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Efficient Exact Inference in Planar Ising Models
"... We give polynomialtime algorithms for the exact computation of lowestenergy states, worst margin violators, partition functions, and marginals in certain binary undirected graphical models. Our approach provides an interesting alternative to the wellknown graph cut paradigm in that it does not im ..."
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We give polynomialtime algorithms for the exact computation of lowestenergy states, worst margin violators, partition functions, and marginals in certain binary undirected graphical models. Our approach provides an interesting alternative to the wellknown graph cut paradigm in that it does not impose any submodularity constraints; instead we require planarity to establish a correspondence with perfect matchings in an expanded dual graph. Maximummargin parameter estimation for a boundary detection task shows our approach to be efficient and effective. 1
Packet Recycling: Eliminating Packet Losses due to Network Failures
"... This paper presents Packet Recycling (PR), a technique that takes advantage of cellular graph embeddings to reroute packets that would otherwise be dropped in case of link or node failures. The technique employs only one bit in the packet header to cover any single link failures, and in the order o ..."
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This paper presents Packet Recycling (PR), a technique that takes advantage of cellular graph embeddings to reroute packets that would otherwise be dropped in case of link or node failures. The technique employs only one bit in the packet header to cover any single link failures, and in the order of log2 (d) bits to cover all nondisconnecting failure combinations, where d is the diameter of the network. We show that our routing strategy is effective and that its path length stretch is acceptable for realistic topologies. The packet header overhead incurred by PR is very small, and the extra memory and packet processing time required to implement it at each router are insignificant. This makes PR suitable for losssensitive, missioncritical network applications. Categories and Subject Descriptors
Testing Planarity of Partially Embedded Graphs
, 2009
"... We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in man ..."
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Cited by 14 (4 self)
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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomialtime solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “oncas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach lineartime complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NPhard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.
Maximum matching in graphs with an excluded minor
 Proceedings of the Eighteenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA) 108–117
, 2007
"... Abstract We present a new randomized algorithm for findinga maximum matching in Hminor free graphs. Forevery fixed H, our algorithm runs in O(n3!/(!+3)) < O(n1.326) time, where n is the number of verticesof the input graph and! < 2.376 is the exponentof matrix multiplication. This improves up ..."
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Cited by 7 (5 self)
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Abstract We present a new randomized algorithm for findinga maximum matching in Hminor free graphs. Forevery fixed H, our algorithm runs in O(n3!/(!+3)) < O(n1.326) time, where n is the number of verticesof the input graph and! < 2.376 is the exponentof matrix multiplication. This improves upon the previous O(n1.5) time bound obtained by applying the O(mn1/2)time algorithm of Micali and Vazirani on thisimportant class of graphs. For graphs with bounded genus, which are special cases of Hminor free graphs, we present a randomized algorithm for finding a maximum matching in O(n!/2) < O(n1.19) time. This extends a previous randomized algorithm of Mucha and Sankowski, having the same running time, that finds a maximum matching ina planar graphs. We also present a deterministic algorithm with arunning time of O(n1+!/2) < O(n2.19) for counting thenumber of perfect matchings in graphs with bounded genus. This algorithm combines the techniques usedby the algorithms above with the counting technique of Kasteleyn. Using this algorithm we can also count,within the same running time, the number of Tjoinsin planar graphs. As special cases, we get algorithms for counting Eulerian subgraphs (T = OE) and oddsubgraphs ( T = V) of planar graphs. 1 Introduction A matching in a graph is a set of pairwise disjointedges. A perfect matching in a graph with n verticesis a matching of size n/2, and a maximum matchingis a matching of largest possible size. The problems
Analyzing the Facebook Friendship Graph
 In Proceedings of the 1st Workshop on Mining the Future Internet
, 2010
"... Abstract. Online Social Networks (OSN) during last years acquired a huge and increasing popularity as one of the most important emerging Web phenomena, deeply modifying the behavior of users and contributing to build a solid substrate of connections and relationships among people using the Web. In t ..."
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Abstract. Online Social Networks (OSN) during last years acquired a huge and increasing popularity as one of the most important emerging Web phenomena, deeply modifying the behavior of users and contributing to build a solid substrate of connections and relationships among people using the Web. In this preliminary work paper, our purpose is to analyze Facebook, considering a significant sample of data reflecting relationships among subscribed users. Our goal is to extract, from this platform, relevant information about the distribution of these relations and exploit tools and algorithms provided by the Social Network Analysis (SNA) to discover and, possibly, understand underlying similarities between the developing of OSN and reallife social networks.
Planarity testing and optimal edge insertion with embedding constraints
, 2008
"... The planarization method has proven to be successful in graph drawing. The output, a combinatorial planar embedding of the socalled planarized graph, can be combined with stateoftheart planar drawing algorithms. However, many practical applications have additional constraints on the drawings tha ..."
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Cited by 6 (2 self)
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The planarization method has proven to be successful in graph drawing. The output, a combinatorial planar embedding of the socalled planarized graph, can be combined with stateoftheart planar drawing algorithms. However, many practical applications have additional constraints on the drawings that result in restrictions on the set of admissible planar embeddings. In this paper, we consider embedding constraints that restrict the admissible order of incident edges around a vertex. Such constraints occur in applications, e.g., from side or port constraints. We introduce a set of hierarchical embedding constraints that include grouping, oriented, and mirror constraints, and show how these constraints can be integrated into the planarization method. For this, we first present a linear time algorithm for testing if a given graph G is ecplanar, i.e., admits a planar embedding satisfying the given embedding constraints. In the case that G is ecplanar, we provide a linear time algorithm for computing the corresponding ecembedding. Otherwise, an ecplanar subgraph is computed. The critical part is to reinsert the deleted edges subject to the embedding constraints so that the number of crossings is kept small. For this, we present a linear time algorithm which is able to insert an edge into an ecplanar graph H so that the insertion is crossing minimal among all ecplanar embeddings of H. As a side result, we characterize the set of all possible ecplanar embeddings using BC and SPQRtrees.
A new approach to exact crossing minimization
 In Proc. ESA ’08, volume 5193 of LNCS
, 2008
"... Abstract. The crossing number problem is to find the smallest number of edge crossings necessary when drawing a graph into the plane. Eventhough the problem is NPhard, we are interested in practically efficient algorithms to solve the problem to provable optimality. In this paper, we present a nove ..."
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Abstract. The crossing number problem is to find the smallest number of edge crossings necessary when drawing a graph into the plane. Eventhough the problem is NPhard, we are interested in practically efficient algorithms to solve the problem to provable optimality. In this paper, we present a novel integer linear programming (ILP) formulation for the crossing number problem. The former formulation [4] had to transform the crossing number polytope into a higherdimensional polytope. The key idea of our approach is to directly consider the natural crossing number polytope and cut it with multiple linearordering polytopes. This leads to a more compact formulation, both in terms of variables and constraints. We describe a BranchandCut algorithm, together with a combinatorial column generation scheme, in order to solve the crossing number problem to provable optimality. Our experiments show that the new approach is more effective than the old one, even when considering a heavily improved version of the former formulation (also presented in this paper). For the first time, we are able to solve graphs with a crossing number of up to 37. 1
Linear time planarity testing and embedding of strongly connected cyclic level graphs
"... A level graph is a directed acyclic graph with a level assignment for each node. Such graphs play a prominent role in graph drawing. They express strict dependencies and occur in many areas, e.g., in scheduling problems and program inheritance structures. In this paper we extend level graphs to cyc ..."
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A level graph is a directed acyclic graph with a level assignment for each node. Such graphs play a prominent role in graph drawing. They express strict dependencies and occur in many areas, e.g., in scheduling problems and program inheritance structures. In this paper we extend level graphs to cyclic level graphs. Such graphs occur as repeating processes in cyclic scheduling, visual data mining, life sciences, and VLSI. We provide a complete study of strongly connected cyclic level graphs. In particular, we present a linear time algorithm for the planarity testing and embedding problem, and we characterize forbidden subgraphs. Our results generalize earlier work on level graphs.