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The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1 / 2 -approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continues this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively. Unfortunately, a recent work Konrad and Naidu (Approximation, randomization, and combinatorial optimization. Algorithms and techniques, APPROX/RANDOM 2021, August 16–18, 2021. LIPIcs, vol 207, pp 19:1–19:18, 2021. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.19) shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than 1 / 2 -approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms were previously known, and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs. The main significance of our results is not in the numerical improvements, but in demonstrating the potential of non-maximal-matching-first algorithms. [ABSTRACT FROM AUTHOR]

In a recent paper, Avis, Bremner, Tiwary and Watanabe gave a method for constructing linear programs (LPs) based on algorithms written in a simple programming language called Sparks. If an algorithm produces the solution x to a problem in polynomial time and space then the LP constructed is also of polynomial size and its optimum solution contains x as well as a complete execution trace of the algorithm. Their method led us to the construction of a compiler called sparktope which we describe in this paper. This compiler allows one to generate polynomial sized LPs for problems in P that have exponential extension complexity, such as matching problems in non-bipartite graphs. In this paper, we describe sparktope, the language Sparks, and the assembler instructions and LP constraints it produces. This is followed by two concrete examples, the makespan problem and the problem of testing if a matching in a graph is maximum, both of which are known to have exponential extension complexity. Computational results are given. In discussing these examples, we make use of visualization techniques included in sparktope that may be of independent interest. The extremely large linear programs produced by the compiler appear to be quite challenging to solve using currently available software. Since the optimum LP solutions can be computed independently they may be useful as benchmarks. Further enhancements of the compiler and its application are also discussed. [ABSTRACT FROM AUTHOR]

In a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser proposed two algorithms for this task. The second algorithm has been intensely studied, but due to technical difficulties, the first algorithm has received less attention. Empirical results by Karp and Sipser suggest that the first algorithm is superior. In this paper we show that this is indeed the case, at least for random k‐regular graphs. We show that w.h.p. the first algorithm will find a matching of size n/2−O(logn) in a random k‐regular graph, k = O(1). We also show that the algorithm can be adapted to find a maximum matching in O(n) time w.h.p. This is to be compared with O(n3/2) time for the worst‐case. [ABSTRACT FROM AUTHOR]

Nowadays, large cities face numerous problems caused by the rapidly increasing number of vehicles on road. Carpooling has been shown as an efficient solution to ease the traffic pressure. The objective of our carpool scheduling problem is to minimise the number of carpools needed to transport users. Previous research on the similar topic introduces static capacity constraint which limits the carpool size to the vehicle's capacity. However, it is unnecessary since a seat in the vehicle can be occupied by several passengers if their routes are not overlapped. In this paper, we remove this constraint, which allows a vehicle to serve more users than its capacity. We propose a greedy approach based on the iterative matching and merging. Specifically, starting from a set of single-user carpools, the algorithm iteratively checks the merge-ability between carpools, and then applies the maximum matching algorithm to maximise the number of carpools merged. In addition, two methods are proposed to reduce the time complexity of merge-ability checking process. Furthermore, we improve the time efficiency of our algorithms by exploiting geometry properties. Experimental results on synthetic and real-world datasets show that our algorithms have better performances than baseline. [ABSTRACT FROM AUTHOR]

A critical prerequisite for controlling complex networks is to find a driver node set with a structural controllability guarantee. This paper introduces the control capacity for a driver node set and solves the problem of finding a complete minimum driver node set that not only guarantees network structural controllability but achieves the desired level of control capacity. A novel algorithmic framework is proposed which is based on the concept of equivalent set and approximate matching replacement technique. The proposed algorithmic framework is shown to outperform the state-of-the-art approaches in the literature. The validity of the proposed algorithm is analyzed and the performance is evaluated by experiments on artificial and real-world complex networks. [ABSTRACT FROM AUTHOR]

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph $$G= (A\, \dot{\cup }\, P, E)$$ with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between $$\textsf {NP}$$ -hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota $$u_{\max }$$ as basis, and we prove that this dependence is necessary unless $$\textsf {FPT}= \textsf {W}[1]$$ . The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee $$u_{\max }+1$$ , which is asymptotically best possible unless $$\textsf {P}= \textsf {NP}$$ . Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas. [ABSTRACT FROM AUTHOR]

A connected graph $$G$$ is said to be factor-critical if $$G-v$$ has a perfect matching for every vertex $$v$$ of $$G$$ . Lovász proved that every factor-critical graph has an ear decomposition. In this paper, the ear decomposition of the factor-critical graphs $$G$$ satisfying that $$G-v$$ has a unique perfect matching for any vertex $$v$$ of $$G$$ with degree at least 3 is characterized. From this, the number of maximum matchings of factor-critical graphs with the special ear decomposition is obtained. [ABSTRACT FROM AUTHOR]

Recently, an ancient slime mold, Physarum polycephalum, has been proved being capable of finding shortest path in physical maze environment, which inspires researchers to extract the core foraging mechanism -- positive feedback -- to simulate or model such intelligent behavior. Among most wellknown mathematical models developed so far, physarum solver, has been adopted and extended to solve a plethora of optimization problems in different fields, including computer science, operations research and transportation etc. In this paper, we first adopt and apply a variant of modified physarum solver, called iterative physarum model, to solve bipartite matching problem. Specifically, the maximum bipartite matching problem is first equivalently transformed to single-source single-sink maximum flow problem. Then the iterative physarum model is used to solve the maximum flow problem adaptively. As iterative physarum solver does not involve solving the systems of linear equations associated with global network flow balance constraints, time complexity of updating node status for one iteration can be reduced from O(n3) to O(m), where n and m are numbers of nodes and edges in the graph, respectively. Extensive numerical studies on both sparse and complete bipartite graphs demonstrate the validity and efficiency of this method [ABSTRACT FROM AUTHOR]

Controlling complex networks are one of the ultimate goals in network science. Previous works have found there exist two distinct control modes when controlling the dense networks: distributed and centralized modes. How to change the control mode of a network is a challenging problem. This paper presents an efficient algorithm to alter a network from distributed mode to centralized mode by reversing the direction of a few edges. We first analyze four possible cases when reversing an edge and then design an efficient algorithm to change the control mode of a network. We prove that our algorithm does not affect the control scheme of the network after mode change. We evaluate the performance of our algorithm on both synthetic and real networks. The results show that the control mode of most networks can be easily changed by reversing very few edges. Furthermore, the number of the possible driver nodes of the network after mode change is dramatically decreased, which means these networks are easier to control. Our algorithm provides the ability to design the desired control mode of a network for different control scenarios, which may be used in many applications. • We present a novel algorithm to change the control mode of a network by reversing the edges. • The control scheme remains unchanged when we change a network from distributed mode to centralized mode. • The results showed our algorithms are efficient and effective to change the control mode of complex networks. [ABSTRACT FROM AUTHOR]

In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in an arbitrary order, and should be irrevocably accepted or rejected. Here, the goal of an online algorithm is to maximize the number of accepted edges while maintaining a feasible matching at any point in time. For this model, the standard greedy heuristic is 1 2 -competitive, and on the other hand, no algorithm that outperforms this ratio is currently known, even for very simple graphs. We present a clean Min-Index framework for devising a family of randomized algorithms, and provide a number of positive and negative results in this context. Among these results, we present a 5 9 -competitive algorithm when the underlying graph is a forest, and prove that this ratio is best possible within the Min-Index framework. In addition, we prove a new general upper bound of 2 3 + 1 / ϕ 2 ≈ 0.5914 on the competitiveness of any algorithm in the edge arrival model. Interestingly, while this result slightly falls short of the currently best 1 1 + ln 2 ≈ 0.5906 bound by Epstein et al. (Inf Comput 259(1):31–40, 2018), it holds even for an easier model in which vertices along with their adjacent edges arrive online. As a result, we improve on the currently best upper bound of 0.6252 for the latter model, due to Wang and Wong (in: Proceedings of the 42nd ICALP, 2015). [ABSTRACT FROM AUTHOR]

Since Tinhofer proposed the MinGreedy algorithm for maximum cardinality matching in 1984, several experimental studies found the randomized algorithm to perform excellently for various classes of random graphs and benchmark instances. In contrast, only few analytical results are known. We show that MinGreedy cannot improve on the trivial approximation ratio of $$\frac{1}{2}$$ whp., even for bipartite graphs. Our hard inputs seem to require a small number of high-degree nodes. This motivates an investigation of greedy algorithms on graphs with maximum degree $$\varDelta $$ : we show that MinGreedy achieves a $${\frac{{\varDelta }-1}{2{\varDelta }-3}} $$ -approximation for graphs with $${\varDelta } {=} 3$$ and for $$\varDelta $$ -regular graphs, and a guarantee of $${\frac{{\varDelta }-1/2}{2{\varDelta }-2}} $$ for graphs with maximum degree $${\varDelta } $$ . Interestingly, our bounds even hold for the deterministic MinGreedy that breaks all ties arbitrarily. Moreover, we investigate the limitations of the greedy paradigm, using the model of priority algorithms introduced by Borodin, Nielsen, and Rackoff. We study deterministic priority algorithms and prove a $${\frac{{\varDelta }-1}{2{\varDelta }-3}}$$ -inapproximability result for graphs with maximum degree $${\varDelta } $$ ; thus, these greedy algorithms do not achieve a $$\frac{1}{2} {+} \varepsilon $$ -approximation and in particular the $$\frac{2}{3}$$ -approximation obtained by the deterministic MinGreedy for $${\varDelta } {=} 3$$ is optimal in this class. For k-uniform hypergraphs we show a tight $$\frac{1}{k}$$ -inapproximability bound. We also study fully randomized priority algorithms and give a $$\frac{5}{6}$$ -inapproximability bound. Thus, they cannot compete with matching algorithms of other paradigms. [ABSTRACT FROM AUTHOR]

We prove $$n^{1+\varOmega (1/p)}/p^{O(1)}$$ lower bounds for the space complexity of p-pass streaming algorithms solving the following problems on n-vertex graphs: (1) testing if an undirected graph has a perfect matching (this implies lower bounds for computing a maximum matching or even just the maximum matching size), (2) testing if two specific vertices are at distance at most $$2(p+1)$$ in an undirected graph, (3) testing if there is a directed path from s to t for two specific vertices s and t in a directed graph. The lower bounds hold for $$p = O(\log n / \log \log n)$$ . Prior to our result, it was known that these problems require $$\varOmega (n^2)$$ space in one pass, but no $$n^{1+\varOmega (1)}$$ lower bound was known for any $$p\ge 2$$ . These streaming results follow from a communication complexity lower bound for a communication game in which the players hold two graphs on the same set of vertices. The task of the players is to find out whether the sets of vertices at distance exactly $$p+1$$ from a specific vertex intersect. The game requires a significant amount of communication only if the players are forced to speak in a specific difficult order. This is reminiscent of lower bounds for communication problems such as indexing and pointer chasing. Among other things, our line of attack requires proving an information cost lower bound for a decision version of the classic pointer chasing problem and a direct sum type theorem for the disjunction of several instances of this problem. [ABSTRACT FROM AUTHOR]