metrics for composing binary HeMLN communities using participating community characteristics. Essentially, this results in an extensible family of community computations. We provide extensive experimental results for showcasing the efficiency and analysis flexibility of the proposed computation using popular IMDb and DBLP data sets.Comment: arXiv admin note: substantial text overlap with arXiv:1910.01737, arXiv:1903.0264

A Weight-scaling Algorithm for ff-factors of Multigraphs

We discuss combinatorial algorithms for finding a maximum weight $f$-factor on an arbitrary multigraph, for given integral weights of magnitude at most $W$. For simple bipartite graphs the best-known time bound is $O(n^{2/3}\, m\, \log nW)$ (\cite{GT89}; $n$ and $m$ are respectively the number of vertices and edges). A recent algorithm of Duan and He et al. \cite{DHZ} for $f$-factors of simple graphs comes within logarithmic factors of this bound, $\widetilde{O} (n^{2/3}\, m\, \log W)$. The best-known bound for bipartite multigraphs is $O(\sqrt {\Phi}\, m\, \log \Phi W)$ ($\Phi\le m$ is the size of the $f$-factor, $\Phi=\sum_{v\in V}f(v)/2$). This bound is more general than the restriction to simple graphs, and is even superior on "small" simple graphs, i.e., $\Phi=o(n^{4/3})$. We present an algorithm that comes within a $\sqrt {\log \Phi}$ factor of this bound, i.e., $O(\sqrt {\Phi \log \Phi}\,m \,\log \Phi W)$. The algorithm is a direct generalization of the algorithm of Gabow and Tarjan \cite{GT} for the special case of ordinary matching ($f\equiv 1$). We present our algorithm first for ordinary matching, as the analysis is a simplified version of \cite{GT}. Furthermore the algorithm and analysis both get incorporated without modification into the multigraph algorithm. To extend these ideas to $f$-factors, the first step is "expanding" edges (i.e., replacing an edge by a length 3 alternating path). \cite{DHZ} uses a one-time expansion of the entire graph. Our algorithm keeps the graph small by only expanding selected edges, and "compressing" them back to their original source when no longer needed. Several other ideas are needed, including a relaxation of the notion of "blossom" to e-blossom ("expanded blossom")