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This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial three-dimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published

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Abstract. Automatic formal verification techniques generally require exponential resources with respect to the number of primary inputs of a netlist. In this paper, we present several fully-automated techniques to enable maximal input reductions of sequential netlists. First, we present a novel min-cut based localization refinement scheme for yielding a safely overapproximated netlist with minimal input count. Second, we present a novel form of reparameterization: as a trace-equivalence preserving structural abstraction, which provably renders a netlist with input count at most a constant factor of register count. In contrast to prior research in reparameterization to offset input growth during symbolic simulation, we are the first to explore this technique as a structural transformation for sequential netlists, enabling its benefits to general verification flows. In particular, we detail the synergy between these input-reducing abstractions, and with other transformations such as retiming which – as with traditional localization approaches – risks substantially increasing input count as a byproduct of its register reductions. Experiments confirm that the complementary reduction strategy enabled by our techniques is necessary for iteratively reducing large problems while keeping both proof-fatal design size metrics – register count and input count – within reasonable limits, ultimately enabling an efficient automated solution. 1

This study shows various ways that formal graph theoretic statements map patterns of network ties into substantive hypotheses about social cohesion. If network cohesion is enhanced by multiple connections between members of a group, for example, then the higher the global minimum of the number of independent paths that connect every pair of nodes in the network, the higher the social cohesion. The cohesiveness of a group is also measured by the extent to which it is not disconnected by removal of 1, 2, 3,..., n actors. Menger's Theorem proves that these two measures are equivalent. Within this graph theoretic framework, we evaluate the family of concepts of cohesion and establish the validity of a pair of related measures: 1. Connectivity - the minimum number k of its actors whose removal would not allow the group to remain connected or would reduce the group to but a single member - measures the social cohesion of a group at a general level. 2. Conditional density measures cohesion on a finer scale as a proportion of ties beyond that required by a graph's connectivity k over the number of ties that would force it to k + 1. Calibrated for successive values of k, these two measures combine into an aggregate measure of social cohesion, suitable for both small-and large-scale network studies. Using these measures to define the core of a new methodology of cohesive blocking, we offer hypotheses about the consequences of cohesive blocks for social groups and their members, and explore empirical examples that illustrate the significance, theoretical relevance, and predictiveness of cohesive blocking in a variety of substantively important applications in sociology.

. The maximum flow problem is a fundamental problem in graph theory and combinatorial optimization with a variety of important applications. Known distributed algorithms for this problem do not tolerate faults or adjust to dynamic changes in network topology. This paper presents the first distributed self-stabilizing algorithm for the maximum flow problem. Starting from an arbitrary state, the algorithm computes the maximum flow in a acyclic network in finitely many steps. Since the algorithm is self-stabilizing, it is inherently tolerant to transient faults and can automatically adjust to topology changes and to changes in other parameters of the problem. The paper presents extensive experimental results to indicate that the algorithm requires n 2 moves in an average-case setting. A slight modification of the original algorithm is also presented and it is conjectured that the new algorithm computes a maximum flow in arbitrary networks. Key words. distributed algorithms, fault-toler...

A long-standing open question in information theory is to characterize the unicast capacity of a wireless relay network. The difficulty arises due to the complex signal interactions induced in the network, since the wireless channel inherently broadcasts the signals and there is interference among transmissions. Recently, Avestimehr, Diggavi and Tse proposed a linear binary deterministic model that takes into account the shared nature of wireless channels, focusing on the signal interactions rather than the background noise. They generalized the min-cut max-flow theorem for graphs to networks of deterministic channels and proved that the capacity can be achieved using information theoretical tools. They showed that the value of the minimum cut is in this case the minimum rank of all the binary adjacency matrices describing source-destination cuts. However, since there exists an exponential number of cuts, identifying the capacity through exhaustive search becomes infeasible. In this paper, we develop a polynomial time algorithm that discovers the relay encoding strategy to achieve the min-cut value in binary linear deterministic (wireless) networks, for the case of a unicast connection. Our algorithm crucially uses a notion of linear independence between edges to calculate the capacity in polynomial time. Moreover, we can achieve the capacity by using very simple onebit processing at the intermediate nodes, thereby constructively yielding finite length strategies that achieve the unicast capacity of the linear deterministic (wireless) relay network. 1

Algorithms are presented for the all-pairs min-cut problem in bounded tree-width, planar and sparse networks. The approach used is to preprocess the input n-vertex network so that, afterwards, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute mincuts subsequently. In particular, after an O(n log n) preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for such networks the all-pairs min-cut problem can be solved in time O(n 2 ). This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, fl, of the input network. The parameter fl varies between 1 and \Theta(n); the algorithms perform well when fl = o(n). The value of a min-cut can be found in t...

A layered approach for identifying communities in the Web is presented and explored by applying the Flake Exact Community Identification Algorithm to the UK academic Web. Although community or topic identification is a common task in information retrieval, a new perspective is developed by: (a) the application of Alternative Document Models, shifting the focus from individual pages to aggregated collections based upon Web directories, domains and entire sites; (b) the removal of internal site links; and (c) the adaptation of a new fast algorithm to allow fully automated community identification using all possible single starting points. The overall topology of the graphs in the three least aggregated layers was first investigated and found to include a large number of isolated points but, surprisingly, with most of the remainder being in one huge connected component, exact proportions varying by layer. The community identification process then found that the number of communities far exceeded the number of topological components, indicating that community identification is a potentially useful technique, even with random starting points. Both the number and size of communities identified was dependant on the parameter of the algorithm, with very different results being obtained in each case. In conclusion, the UK academic Web is embedded with layers of non-trivial communities and, if it is not unique in this, then there is the promise of (a) improved results for information retrieval algorithms that can exploit this additional structure, and (b) the application of the technique directly to partially automate Web metrics tasks such as that of finding all pages related to a given subject hosted by a single country’s universities.

We introduce a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearly-linear time. Using this approach, we develop the fastest known algorithm for computing approximately maximum s-t flows. For a graph having n vertices and m edges, our algorithm computes a (1−ɛ)approximately maximum s-t flow in time 1 Õ ( mn 1/3 ɛ −11/3). A dual version of our approach computes a (1 + ɛ)-approximately minimum s-t cut in time Õ ( m + n 4/3 ɛ −16/3) , which is the fastest known algorithm for this problem as well. Previously, the best dependence on m and n was achieved by the algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute approximately maximum s-t flows in time Õ ( m √ nɛ −1) , and approximately minimum s-t cuts in time Õ ( m + n 3/2 ɛ −3). Research partially supported by NSF grant CCF-0843915.