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The Complexity of Path Coloring and Call Scheduling
 Theoretical Computer Science
, 2000
"... Modern highperformance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests. In alloptical networks with wavelengthdivision multiplexing, connection requests must be assigned paths and colors (wavelengths) such that i ..."
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Cited by 22 (6 self)
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Modern highperformance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests. In alloptical networks with wavelengthdivision multiplexing, connection requests must be assigned paths and colors (wavelengths) such that intersecting paths receive different colors, and the goal is to minimize the number of colors used. This path coloring problem is proved NPhard for undirected and bidirected ring networks. Path coloring in undirected tree networks is shown to be equivalent to edge coloring of multigraphs, which implies a polynomialtime optimal algorithm for trees of constant degree as well as NPhardness and an approximation algorithm with absolute approximation ratio 4:3 and asymptotic approximation ratio 1:1 for trees of arbitrary degree. For bidirected trees, path coloring is shown to be NPhard even in the binary case. A polynomialtime optimal algorithm is given for path coloring in undirected or bidir...
The Maximum EdgeDisjoint Paths Problem In Bidirected Trees
 SIAM Journal on Discrete Mathematics
, 1998
"... . A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edgedisjoint paths problem is to select a maximumcardinality subse ..."
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Cited by 17 (3 self)
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. A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edgedisjoint paths problem is to select a maximumcardinality subset of the paths such that the selected paths are edgedisjoint. This problem can be solved optimally in polynomial time for bidirected trees of constant degree, but is MAXSNPhard for bidirected trees of arbitrary degree. For every fixed " ? 0, a polynomialtime (5=3+ ")approximation algorithm is presented. Key words. approximation algorithms, edgedisjoint paths, bidirected trees AMS subject classifications. 68Q25, 68R10 1. Introduction. Research on disjoint paths problems in graphs has a long history [12]. In recent years, edgedisjoint paths problems have been brought into the focus of attention by advances in the field of communication networks. Many modern network architectures estab...
The PermutationPath Coloring Problem on Trees
, 2000
"... In this paper we first show that the permutationpath coloring problem is NPhard even for very restrictive instances like involutions, which are permutations that contain only cycles of length at most two, on both binary trees and on trees having only two vertices with degree greater than two, and ..."
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Cited by 6 (0 self)
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In this paper we first show that the permutationpath coloring problem is NPhard even for very restrictive instances like involutions, which are permutations that contain only cycles of length at most two, on both binary trees and on trees having only two vertices with degree greater than two, and for circular permutations which are permutations that contain exactly one cycle on trees with maximum degree greater or equal to 4. We calculate a lower bound on the average complexity of the permutationpath coloring problem on arbitrary networks. Then we give combinatorial and asymptotic results for the permutationpath coloring problem on linear networks in order to show that the average number of colors needed to color any permutation on a linear network on n vertices is n=4 o(n). We extend these results and obtain an upper bound on the average complexity of the permutationpath coloring problem on arbitrary trees, obtaining exact results in the case of generalized star trees. Finally we explain how to extend these results for the involutionspath coloring problem on arbitrary trees.
A short proof of the NPcompleteness of circular arc coloring
, 2003
"... Coloring circular arcs was shown to be NPcomplete by Garey, Johnson, Miller and Papadimitriou [5]. Here we present a simpler proof of this result. ..."
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Cited by 2 (0 self)
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Coloring circular arcs was shown to be NPcomplete by Garey, Johnson, Miller and Papadimitriou [5]. Here we present a simpler proof of this result.
Resource Allocation in Bounded Degree Trees ∗
, 2007
"... We study the bandwidth allocation problem (bap) in bounded degree trees. In this problem we are given a tree and a set of connection requests. Each request consists of a path in the tree, a bandwidth requirement, and a weight. Our goal is to find a maximum weight subset S of requests such that, for ..."
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Cited by 1 (0 self)
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We study the bandwidth allocation problem (bap) in bounded degree trees. In this problem we are given a tree and a set of connection requests. Each request consists of a path in the tree, a bandwidth requirement, and a weight. Our goal is to find a maximum weight subset S of requests such that, for every edge e, the total bandwidth of requests in S whose path contains e is at most 1. We also consider the storage allocation problem (sap), in which it is also required that every request in the solution is given the same contiguous portion of the resource in every edge in its path. We present a deterministic approximation algorithm for bap in bounded degree trees with ratio (2 √ e − 1)/ ( √ e − 1) + ε < 3.542. Our algorithm is based on a novel application of the local ratio technique in which the available bandwidth is divided into narrow strips and requests with very small bandwidths are allocated in these strips. We also present a randomized (2 + ε)approximation algorithm for bap in bounded degree trees. The best previously known ratio for bap in general trees is 5. We present a reduction from sap to bap that works for instances where the tree is a line and the bandwidths are very small. It follows that there exists a deterministic 2.582approximation algorithm and a randomized (2 + ε)approximation algorithm for sap in the line. The best previously known ratio for this problem is 7.
A short proof of the NPcompleteness of circular arc coloring
"... Coloring circular arcs was shown to be NPcomplete by Garey, Johnson, Miller and Papadimitriou [5]. Here we present a simpler proof of this result. ..."
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Coloring circular arcs was shown to be NPcomplete by Garey, Johnson, Miller and Papadimitriou [5]. Here we present a simpler proof of this result.
The PermutationPath Coloring Problem on Trees
, 2000
"... The paper deals with the problem of routing a set of communication requests representing a permutation of the nodes of an alloptical tree shaped network employing the wavelength division multiplexing (or WDM) technology. In such networks, information between nodes is transmitted as light on fibe ..."
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The paper deals with the problem of routing a set of communication requests representing a permutation of the nodes of an alloptical tree shaped network employing the wavelength division multiplexing (or WDM) technology. In such networks, information between nodes is transmitted as light on fiberoptic lines without being converted to electronic form in between, and different messages may use the same link concurrently if and only if they are assigned distinct wavelengths. Thus, the goal of the routing problem on these networks is to assign a wavelength to each communication request in order to minimize the number of wavelengths needed to perform all communications in only one round. Such a routing problem can be modeled as a permutationpath coloring problem on trees. An instance of the permutationpath coloring problem on trees is given by a directed symmetric tree graph T on n nodes and a permutation oe of the vertex set of T . Moreover, we associate with each pair (i; oe(i)), i 6= oe(i), 1 i n, the unique directed path on T from vertex i to vertex oe(i). Thus, the permutationpath coloring problem for this instance consists in assigning the minimum number of colors to such a permutationset of paths in such a way that any two paths sharing a same arc of the tree are assigned different colors. In fact, the colors in the latest problem represents the wavelengths in the former one. In this paper we first show that the permutationpath coloring problem is NPhard even in the case of involutions (resp. circular permutations) , that are permutations which contain only cycles of length at most two (resp.
Tree Networks, NPcompleteness.
"... On the complexity of routing permutations on trees by arcdisjoint paths (extended abstract) ..."
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On the complexity of routing permutations on trees by arcdisjoint paths (extended abstract)