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20
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 139 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Graph Minor Theory
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2005
"... A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching ..."
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Cited by 17 (0 self)
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A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching generalization of this, conjectured by Wagner: If a class of graphs is minorclosed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs: If a minorclosed class of graphs does not contain all graphs, then every graph in it is glued together in a treelike fashion from graphs that can almost be embedded in a fixed surface. We describe the precise formulation of the main results and survey some of its applications to algorithmic and structural problems in graph theory.
Algorithmic Aspects Of Ordered Structures
, 1992
"... In this work we relate the theory of quasiorders to the theory of algorithms over some combinatorial objects. First we develope the theory of wellquasiorderings, wqo's, and relate it to the theory of worstcase complexity. Then we give a general 01law for hereditary properties that has implicat ..."
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Cited by 9 (2 self)
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In this work we relate the theory of quasiorders to the theory of algorithms over some combinatorial objects. First we develope the theory of wellquasiorderings, wqo's, and relate it to the theory of worstcase complexity. Then we give a general 01law for hereditary properties that has implications for average case complexity. This result on averagecase complexity is applied to the class of finite graphs equipped with the induced subgraph relation. We obtain that a wide class of problems, including e.g. perfectness, has average constant time algorithms. Then we show, by extending a result of Damaschke on cographs, that the classes of finite orders resp. graphs with bounded decomposition diameter form wqo's with respect to the induced suborder resp. induced subgraph relation. This leads to linear time algorithms for the recognition of any hereditary property on these objects. Our main result is then that the set of finite posets is a wqo with respect to a certain relation , calle...
Graph Minors and Graphs on Surfaces
, 2001
"... Graph minors and the theory of graphs embedded in surfaces are ..."
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Cited by 8 (3 self)
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Graph minors and the theory of graphs embedded in surfaces are
Terminal backup, 3d matching, and covering cubic graphs
 In Proceedings of the 39th Annual ACM Symposium on Theory of Computing. ACM
, 2007
"... Abstract. We define a problem called Simplex Matching and show that it is solvable in polynomial time. While Simplex Matching is interesting in its own right as a nontrivial extension of nonbipartite mincost matching, its main value lies in many (seemingly very different) problems that can be solve ..."
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Cited by 7 (4 self)
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Abstract. We define a problem called Simplex Matching and show that it is solvable in polynomial time. While Simplex Matching is interesting in its own right as a nontrivial extension of nonbipartite mincost matching, its main value lies in many (seemingly very different) problems that can be solved using our algorithm. For example, suppose that we are given a graph with terminal nodes, nonterminal nodes, and edge costs. Then, the Terminal Backup problem, which consists of finding the cheapest forest connecting every terminal to at least one other terminal, is reducible to Simplex Matching. Simplex Matching is also useful for various tasks that involve forming groups of at least two members, such as project assignment and variants of facility location. In an instance of Simplex Matching, we are given a hypergraph H with edge costs and edge size at most 3. We show how to find the mincost perfect matching of H efficiently if the edge costs obey a simple and realistic inequality that we call the Simplex Condition. The algorithm we provide is relatively simple to understand and implement but difficult to prove correct. In the process of this proof we show some powerful new results about covering cubic graphs with simple combinatorial objects.
Algorithmic graph minor theory: Improved grid minor bounds and wagner’s contraction
 Proceedings of the Third International Conference on Distributed Computing and Internet Technology
, 2006
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On possible counterexamples to Negami's planar cover conjecture
 J. Graph Theory
, 1999
"... A simple graph H is a cover of a graph G if there exists a mapping ϕ from H onto G such that ϕ maps the neighbors of every vertex v in H bijectively to the neighbors of ϕ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective p ..."
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Cited by 5 (3 self)
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A simple graph H is a cover of a graph G if there exists a mapping ϕ from H onto G such that ϕ maps the neighbors of every vertex v in H bijectively to the neighbors of ϕ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, and the first author that the conjecture holds as long as the graph K1,2,2,2 has no finite planar cover. However, those results seem to say little about counterexamples if the conjecture was not true. We show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami’s conjecture. Moreover, we exhibit a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in our list.
Contributions to Parameterized Complexity
, 2003
"... This thesis is presented in two parts. In Part One we concentrate on algorithmic aspects of parameterized complexity. We explore ways in which the concepts and algorithmic techniques of parameterized complexity can be fruitfully brought to bear on a (classically) wellstudied problem area, that of s ..."
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Cited by 4 (3 self)
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This thesis is presented in two parts. In Part One we concentrate on algorithmic aspects of parameterized complexity. We explore ways in which the concepts and algorithmic techniques of parameterized complexity can be fruitfully brought to bear on a (classically) wellstudied problem area, that of scheduling problems modelled on partial orderings. We develop e#cient and constructive algorithms for parameterized versions of some classically intractable scheduling problems.
Embedding a Graph Into the Torus in Linear Time
, 1994
"... A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are ..."
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Cited by 4 (0 self)
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A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are given an embedding of K in some surface. The embedding extension problem asks whether it is embedding extension problem possible to extend the embedding of K to an embedding of G in the same surface, and any such embedding is an embedding extension of K to G. An embedding extension obstruction for embedding extensions is a subgraph\Omega of G \Gamma E(K) such that obstruction the embedding of K cannot be extended to K [ \Omega\Gamma The obstruction is small small if K [\Omega is homeomorphic to a graph with a small number of edges. If\Omega is small, then it is easy to verify (for example, by checking all the possibilities Supported in part by the Ministry of Science and Technolo...
NonPlanar Extensions Of Planar Graphs
, 1999
"... A graph G is almost 4connected if it is simple, 3connected, has at least five vertices, and V (G) cannot be partitioned into three sets A; B; C in such a way that jCj = 3, jAj 2, jBj 2, and no edge of G has one end in A and the other end in B. A graph K is a subdivision of a graph G if K is ob ..."
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Cited by 2 (0 self)
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A graph G is almost 4connected if it is simple, 3connected, has at least five vertices, and V (G) cannot be partitioned into three sets A; B; C in such a way that jCj = 3, jAj 2, jBj 2, and no edge of G has one end in A and the other end in B. A graph K is a subdivision of a graph G if K is obtained from G by replacing its edges by internally disjoint nonzero length paths with the same ends, called segments. Let G; H be almost 4connected graphs such that G is planar, H is nonplanar, and H has a subgraph isomorphic to a subdivision of G. If K is a subgraph of H, then a Kpath in H is a path with at least one edge, both ends in K, and otherwise disjoint from K. We prove that there exists a subgraph K of H isomorphic to a subdivision of G such that one of the following conditions holds. (i) There exists a Kpath in H such that no region boundary of K contains both ends of the path. (ii) There exist two disjoint Kpaths with ends s 1 ; t 1 and s 2 ; t 2 , respectively, such that the...