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18
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Drawing Graphs Nicely Using Simulated Annealing
, 1996
"... The paradigm of simulated annealing is applied to the problem of drawing graphs "nicely." Our algorithm deals with general graphs with straighline edges, and employs several simple criteria for the aesthetic quality of the result. The algorithm is flexible, in that the relative weights of the crite ..."
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Cited by 174 (11 self)
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The paradigm of simulated annealing is applied to the problem of drawing graphs "nicely." Our algorithm deals with general graphs with straighline edges, and employs several simple criteria for the aesthetic quality of the result. The algorithm is flexible, in that the relative weights of the criteria can be changed. For graphs of modest size it produces good results, competitive with those produced by other methods, notably, the "spring method" and its variants.
Ensuring the Drawability of Extended Euler Diagrams for up to 8 Sets
 PROC. DIAGRAMS 2004. LNAI 2980
, 2003
"... This paper shows by a constructive method the existence of a diagrammatic representation called extended Euler diagrams for any collection of sets X1 , ..., Xn , n 9. These diagrams are adapted for representing sets inclusions and intersections: each set X i and each non empty intersection of a sub ..."
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Cited by 16 (2 self)
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This paper shows by a constructive method the existence of a diagrammatic representation called extended Euler diagrams for any collection of sets X1 , ..., Xn , n 9. These diagrams are adapted for representing sets inclusions and intersections: each set X i and each non empty intersection of a subcollection of X1 , ..., Xn is represented by a unique connected region of the plane. Starting with an abstract description of the diagram, we define the dual graph G and reason with the properties of this graph to build a planar representation of the X1 , ..., Xn . These diagrams will be used to visualize the results of a complex request on any indexed video databases. In fact, such a representation allows the user to perceive simultaneously the results of his query and the relevance of the database according to the query.
Subdivision Drawings of Hypergraphs
"... We introduce the concept of subdivision drawings of hypergraphs. In a subdivision drawing each vertex corresponds uniquely to a face of a planar subdivision and, for each hyperedge, the union of the faces corresponding to the vertices incident to that hyperedge is connected. Vertexbased Venn diag ..."
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Cited by 8 (3 self)
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We introduce the concept of subdivision drawings of hypergraphs. In a subdivision drawing each vertex corresponds uniquely to a face of a planar subdivision and, for each hyperedge, the union of the faces corresponding to the vertices incident to that hyperedge is connected. Vertexbased Venn diagrams and concrete Euler diagrams are both subdivision drawings. In this paper we study two new types of subdivision drawings which are more general than concrete Euler diagrams and more restricted than vertexbased Venn diagrams. They allow us to draw more hypergraphs than the former while having better aesthetic properties than the latter.
Algorithms for the hypergraph and the minor crossing number problems
 In Proc. ISAAC’07, volume 4835 of LNCS
, 2007
"... Abstract. We consider the problems of hypergraph and minor crossing minimization, and point out a relation between these two problems which has not been exploited before. We present some complexity results regarding the corresponding edge and node insertion problems. Based on these results, we give ..."
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Cited by 6 (4 self)
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Abstract. We consider the problems of hypergraph and minor crossing minimization, and point out a relation between these two problems which has not been exploited before. We present some complexity results regarding the corresponding edge and node insertion problems. Based on these results, we give the first embeddingbased heuristics to tackle both problems and present a short experimental study. Furthermore, we give the first exact ILP formulation for both problems. 1
On Planar Supports for Hypergraphs
"... A graph G is a support for a hypergraph H = (V, S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si ∈ S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [9] proved that it is NPcomplete to decide ..."
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Cited by 5 (0 self)
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A graph G is a support for a hypergraph H = (V, S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si ∈ S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [9] proved that it is NPcomplete to decide if a given hypergraph has a planar support. In contrast, there are polynomial time algorithms to test whether a given hypergraph has a planar support that is a path, cycle, or tree. In this paper we present an algorithm which tests in polynomial time if a given hypergraph has a planar support that is a tree where the maximal degree of each vertex is bounded. Our algorithm is constructive and computes a support if it exists. Furthermore, we prove that it is already NPhard to decide if a hypergraph has a 2outerplanar support. 1
PathBased Supports for Hypergraphs
, 2010
"... A pathbased support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a Hamiltonian subgraph. While it is N Pcomplete to compute a pathbased support with the minimum number of edges or to decide whether there is a planar pathbased support, we show that a ..."
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Cited by 4 (1 self)
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A pathbased support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a Hamiltonian subgraph. While it is N Pcomplete to compute a pathbased support with the minimum number of edges or to decide whether there is a planar pathbased support, we show that a pathbased tree support can be computed in polynomial time if it exists.
Multidimensional Interval Routing Schemes
, 2001
"... Routing messages between pairs of nodes is one of the most fundamental tasks in any distributed computing system. An Interval Routing Scheme (IRS) is a wellknown, spaceefficient routing strategy for routing messages in a network. In this scheme, each node of the network is assigned an integer labe ..."
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Cited by 2 (1 self)
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Routing messages between pairs of nodes is one of the most fundamental tasks in any distributed computing system. An Interval Routing Scheme (IRS) is a wellknown, spaceefficient routing strategy for routing messages in a network. In this scheme, each node of the network is assigned an integer label and each link at each node is labeled with an interval. The interval assigned to a link l at a node v indicates the set of destination addresses of the messages which should be forwarded through l at v. When studying interval routing schemes, there are two main problems to be considered: a) Which classes of networks do support a specific routing scheme? b) Assuming that a given network supports IRS, how good are the paths traversed by messages? The first problem is known as the characterization problem and has been studied for several types of IRS. In this thesis, we study the characterization problem for various schemes in which the labels assigned to the vertices are dary integer tuples (ddimensional IRS) and the label assigned to each link of the network is a list of d 1dimensional intervals. This is known as Multidimensional IRS (MIRS) and is an extension of the the original IRS. We completely characterize the class of network which support MIRS for linear (which has no cyclic intervals) and strict (which has no intervals assigned to a link at a node v containing the label of v) MIRS. In real networks usually the costs of links may vary over time (dynamic cost links). We also give a complete characterization for the class of networks which support a certain type of MIRS which routes all messages on shortest paths in a network with dynamic cost links. The main criterion used to measure the quality of routing (the second problem) is the length of routing paths. In this the...
Blocks of Hypergraphs  applied to Hypergraphs and Outerplanarity
, 2010
"... A support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a connected subgraph. We show how to test in polynomial time whether a given hypergraph has a cactus support, i.e. a support that is a tree of edges and cycles. While it is N Pcomplete to decide w ..."
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Cited by 2 (1 self)
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A support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a connected subgraph. We show how to test in polynomial time whether a given hypergraph has a cactus support, i.e. a support that is a tree of edges and cycles. While it is N Pcomplete to decide whether a hypergraph has a 2outerplanar support, we show how to test in polynomial time whether a hypergraph that is closed under intersections and differences has an outerplanar or a planar support. In all cases our algorithms yield a construction of the required support if it exists. The algorithms are based on a new definition of biconnected components in hypergraphs.
Directed Hypergraph Planarity
, 2001
"... Directed hypergraphs are generalizations of digraphs and can be used to model binary relations among subsets of a given set. Planarity of hypergraphs was studied by Johnson and Pollak; planarity of directed hypergraphs was studied by Makinen, being assumed a restricted definition. In this paper we e ..."
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Cited by 1 (1 self)
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Directed hypergraphs are generalizations of digraphs and can be used to model binary relations among subsets of a given set. Planarity of hypergraphs was studied by Johnson and Pollak; planarity of directed hypergraphs was studied by Makinen, being assumed a restricted definition. In this paper we extend the planarity concept to directed hypergraphs.It is well known that the planarity of a digraph relies on the planarity of its underlying graph. However, for directed hypergraphs, this property cannot be applied and we propose a new approach which generalizes the usual concept. We also show that the recognition of the planarity for directed hypergraphs is linear.