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28
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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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
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 140 (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.
Finding Regular Simple Paths In Graph Databases
, 1989
"... We consider the following problem: given a labelled directed graph G and a regular expression R, find all pairs of nodes connected by a simple path such that the concatenation of the labels along the path satisfies R. The problem is motivated by the observation that many recursive queries in relatio ..."
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Cited by 109 (5 self)
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We consider the following problem: given a labelled directed graph G and a regular expression R, find all pairs of nodes connected by a simple path such that the concatenation of the labels along the path satisfies R. The problem is motivated by the observation that many recursive queries in relational databases can be expressed in this form, and by the implementation of a query language, G+ , based on this observation. We show that the problem is in general intractable, but present an algorithm than runs in polynomial time in the size of the graph when the regular expression and the graph are free of conflicts. We also present a class of languages whose expressions can always be evaluated in time polynomial in the size of both the graph and the expression, and characterize syntactically the expressions for such languages. Key words. Labelled directed graphs, NPcompleteness, polynomialtime algorithms, regular expressions, simple paths AMS(MOS) subject classifications. 68P, 6...
Hadwiger’s conjecture for K6free graphs
 COMBINATORICA
, 1993
"... In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ..."
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Cited by 36 (2 self)
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In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ≥ 5 it has remained open. Here we show that when t = 5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger’s conjecture when t = 5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger’s conjecture when t = 5, because it implies that apex graphs are 5colourable.
Rectilinear Paths among Rectilinear Obstacles
 Discrete Applied Mathematics
, 1996
"... Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations resear ..."
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Cited by 23 (3 self)
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Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations research. In this survey we emphasize its applications to VLSI design and limit ourselves to the rectilinear domain in which the goal path to be computed and the underlying obstacles are all rectilinearly oriented, i.e., the segments are either horizontal or vertical. We consider different routing environments, and various optimization criteria pertaining to VLSI design, and provide a survey of results that have been developed in the past, present current results and give open problems for future research. 1 Introduction Given a set of obstacles and two distinguished points in the plane, the problem of finding a collision free path subject to a certain optimization function is a fundamental probl...
An Improved Linear Edge Bound for Graph Linkages
 EUROP. J. COMBINATORICS
, 2004
"... A graph is said to be klinked if it has at least 2k vertices and for every sequence s1,...,s k,t 1,...,t k of distinct vertices there exist disjoint paths P1,...,P k such that the ends of P i are s i and t i . Bollobas and Thomason showed that if a simple graph G on n vertices is 2kconnected and ..."
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Cited by 21 (2 self)
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A graph is said to be klinked if it has at least 2k vertices and for every sequence s1,...,s k,t 1,...,t k of distinct vertices there exist disjoint paths P1,...,P k such that the ends of P i are s i and t i . Bollobas and Thomason showed that if a simple graph G on n vertices is 2kconnected and G has at least 11kn edges, then G is klinked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2kconnectivity suffice, and then with more effort improve the edge bound to 5kn.
Parameterized tractability of edgedisjoint paths on directed acyclic graphs
 Proceedings of the 11th Annual European Symposium on Algorithms, ESA ’03, volume 2832 of Lecture Notes in Computer Science
, 2003
"... Given a graph and terminal pairs (si, ti), i ∈ [k], the edgedisjoint paths problem is to determine whether there exist siti paths, i ∈ [k], that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NPcomplete and solvable in time n O(k) where n is the number of n ..."
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Cited by 12 (1 self)
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Given a graph and terminal pairs (si, ti), i ∈ [k], the edgedisjoint paths problem is to determine whether there exist siti paths, i ∈ [k], that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NPcomplete and solvable in time n O(k) where n is the number of nodes. It has been a longstanding open question whether it is fixedparameter tractable in k, i.e. whether it admits an algorithm with running time of the form f(k) n O(1). We resolve this question in the negative: we show that the problem is W [1]hard, hence unlikely to be fixedparameter tractable. In fact it remains W [1]hard even if the demand graph consists of two sets of parallel edges. On a positive side, we give an O(m+k O(1) k! n) algorithm for the special case when G is acyclic and G + H is Eulerian, where H is the demand graph. We generalize this result (1) to the case when G + H is “nearly ” Eulerian, (2) to an analogous special case of the unsplittable flow problem, a generalized version of disjoint paths that has capacities and demands. Keywords. Disjoint paths, fixedparameter tractability, W[1]hardness, Eulerian graphs, unsplittable flow.
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.
The 1FixedEndpoint Path Cover Problem is Polynomial on Interval Graphs
, 2009
"... We consider a variant of the path cover problem, namely, the kfixedendpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T of k vertices of V(G),akfixedendpoint path cover of G with respect to T is a set of vertexdisjoint paths P that covers the vertic ..."
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Cited by 5 (1 self)
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We consider a variant of the path cover problem, namely, the kfixedendpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T of k vertices of V(G),akfixedendpoint path cover of G with respect to T is a set of vertexdisjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. ThekPC problem is to find a kfixedendpoint path cover of G of minimum cardinality; note that, if T is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1fixedendpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NPcomplete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112:49– 64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomialtime algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.
Rooted Routing in the Plane
, 1993
"... this paper, we discuss a lineartime algorithm for instances of krealizations in which G is a planar graph (a graph is planar if it can be drawn in the plane so that its edges do not cross). This algorithm is due to Reed, Robertson, Schrijver, and Seymour [4] (see also [5]). We will also discuss ho ..."
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this paper, we discuss a lineartime algorithm for instances of krealizations in which G is a planar graph (a graph is planar if it can be drawn in the plane so that its edges do not cross). This algorithm is due to Reed, Robertson, Schrijver, and Seymour [4] (see also [5]). We will also discuss how to generalize this algorithm to more complicated surfaces and make some remarks about Robertson and Seymour's algorithm for krealizations in arbitrary graphs.