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Approximation algorithms for edgedisjoint paths and unsplittable flow
 Efficient Approximation and Online Algorithms
, 2006
"... Abstract. In the maximum edgedisjoint paths problem (MEDP) the input consists of a graph and a set of requests (pairs of vertices), and the goal is to connect as many requests as possible along edgedisjoint paths. We give a survey of known results about the complexity and approximability of MEDP ..."
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Abstract. In the maximum edgedisjoint paths problem (MEDP) the input consists of a graph and a set of requests (pairs of vertices), and the goal is to connect as many requests as possible along edgedisjoint paths. We give a survey of known results about the complexity and approximability of MEDP and sketch some of the main ideas that have been used to obtain approximation algorithms for the problem. We consider also the generalization of MEDP where the edges of the graph have capacities and each request has a profit and a demand, called the unsplittable flow problem.
Routing Permutations and 21 Routing Requests in the Hypercube
, 2000
"... Let H n be the directed symmetric ndimensional hypercube. Using the computer, we show that for any permutation of the vertices of H 4 , there exists a system of pairwise arcdisjoint directed paths from each vertex to its target in the permutation. This veries Szymanski's conjecture [8] for n ..."
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Let H n be the directed symmetric ndimensional hypercube. Using the computer, we show that for any permutation of the vertices of H 4 , there exists a system of pairwise arcdisjoint directed paths from each vertex to its target in the permutation. This veries Szymanski's conjecture [8] for n = 4.
Routing Permutations in the Hypercube
, 1999
"... We study an ndimensional directed symmetric hypercube Hn , in which every pair of adjacent vertices is connected by two arcs of opposite directions. Using the computer, we show that for H4 and for any permutation on its vertices, there exists a system of pairwise arcdisjoint directed paths from ea ..."
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We study an ndimensional directed symmetric hypercube Hn , in which every pair of adjacent vertices is connected by two arcs of opposite directions. Using the computer, we show that for H4 and for any permutation on its vertices, there exists a system of pairwise arcdisjoint directed paths from each vertex to its target in the permutation. This gives the answer to Szymanski's conjecture [Szy89] for dimension 4. In addition to this study, we consider in Hn the socalled 21 routing requests, that is routing requests where any vertex of Hn can be used twice as a source, but only once as a target. We give two such routing requests which cannot be routed in H3 . Moreover, we show that for any dimension n 3, it is possible to find a 21 routing request gn such that gn cannot be routed in Hn : in other words, for any n 3, Hn is not (21)rearrangeable. Keywords : Hypercubes, routing permutations, Szymanski's conjecture, 21 routing requests. 1 Introduction The directed symmetric hyper...
On the Oblivious Circuit Switching in MultiLink Binary Hypercubes
"... This paper considers the necessary conditions for the existence of oblivious circuit switching in multilink binary hypercubes. It is also shown that 1(2,3,4,7)link edgedisjoint, manyone routing exists in the 1(2,3,4,5)cube. Modular destination graphs are decomposed for the said number of links, ..."
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This paper considers the necessary conditions for the existence of oblivious circuit switching in multilink binary hypercubes. It is also shown that 1(2,3,4,7)link edgedisjoint, manyone routing exists in the 1(2,3,4,5)cube. Modular destination graphs are decomposed for the said number of links, where the physical paths are obtained using quasidimensionorder (cyclic) routing. The basic requirements in obtaining manyone modular destination graphs are discussed. As a result of the combination of two manyone graphs of the same dimension, it directly follows that 1(2,3,4,7)link, edgedisjoint permutation routing exists in the 2(4,6,8,10)cube. Also, by combining two, manyone graphs of different dimensions, one can obtain (2,3,4,7)link edgedisjoint permutation routing in the (3,5,7,9)cube.
Embedding multidimensional grids into optimal hypercubes
, 2014
"... Let G and H be graphs, with V (H)  ≥ V (G), and f: V (G) → V (H) a one to one map of their vertices. Let dilation(f) = max{distH(f(x), f(y)) : xy ∈ E(G)}, where distH(v, w) is the distance between vertices v and w of H. Now let B(G,H) = minf{dilation(f)}, over all such maps f. The parameter B ..."
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Let G and H be graphs, with V (H)  ≥ V (G), and f: V (G) → V (H) a one to one map of their vertices. Let dilation(f) = max{distH(f(x), f(y)) : xy ∈ E(G)}, where distH(v, w) is the distance between vertices v and w of H. Now let B(G,H) = minf{dilation(f)}, over all such maps f. The parameter B(G,H) is a generalization of the classic and well studied “bandwidth ” of G, defined as B(G,P (n)), where P (n) is the path on n points and n = V (G). Let [a1 × a2 × · · · × ak] be the kdimensional grid graph with integer values 1 through ai in the i’th coordinate. In this paper, we study B(G,H) in the case when G = [a1 × a2 × · · · × ak] and H is the hypercube Qn of dimension n = dlog2(V (G))e, the hypercube of smallest dimension having at least as many points as G. Our main result is that B([a1 × a2 × · · · × ak], Qn) ≤ 3k, provided ai ≥ 222 for each 1 ≤ i ≤ k. For such G, the bound 3k improves on the previous best upper bound 4k + O(1). Our methods include an application of Knuth’s result on twoway rounding and of the existence of spanning regular cyclic caterpillars in the hypercube. 1