Maximum Bipartite Matching in 𝑛 2+𝑜(1) Time via a Combinatorial Algorithm

In this paper, we propose a depth-first search (DFS) algorithm for finding maximum matching in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating-paths in the super-vertices shrinking from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending the time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with m edges and n vertices in 𝑂(𝑚𝑛) time with space complexity 𝑂(𝑛).

Maximum matchings in general graphs through randomization

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in [EQUATION] time, [EQUATION] per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs [29, 18]. Here m, n, and N bound the number of edges, vertices, and magnitude of any integer edge weight. Our result improves on a 25-year old algorithm of Gabow and Tarjan, which runs in [EQUATION] log(nN)) time.

We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O ( n ) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S ; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m -edge graph and test the independence of the maintained set in time O (√m) per update. We use the domination data structures as part of our enumeration algorithms.

A vertex-subset graph problem \(\mathcal {Q}\) defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions \(S_s\) and \(S_t\) of size k, whether it is possible to transform \(S_s\) into \(S_t\) by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by \(\textsc {ISR}\) and the latter by \(\textsc {DSR}\). Both \(\textsc {ISR}\) and \(\textsc {DSR}\) are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that \(\textsc {ISR}\) is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that \(\textsc {ISR}\) is fixed-parameter tractable on planar graphs and graphs of bounded degree. For \(\textsc {DSR}\), we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes degenerate and nowhere dense graphs.

The Group Steiner Tree (GST) problem is a classical problem in combinatorial optimization and theoretical computer science. In the Edge-Weighted Group Steiner Tree (EW-GST) problem, we are given an undirected graph G = (V, E) on n vertices with edge costs c : E → ℝ≥ 0, a source vertex s and a collection of subsets of vertices, called groups, S1, . . . , Sk ⊆ V. The goal is to find a minimum-cost tree H ⊆ G that connects s to some vertex from each group Si, for all i = 1, 2, . . . , k. The Node-Weighted Group Steiner Tree (NW-GST) problem has the same setting, but the costs are associated with nodes. The goal is to find a minimum-cost node set X ⊆ V such that G[X] connects every group to the source.When G is a tree, both EW-GST and NW-GST admit a polynomial-time O(log n log k) approximation algorithm due to the seminal result of [Garg et al., SODA'98 and J. Algorithm]. The matching hardness of log2−ϵn is known even for tree instances of EW-GST and NW-GST [Halperin and Krauthgamer STOC'03]. In general graphs, most of polynomial-time approximation algorithms for EW-GST reduce the problem to a tree instance using the metric-tree embedding, incurring a loss of O(log n) on the approximation factor [Bartal, FOCS'96; Fakcharoenphol et al., FOCS'03 and JCSS]. This yields an approximation ratio of O(log2n log k) for EW-GST. Using metric-tree embedding, this factor cannot be improved: The loss of Ω(log n) is necessary on some input graphs (e.g., grids and expanders). There are alternative approaches that avoid metric-tree embedding, e.g., the algorithm of [Chekuri and Pal, FOCS'05], which gives a tight approximation ratio, but none of which achieves polylogarithmic approximation in polynomial-time. This state of the art shows a clear lack of understanding of GST in general graphs beyond the metric-tree embedding technique. For NW-GST (for which the metric-tree embedding does not apply), not even a polynomial-time polylogarithmic approximation algorithm is known.In this paper, we present O(log n log k) approximation algorithms that run in time nO(tw(G)2) for both NW-GST and EW-GST1, where tw(G) denotes the treewidth of graph G. The key to both results is a different type of "tree-embedding" that produces a tree of much bigger size, but does not cause any loss on the approximation factor. Our embedding is inspired by dynamic programming, a technique which is typically not applicable to Group Steiner problems.

We consider the fine-grained complexity of sparse graph problems that currently have O(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (Minimum Weight Cycle), Radius, Eccentricities, BC (Betweenness Centrality), etc. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in the O(mn) class. There are many sub-cubic reductions between graph problems in the O(mn) class, but surprisingly few of these preserve sparsity. In the directed case, our results give a partial order on a large collection of problems in the O(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to unweighted APSP. We develop a new ‘bit-sampling’ method for these sparse reductions on undirected graphs, which also gives rise to improved or simpler algorithms for cycle finding problems in undirected graphs. We formulate the the notion of MWC hardness, which is based on the assumption that a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in the O(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, Eccentricities and BC are MWC hard. Our sparse reductions give MWC hardness a status for the O(mn) class similar to 3SUM hardness for the quadratic class, since they show sub-mn hardness for a large collection of fundamental and well-studied graph problems that have maintained an O(mn) time bound for over half a century. We also identify Eccentricities and BC as key problems in the O(mn) class which are simultaneously MWC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than nk. Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which the O(mn) time algorithms are the ones typically used in practice, and not the O(n3) time algorithms.

Branch-and-cut approaches for [formula omitted]-Cluster Editing

In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs, and one most suitable for bounded-degree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant complexity, while the complexity of testing bounded-degree graphs is \(\tilde{\Theta}(\sqrt{n})\), where n is the number of vertices in the graph. Thus there is a large gap between the complexity of testing in the two cases.In this work we bridge the gap described above. In particular, we study the problem of testing bipartiteness in a model that is suitable for all densities. We present an algorithm whose complexity is \(\tilde{O}(\min(\sqrt{n},n^2/m))\) where m is the number of edges in the graph, and match it with an almost tight lower bound.KeywordsAverage DegreeRegular GraphGeneral GraphQuery ComplexityTesting AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Does Surgical Technique in Aortic Valve Replacement Surgery Have an Impact on Rates of Postoperative Conduction Defects Requiring Permanent Pacemaker Implantation

The maximum cut (MAX-CUT) problem is to find a bipartition of the vertices in a given graph such that the number of edges with ends in different sets reaches the largest. Though, several experimental investigations have shown that evolutionary algorithms (EAs) are efficient for this NP-complete problem, there is little theoretical work about EAs on the problem. In this paper, we theoretically investigate the performance of EAs on the MAX-CUT problem. We find that both the (1+1) EA and the (1+1) EA*, two simple EAs, efficiently achieve approximation solutions of (m/2)+(1/4)s(G) and (m/2)+(1/2)(√{8m+1}-1), where m and s(G) are respectively the number of edges and the number of odd degree vertices in the input graph. We also reveal that for a given integer k the (1+1) EA* finds a cut of size at least k in expected runtime O(nm+1/δ(4k)) and a cut of size at least (m/2)+k in expected runtime O(n(2)m+1/δ((64/3)k(2))), where δ is a constant mutation probability and n is the number of vertices in the input graph. Finally, we show that the (1+1) EA and the (1+1) EA* are better than some local search algorithms in one instance, and we also show that these two simple EAs may not be efficient in another instance.

Background: The technique of suturing perineal trauma following childbirth may have a significant effect on the extent and degree of postpartum maternal morbidity. The aim of this study was to compare continuous and interrupted suturing methods of right mediolateral episiotomy in terms of puerperal morbidity in women at Kamla Nehru State Hospital For Mother And Child, Shimla. Methods: A randomized controlled trial was conducted in the Department of Obstetrics and Gynecology, Kamla Nehru State Hospital for Mother and Child, IGMC Shimla for a period of one year where two hundred seventy pregnant females at term were recruited for study after informed consent and satisfying inclusion and exclusion criteria. Only two hundred seven cases could complete the study. Rest was lost to follow up and hence were dropped from study. They were randomized in two groups: group A where the episiotomy was stitched by the continuous technique; group B had their episiotomy was repaired by the interrupted method. Findings: Continuous suturing gave better results in terms of healing (REEDA score) as compared to interrupted suturing technique at two hours, twenty four hours and forty eight hours postpartum (p value<0.001). Less pain was experienced by patients in the continuous group as compared to patients in the interrupted group at two hours, twenty four hours, forty eight hours postpartum in lying, sitting position and while walking (p value<0.001). The time required to repair episiotomy, number of suture packets required was less in continuous as compared to interrupted episiotomy (p value<0.001). Interpretation: It was concluded that continuous suturing is a better method compared to the conventional interrupted technique of repair of RMLE. The continuous technique causes less discomfort to women in the postpartum period as compared to the interrupted method. Not only does it cause less pain immediately and in later hours and days postpartum but, it also gives better healing results. Keywords: Episiotomy, continuous versus interrupted suturing, right mediolateral episiotomy

Solving combinatorial optimization problems with single seekers society algorithm

We provide an algorithm that maintains, against an adaptive adversary, a (1 — ε )-approximate maximum matching in n-node m-edge general (not necessarily bipartite) undirected graph undergoing edge deletions with high probability with (amortized) O (poly(ε-1, log n )) time per update. We also obtain the same update time for maintaining a fractional approximate weighted matching (and hence an approximation to the value of the maximum weight matching) and an integral approximate weighted matching in dense graphs.1 Our unweighted result improves upon the prior state-of-the-art which includes a poly(log n ) · 2O (1/ɛ2) update time [Assadi-Bernstein-Dudeja 2022] and an update time [Gupta-Peng 2013], and our weighted result improves upon the log n ) update time due to [Gupta-Peng 2013].

The problem of computing a matching of maximum weight in a given edge-weighted graph is not known to be P-hard or in RNC. This paper presents four parallel approximation algorithms for this problem. The first is an RNC-approximation scheme, i.e., an RNC algorithm that computes a matching of weight at least 1 - Ɛ times the maximum for any given constant Ɛ > 0. The second one is an NC approximation algorithm achieving an approximation ratio of 1/2+Ɛ for any fixed Ɛ > 0. The third and fourth algorithms only need to know the total order of weights, so they are useful when the edge weights require a large amount of memories to represent. The third one is an NC approximation algorithm that finds a matching of weight at least 2/3Δ+2 times the maximum, where Δ is the maximum degree of the graph. The fourth one is an RNC algorithm that finds a matching of weight at least 1/2Δ+4 times the maximum on average, and runs in O(logΔ) time, not depending on the size of the graph.