Faster Algorithms for Semi-Matching Problems

We consider the problem of finding \textit{semi-matching} in bipartite graphs which is also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted case. For the weighted case, we give an $O(nm\log n)$-time algorithm, where $n$ is the number of vertices and $m$ is the number of edges, by exploiting the geometric structure of the problem. This improves the classical $O(n^3)$ algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi [Communications of the ACM 1974]. For the unweighted case, the bound could be improved even further. We give a simple divide-and-conquer algorithm which runs in $O(\sqrt{n}m\log n)$ time, improving two previous $O(nm)$-time algorithms by Abraham [MSc thesis, University of Glasgow 2003] and Harvey, Ladner, Lov\'asz and Tamir [WADS 2003 and Journal of Algorithms 2006]. We also extend this algorithm to solve the \textit{Balance Edge Cover} problem in $O(\sqrt{n}m\log n)$ time, improving the previous $O(nm)$-time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC 2008].Comment: ICALP 201

Minimum Cost Flows in Graphs with Unit Capacities

We consider the minimum cost flow problem on graphs with unit capacities and its special cases. In previous studies, special purpose algorithms exploiting the fact that capacities are one have been developed. In contrast, for maximum flow with unit capacities, the best bounds are proven for slight modifications of classical blocking flow and push-relabel algorithms. In this paper we show that the classical cost scaling algorithms of Goldberg and Tarjan (for general integer capacities) applied to a problem with unit capacities achieve or improve the best known bounds. For weighted bipartite matching we establish a bound of O(sqrt{rm}log C) on a slight variation of this algorithm. Here r is the size of the smaller side of the bipartite graph, m is the number of edges, and C is the largest absolute value of an arc-cost. This simplifies a result of [Duan et al. 2011] and improves the bound, answering an open question of [Tarjan and Ramshaw 2012]. For graphs with unit vertex capacities we establish a novel O(sqrt{n}mlog(nC)) bound. We also give the first cycle canceling algorithm for minimum cost flow with unit capacities. The algorithm naturally generalizes the single source shortest path algorithm of [Goldberg 1995]

Node-balancing by edge-increments

Suppose you are given a graph $G=(V,E)$ with a weight assignment $w:V\rightarrow\mathbb{Z}$ and that your objective is to modify $w$ using legal steps such that all vertices will have the same weight, where in each legal step you are allowed to choose an edge and increment the weights of its end points by $1$. In this paper we study several variants of this problem for graphs and hypergraphs. On the combinatorial side we show connections with fundamental results from matching theory such as Hall's Theorem and Tutte's Theorem. On the algorithmic side we study the computational complexity of associated decision problems. Our main results are a characterization of the graphs for which any initial assignment can be balanced by edge-increments and a strongly polynomial-time algorithm that computes a balancing sequence of increments if one exists.Comment: 10 page

Network flow algorithms and applications

This paper looks at several methods for solving network flow problems. The first chapter gives a brief background for linear programming (LP) problems. It includes basic definitions and theorems. The second chapter gives an overview of graph theory including definitions, theorems, and examples. Chapters 3-5 are the heart of this thesis. Chapter 3 includes algorithms and applications for maximum flow problems. It includes a look at a very important theorem. Maximum Flow/Minimum Cut Theorem. There is also a section on the Augmenting Path Algorithm. Chapter 4 Deals with shortest path problem. It includes Dijsksta\u27s Algorithm and the All-Pairs Labeling Algorithm. Chapter 5 includes information on algorithms and applications for the minimum cost flow(MCF)problem. The algorithms covered include the Cycle Canceling,Successive ShortestPath,and Primal-Dual Algorithms. Each of these chapters 3-5 contain definitions,theorems,and algorithms to solve network flow problems. Throughout the paper the computer program LINDO is used. It serves a couple of functions. First it is a way of checking each solution. The second use is to expose the reader to a very valuable tool in linear programming

An Experiment in Ping-Pong Protocol Verification by Nondeterministic Pushdown Automata

An experiment is described that confirms the security of a well-studied class of cryptographic protocols (Dolev-Yao intruder model) can be verified by two-way nondeterministic pushdown automata (2NPDA). A nondeterministic pushdown program checks whether the intersection of a regular language (the protocol to verify) and a given Dyck language containing all canceling words is empty. If it is not, an intruder can reveal secret messages sent between trusted users. The verification is guaranteed to terminate in cubic time at most on a 2NPDA-simulator. The interpretive approach used in this experiment simplifies the verification, by separating the nondeterministic pushdown logic and program control, and makes it more predictable. We describe the interpretive approach and the known transformational solutions, and show they share interesting features. Also noteworthy is how abstract results from automata theory can solve practical problems by programming language means.Comment: In Proceedings MARS/VPT 2018, arXiv:1803.0866

Efficient Solution of Minimum Cost Flow Problems for Large-scale Transportation Networks

With the rapid advance of information technology in the transportation industry, of which intermodal transportation is one of the most important subfields, the scale and dimension of problem sizes and datasets is rising significantly. This trend raises the need for study on improving the efficiency, profitability and level of competitiveness of intermodal transportation networks while exploiting the rich information of big data related to these networks. Therefore, this dissertation aims to investigate intermodal transportation network design problems, especially practical optimization problems, and to develop more realistic and effective models and solution approaches that will assist network operators and/or decision makers of the intermodal transportation system. This dissertation focuses on developing a novel strategy for solving the Minimum Cost Flow (MCF) problem for large-scale network design problems by adopting a divide-and-conquer policy during the optimization process. The main contribution is the development of an agglomerative clustering based tiling strategy to significantly reduce the computational and peak memory consumption of the MCF model for large-scale networks. The tiling strategy is supported by the regional-division theorem and -approximation regional-division theorem that are proposed and proved in this dissertation. The region-division theorem is a sufficient condition to exactly guarantee the consistency between the local MCF solution of each sub-network obtained by the aforementioned tiling strategy and the global MCF solution of the whole network. Furthermore, the -approximation region-division theorem provides worst-case bounds, so that the practical approximation MCF solution closely approximates the optimal solution in terms of its optimal value. A series of experiments are performed to evaluate the utility of the proposed approach of solving the large-scale MCF problem. The results indicate that the proposed approach is beneficial to save the execution time and peak memory consumption in large-scale MCF problems under different circumstances