M. Karpinski, J. Wirtgen and A. Zelikovsky, An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, ECCC Technical Report TR 97-017 (1997).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the bounds are within a multiplicative constant. Real applications of the index assignment problem arise in video and speech communication over packetswitched networks, where the information has to be split into several packets which can be lost in transmission resulting in poor signal quality ([17], 18] 32] 2] 1.2 Problem Statement We are given a communication system with k channels. Channel i transmits information reliably at a rate log n i bits per second. Each channel either succeeds or fails to transmit the information. If a channel fails, all the information transmitted over ....

Karpinski, M., J. Wirtgen and A. Zelikovsky, \An approximation algorithm for the bandwidth problem on dense graphs", ECCC TR97-017.

....heuristics for minimizing the bandwidth appear to work rather well [9] Some theoretical justification for the success of heuristics is given in [30] where their performance on random graphs is investigated. Algorithms for approximating the bandwidth in special classes of graphs are presented in [15, 18, 16]. However, for general graphs, there were no known algorithms giving nontrivial approximation ratios (such as n ffi for some ffi 1) for the bandwidth, not even for trees. We note that there are known graphs on which the heuristics studied in [30] fail to give approximation ratios better ....

M. Karpinski, J. Wirtgen, A. Zelikovsky. "An approximation algorithm for the bandwidth problem on dense graphs". ECCC TR97-017.

.... problems would imply that P=NP, by results of Arora, Lund, Motwani, Sudan and Szegedy [ALMSS92] The development above was followed by the study of the dense covering problems in Karpinski and Zelikovsky [KZ97b] and the dense bandwidth minimization problems, Karpinski, Wirtgen and Zelikovsky [KWZ97]; as well as metric instances of MAX CUT, Fernandez de la Vega and Kenyon [FdVKe98] It is also a very interesting artifact that the recent successes in design of the polynomial time approximation schemes for dense optimization problems parallel the successes of the past attacks on dense ....

....approximation ratio algorithms, among them log n approximation algorithm for the caterpillars ( HMM91] We consider here the BANDWIDTH problem on the everywhere dense graphs. Using a randomized placing technique combined with the special perfect matching design Karpinski, Wirtgen and Zelikovsky [KWZ97] proved Proposition 10. KWZ97] There exists a randomized polynomial time approximation algorithm for the BANDWIDTH problem on everywhere dense Using a more constrained nature of DBANDWIDTH the similar techniques yield. Proposition 11. KWZ97] There exists a randomized polynomial time ....

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M. Karpinski, J. Wirtgen and A. Zelikovsky, An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, ECCC Technical Report TR 97-017 (1997).

....between the numbers of adjacent vertices is minimal. The problem has a long and varied history and is known to be NP hard Papadimitriou [Pa 76] Recently for ffi dense graphs a constant ratio approximation algorithm for this problem has been constructed in Karpinski, Wirtgen and Zelikovsky [KWZ 97] In this paper we prove that the bandwidth problem on the dense instances remains NP hard. Dept. of Computer Science, University of Bonn, 53117 Bonn. Research partially supported by DFG Grant KA 673 4 1, by the ESPRIT BR Grants 7097 and EC US 030, and by the Max Planck Research Prize. Email: ....

....log n approximation algorithm. A caterpillar is a special kind of a tree consisting of a simple chain, the body, with an arbitrary number of simple chains, the hairs, attached to the body by coalescing an endpoint of the added chain with a vertex of the body. Karpinski, Wirtgen and Zelikovsky [KWZ 97] designed a 3 approximation algorithm for ffi dense graphs. Definition 1 ( AKK 95] We call a graph dense, if the number of edges is in Omega Gamma n 2 ) A graph G is ffi dense, if the minimum degree ffi (G) is at least ffi n. We call it everywhere dense, if it is ffi dense for some ffi ....

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Karpinski, M., Wirtgen, J., Zelikovsky, A., An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, Technical Report ECCC TR 97-017, 1997.

....simple chain, the body, with an arbitrary number of simple chains, the hairs, attached to the body by coalescing an endpoint of the added chain with a vertex of the body. For this special class of trees the bandwidth problem was also shown to be NP hard [Mo 86] Karpinski, Wirtgen and Zelikovsky [KWZ 97] constructed a 3 approximation algorithm for ffi dense graphs. A graph G is ffi dense, if the minimum degree ffi(G) is at least ffi n. We call it everywhere dense, if it is ffi dense for some ffi 0. The design of approximation algorithms for NP hard optimization problems became an important ....

....of the same literal in one clause. It then becomes somewhat more complicated to describe how many stars and nodes are at the different positions. 3 Open problems An important computational problem still remains open about the existence of a PTAS for the bandwidth problem on dense graphs (cf. KWZ 97] Another important question is to improve both upper and lower approximation bounds on the general bandwidth problem, closing a large gap between O(1) and log O(1) n (cf. Fe 97] ....

Karpinski, M., Wirtgen, J., Zelikovsky, A., An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, Technical Report TR-97-017, ECCC, 1997.

....between the numbers of adjacent vertices is minimal. The problem has a long and varied history and is known to be NP hard Papadimitriou [Pa 76] Recently for dense graphs a constant ratio approximation algorithm for this problem has been constructed in Karpinski, Wirtgen and Zelikovsky [KWZ 97] In this paper we prove that the bandwidth problem on the dense instances remains NP hard. Dept. of Computer Science, University of Bonn, 53117 Bonn. Research partially supported by DFG Grant KA 673 4 1, by the ESPRIT BR Grants 7097 and EC US 030, and by the Max Planck Research Prize. Email: ....

....log n approximation algorithm. A caterpillar is a special kind of a tree consisting of a simple chain, the body, with an arbitrary number of simple chains, the hairs, attached to the body by coalescing an endpoint of the added chain with a vertex of the body. In Karpinski, Wirtgen and Zelikovsky [KWZ 97] introduced a 3 approximation algorithm for everywhere ffi dense graphs. Definition 1 ( AKK 95] We call a graph G (everywhere) ffi dense, if the minimum degree ffi(G) is at least ffin. We call it dense in average, if the number of edges is in Omega Gamma n 2 ) In this paper we show that ....

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Karpinski, M., Wirtgen, J., Zelikovsky, A., An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, Technical Report TR-97-017, ECCC, 1997.

....to construct a dominating set. Lemma 3.3 Let G = V; E) be a graph with minimum degree d(n) A set of k = Theta(log(n)n=d(n) randomly chosen vertices R forms a dominating set with high probability. Clearly if the minimum degree is in Theta(n) then k can be chosen to be Theta(log n) KWZ 97] DDLW 98] Such graphs are called dense. Proof: The probability, that one particular vertex v will be dominated by one of the randomly chosen vertices, is at least d(n) n. Define the random variable X v;i to be 1, if the ith random vertex dominates this vertex and X v = k X i X v;i v = ....

.... time algorithm which finds a layout f , such that B(f; G) log (k) log (k 1) B(G) where log (k) is the k times iterated logarithm (log (1) log; log (k 1) log log (k) This theorem gives a somewhat worse approximation ratio (a 3 approximation) for dense graphs, than given in [KWZ 97] But the running time is significantly more practical. Acknowledgment I thank Gunter Blache, Carsten Dorgerloh and Marek Karpinski for a number of interesting discussions on the subject of this paper. ....

Karpinski, M., Wirtgen, J., Zelikovsky, A., An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, Proc. 1 st RALCOM (1997).

.... instances would imply that P=NP by results of Arora, Lund, Motwani, Sudan, and Szegedy [ALMSS92] The development above was followed by the study of the dense covering problems, Karpinski and Zelikovsky [KZ97b] and the dense bandwidth minimization problems, Karpinski, Wirtgen and Zelikovsky [KWZ97]. It is also a very interesting artifact that the recent successes in design of the polynomial time approximation schemes for dense optimization problems parallel the successes of the past attacks on dense approximate counting problems, Broder [B86] Jerrum and Sinclair [JS89] Dyer, Frieze, ....

....n ffl approximation algorithms known for the BANDWIDTH problem even if restricted to trees. We consider here the BANDWIDTH problem on the everywhere dense graphs. Using a randomized placing technique combined with the special perfect matching construction Karpinski, Wirtgen and Zelikovsky [KWZ97] proved. Proposition 10. KWZ97] There exists a randomized polynomial time approximation algorithm for the BANDWIDTH problem on everywhere dense graphs with approximation ratio 3. Using a more constrained nature of DBANDWIDTH the similar techniques yield. Proposition 11. KWZ97] There ....

[Article contains additional citation context not shown here]

M. Karpinski, J. Wirtgen and A. Zelikovsky, An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, ECCC Technical Report TR 97-017 (1997).

....of G fB(f; G)g Clearly the bandwidth of G is the greatest bandwidth of its components. The problem of finding the bandwidth of a graph is NP hard [Pa 76] even for trees with maximum degree 3 [GGJK 78] The general problem is not known to have any sublinear n ffl approximation algorithm (cf. KWZ 97] Ka 97] Smithline [Sm 95] proved that the bandwidth of a complete k ary tree can be computed in polynomial time. For caterpillars [HMM 91] found a polynomial time log n approximation algorithm. A caterpillar is a special kind of a tree consisting of a simple chain, the body, with an ....

....log n approximation algorithm. A caterpillar is a special kind of a tree consisting of a simple chain, the body, with an arbitrary number of simple chains, the hairs, attached to the body by coalescing an endpoint of the added chain with a vertex of the body. Karpinski, Wirtgen and Zelikovsky [KWZ 97] constructed a 3 approximation algorithm for ffi dense graphs. Definition 1.1 We call a graph dense if it has Omega Gamma n 2 ) edges. A graph G is ffi dense, if the minimum degree ffi(G) is at least ffin. We call it everywhere dense, if it is ffi dense for some ffi 0. The design of ....

[Article contains additional citation context not shown here]

Karpinski, M., Wirtgen, J., Zelikovsky, A., An Approximation Algorithm for the Bandwidth Problem on Dense Graphs, Technical Report TR-97017, ECCC, 1997.

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