M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems , ECCC Technical Report TR 97-004,  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Anton V. Eremeev, Omsk Branch of Sobolev Institute of Mathematics, RAS Summary: In this work we study the performance of some algorithms approximating the vertex cover problem (VCP) with respect to density constraints. We consider a modification of the algorithm of Karpinski and Zelikovsky [9] for the weighted VCP and prove a performance guarantee for this algorithm in terms of a new density parameter. Also we investigate the hardness of approximation for the VCP on everywhere dense graph and show that it is NP hard to approximate within a factor less than 7 6 2 . 1. ....

....[5] show that even in case of unweighted VCP on the everywhere dense graph for any 2 (0; 1) there exists such Delta( 1 that obtaining a cover with approximation ratio less than Delta( is NP hard. An example of the function Delta( will be given explicitly further in this paper. In [9] Karpinski and Zelikovsky designed an approximation algorithm for the unweighted case of VCP with a guaranteed performance ratio 2 1 . We shall refer to this algorithm as DVC. In this paper we will study how the concept of density could be extended to the weighted case permitting a similar ....

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Karpinski, M.; Zelikovsky, A. (1996): Approximating Dense Cases of Covering Problems. DIMACS Technical Report TR 96-59. International Computer Science Institute, Berkeley.

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M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems , ECCC Technical Report TR 97-004,

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M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems, ECCC Technical Report TR 97-004,

....trivial approximation algorithms dramatically improving performances of the best known approximation algorithms on general instances. An interesting artifact on their complementary, i.e. dense, instances was also the existence of polynomial time approximation schemes (PTASs) for them [AKK95] [KZ97], see [K01] To appear in Proc. 13th Symp. on Fundamentals of Computation Theory, FCT 01, August 22 24, 2001. Dept. of Computer Science, University of Bonn, 53117 Bonn. Supported in part by DFG grants, DIMACS, and IST grant 14036 (RAND APX) and by the Max Planck Research Prize. Research ....

M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems, ECCC Technical Report TR 97-004,

....of the dense instances of NP hard optimization problems. Starting in 1995, the first polynomial time approximation schemes have been designed for such problems in Arora, Karger and Karpinski [AKK95] Fernandez de la Vega [FdV96] Arora, Frieze and Kaplan [AFK96] and Karpinski and Zelikovsky [KZ97b]. Later on, Goldreich, Goldwasser and Ron [GGR96] and Frieze and Kannan [FK97] gave a constant sample size approximation schemes for some dense optimization problems. Fernandez de la Vega and Karpinski [FdVK97] gave also the first polynomial time approximability characterization for dense ....

.... the fact that the existence of such schemes for general instances of all the above mentioned 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 ....

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M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems, ECCC Technical Report TR 97-004,

....that B(f; G) 2B(G) 5 The Superdense Case Further densification leads to polynomial approximation scheme for the bandwidth minimization problem. We call a simple graph G superdense, if the minimum degree of G is at least n Gamma o(n ffi ) The notion of superdenseness has been introduced in [KZ 97] If G is superdense, B(G) is at least n Gamma o(n ffi ) and therefore any layout f will suffice to be a good approximation: B(f; G) n (1 ffl)n Gamma o(n ffi ) for any ffl 2 O(1) 1 ffl) n Gamma o(n ffi ) 1 ffl)B(G) 6 Further Research and Open Problems There remains ....

Karpinski, M., Zelikovsky, A., Approximating Dense Cases of Covering Problems, Technical Report TR-97-004, ECCC, 1997.

....that B(f; G) 2B(G) 5 The Superdense Case Further densification leads to polynomial approximation scheme for the bandwidth minimization problem. We call a simple graph G superdense, if the minimum degree of G is at least n Gamma o(n ffi ) The notion of superdenseness has been introduced in [KZ 97] If G is superdense, B(G) is at least n Gamma o(n ffi ) and therefore any layout f will suffice to be a good approximation: B(f; G) n (1 ffl)n Gamma o(n ffi ) for any ffl 2 O(1) 1 ffl) n Gamma o(n ffi ) 1 ffl)B(G) 6 Further Research and Open Problems There remains ....

Karpinski, M., Zelikovsky, A., Approximating Dense Cases of Covering Problems, Technical Report TR-97-004, ECCC, 1997.

....of the dense instances of NP hard optimization problems. Recently, the first polynomial time approximation schemes have been designed for these problems in Arora, Karger and Karpinski [AKK95] Fernandez de la Vega [FV96] Arora, Frieze and Kaplan [AFK96] and Karpinski and Zelikovsky [KZ97b]. Later on, Goldreich, Goldwasser and Ron [GGR96] and Frieze and Kannan [FK97] gave a constant sample size approximation schemes for some dense optimization problems. Fernandez de la Vega and Karpinski [FK97] gave also the first polynomial time approximability characterization for dense weighted ....

.... This development was in contrast to the fact that the existence of such schemes for general 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 ....

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M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems, ECCC Technical Report TR 97-004, 1997, to appear in Proc. DIMACS Workshop on Network Design: Connectivity and Facilities Location, Princeton (1997).

....The general SET COVER was proven recently to have a threshold (1 Gamma o(1) ln n for the polynomial time approximation (cf. Feige [F96] which in fact is matching asymptotically the approximation ratio by the well known greedy heuristic algorithm. It is shown in Karpinski and Zelikovsky [KZ96] that the greedy heuristic algorithm can be applied more efficiently towards ther dense SET COVER. Proposition 4. KZ96] For any constant c 0 and any ffl 0, there is a polynomial time approximation algorithm for the ffl dense SET COVER with the approximation ratio c Delta log n. ....

....(cf. Feige [F96] which in fact is matching asymptotically the approximation ratio by the well known greedy heuristic algorithm. It is shown in Karpinski and Zelikovsky [KZ96] that the greedy heuristic algorithm can be applied more efficiently towards ther dense SET COVER. Proposition 4. [KZ96]) For any constant c 0 and any ffl 0, there is a polynomial time approximation algorithm for the ffl dense SET COVER with the approximation ratio c Delta log n. Interestingly, we cannot expect on the lower bound side of the dense SET COVER, its NP hardness, as the results of Papadimitriou and ....

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M. Karpinski and A. Zelikovsky, Approximating Dense Cases of Covering Problems (Preliminary Version), Technical Report TR-96-059, International Computer Science Institute, Berkeley (1996).

....ffl dense if any vertex in G has at least ffljV j neighbors, and we call a graph G weakly ffl dense if the average degree of a vertex in G is at least ffljV j, i.e. P v2V deg(v) jV j ffl Delta jV j Note that strong ffl density implies weak ffl density. In the preliminary draft of this paper [KZ96] as well as in [CT96] it was shown that the vertex cover problem even for strongly ffl dense graphs is still MAX SNP hard. We present an approximation algorithm with the improved performance bounds 2 1 ffl and 2 2 Gamma p 1 Gammaffl for strong and weak ffl dense graphs, respectively. We say ....

....j 2 2 Gamma p 1 Gamma ffl Theorems 4.2 and 4.3 show that the density can help in approximating the vertex cover problem. On the other hand, we cannot expect any polynomial time approximation scheme for the strongly (as well as weakly) dense vertex cover problem, since it is MAX NP hard (see [KZ96, CT96]) Further densification (as for the set cover problem and the Steiner tree problem) implies further decreasing of the approximation complexity. Let us consider the ffi superdense vertex cover problem, i.e. the case of the vertex cover problem when the average vertex degree is at least jV j ....

M. Karpinski and A. Zelikovsky, Approximating dense cases of covering problems. Technical Report TR-96-059, International Computer Science Institute, Berkeley, 1996.

....that B(f; G) 2B(G) 5 The Superdense Case Further densification leads to polynomial approximation scheme for the bandwidth minimization problem. We call a simple graph G superdense, if the minimum degree of G is at least n Gamma o(n ffi ) The notion of superdenseness has been introduced in [KZ 97] If G is superdense, B(G) is at least n Gamma o(n ffi ) and therefore any layout f will suffice to be a good approximation: B(f; G) n (1 ffl)n Gamma o(n ffi ) for any ffl 2 O(1) 1 ffl) n Gamma o(n ffi ) 1 ffl)B(G) 6 Further Research and Open Problems There remains ....

Karpinski, M., Zelikovsky, A., Approximating Dense Cases of Covering Problems, Technical Report TR-97-004, ECCC, 1997.

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