D. R. Karger, R. Motwani, and G. D. S. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18(1):82--98, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....broken by Vishwanathan [9] His algorithm achieves the same performance ratio (1) as ours, but works only for Hamiltonian graphs. In sparse Hamiltonian graphs, Feder, Motwani, and Subi [4] nd even longer paths. The hardness results for this problem are mainly due to Karger, Motwani, and Ramkumar [6]: The longest path problem does not belong to APX and cannot be approximated within 2 1 # V unless NP DTIME O(log 1 # n) any # 0. 2 Preliminaries In the remainder, we consider a connected graph G = V, E) with n = vertices and e = E edges. We write G[W ] for the graph ....

D. Karger, R. Motwani, and G.D.S. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18(1):8298, 1997.

.... NP=P (or another unlikely collapse between complexity classes occur) This idea has been exploited in [18] to prove that Max Independent Set either has a PTAS or no constant factor polynomial time approximation algorithm (unless NP=P) and has been successively pushed further by Karger et al. in [26] to prove that the Longest Path cannot be approximated within O(logn) unless P=NP. In the latter paper the inapproximability result has been obtained by combining the self improvement technique and an L reduction (to prove that the problem is MAX SNP hard) Consequently the easy way to prove ....

D. Karger, R. Motwani, and G. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18(1):82-98, 1997.

....[12] Let us x any c 2 (0; 1=2) In [14] de la Vega Karpinski show, that TSP(1,2) is Max SNP hard, when the input graph is c dense (implicit in their work is a parametrization of the hardness factor by c) Longest path problem: Given a graph, let n be the number of its vertices. Karger et al. [21] have given a polynomial algorithm that nds a path of length ngt n) in a 1 tough graph, i.e. an log n n ) approximation (Hamiltonian graphs are also 1 tough) Alon et al. 2] give a polynomial algorithm that for any constant p 0, nds a path of length p log n, In this paper, saying that ....

....problem. if there is such a path. Vishwanathan [35] has improved this bound for Hamiltonian graphs, by showing a (log n) n(log log n) 2 ) approximation. The problem is very hard, as it has no constant approximation for any constant, unless P = NP, even for graphs with maximum degree 4 [21]. Bazgan et al. 4] have proved the same hardness result on 3 regular Hamiltonian graphs. The same holds for the longest cycle problem. For dense graphs, Karger et al. 21] gave a polynomial algorithm that nds a path of length at least m=n in a graph with n vertices and m edges. This is a 2 ....

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D. Karger, R. Motwani and G. Ramkumar. On Approximating the Longest Path in a Graph. Algorithmica, 18, 82-98, 1997.

....result, e.g. cutwidth [LR99] bandwidth [Fei00] and minimum bisection (see Section 1.4) There are problems which are even harder to approximate. Several problems, such as Label cover, Nearest Lattice Vector (CVP) Nearest Codeword and Longest Path cannot be approximated [ABSS97, KMR97] within 2 log 1 Gammaffl n , for any fixed ffl 0, unless NP DT IME(n polylogn ) for some of these problems, these inapproximability results were slightly improved in [DS99, DKRS99] However, there is still a large gap between the inapproximability results and the approximability ....

D. Karger, R. Motwani, and G. D. S. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18(1):82--98, 1997.

....both theoretically and computationally (see e.g. D, GJ, L1] In spite of recent advances, the problem presents an imminent scienti c challenge. In the past few years the problem was modi ed to that of computing the length of the longest (self avoiding) path, and on approximation of this length [FMS, KMR]. 1 To quote Feder et al. FMS] this is] one problem that resisted all attempts at devising either positive or negative results. Essentially, there is no known algorithm which guarantees approximation ratio better than n polylog(n) and there are no hardness of approximation results that ....

....contain a Hamilton path [KS] Let us note that Theorem 2 cannot be used for approximation of the length of the longest path. First and foremost, it works only in one direction: a graph may have a very long path but a very large mixing time (an n cycle is the simplest example. It was suggested in [KMR] that approximating length of the longest path is as hard problem as approximating a cluque or a chromatic number: no O(n 1 ) approximation ratio can be obtained. On the other hand, approximating the mixing time is much easier. Indeed, mix( can be approximated by conductance via ( while ....

D. Karger, R. Motwani, G. D. S. Ramkumar, On approximating the longest path in a graph, Algorithmica 18 (1997), 82-98

....of either corollary is sufficient to show that 16 M 76 seconds. Computing the longest simple path for a given graph is known to be NP complete. Even worse, there is provably no good heuristic to approximate the longest simple path unless P = NP (the longest path problem is in APX [16]) In order to optimize 1 convergence time, we need to minimize the longest paths in a network. However, we encounter a trade off between the length of possible paths and the degree of connectivity in a network. Specifically, in a highly connected network, the longest path is linear in the ....

David R. Karger, Rajeev Motwani, and G. D. S. Ramkumar, "On approximating the longest path in a graph," Algorithmica, vol. 18, no. 1, pp. 82--98, May 1997.

....where D is the length of the longest path in the network. Computing the longest simple path for a given graph is known to be NPcomplete. Even more problematic, we note that there is provably no good heuristic to approximate the longest simple path unless P = NP (the longest path problem is in APX [16]) In order to optimize T down convergence time, we need to minimize the longest paths in a network. However, we encounter a trade o between the length of possible paths and the degree of connectivity in a network. Speci cally, in a highly connected network, the longest path is linear in the ....

David R. Karger, Rajeev Motwani, and G. D. S. Ramkumar, \On approximating the longest path in a graph," Algorithmica, vol. 18, no. 1, pp. 82-98, May 1997. 22

....(2) Given a surplus cluster with more than one outgoing edge, the choice of the outgoing edge that minimizes DISP in the resultant movement path is not obvious. Indeed we can show: Theorem 4. The PM decision problem is NP complete. Proof: See [TNLH00] 2 Furthermore, by using a result given in [KMR97], we can show that it is not possible to compute in polynomial time a constant factor approximation for the PM problem. Thus, an alternative is to use heuristics which could work well in practice and ecient enough for handling a large dataset. The purpose of the heuristic is to iteratively pick an ....

D. Karger, R. Motwani, and G. D. S. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18:99-110, 1997.

.... case complexity of this problem: finding a Hamilton path in random D regular graphs, for various values of D=n (see e.g. BFF,BFS,P] Closer to the subject of this paper, in the past years the problem was modified to determining the length of the longest path, and on approximation of this length [FMS,KMR]. To quote Feder et al. FMS] this is] one problem that resisted all attempts at devising either positive or negative results. Essentially, there is no known algorithm which guarantees approximation ratio better than n polylog(n) and there are no hardness of approximation results that ....

....cost O i n k log n k 2 j : Naturally, the theorem cannot be used for approximation of the length of the longest path. First and foremost, it works only in one direction: a graph may have a very long path but a very large mixing time (an n cycle is the simplest example. It was suggested in [KMR] that approximating length of the longest path is as hard problem as approximating a cluque or a chromatic number: no O(n 1 Gammaffl ) approximation ratio can be obtained. On the other hand, approximating the mixing time is much easier. Indeed, mix( Gamma) can be approximated by conductance via ....

D. Karger, R. Motwani, G. D. S. Ramkumar, On approximating the longest path in a graph, Algorithmica 18 (1997), 82-98.

....of either corollary is sufficient to show that T down 15jpj seconds. Computing the longest simple path for a given graph is known to be NP complete. Even worse, there is provably no good heuristic to approximate the longest simple path unless P = NP (the longest path problem is in APX [17]) In order to optimize T down convergence time, we need to minimize the longest paths in a network. However, we encounter a trade off between the length of possible paths and the degree of connectivity in a network. Specifically, in a highly connected network, the longest path is linear in the ....

David R. Karger, Rajeev Motwani, and G. D. S. Ramkumar, "On approximating the longest path in a graph," Algorithmica, vol. 18, no. 1, pp. 82--98, May 1997.

.... a largest set of vertices that are pairwise adjacent (Hastad, 1996) Minimum Vertex Coloring: Given a graph, color the vertices with a minimum number of colors so that adjacent vertices receive distinct colors (Lund and Yannakakis, 1994) Longest Path: Given a graph, find a longest simple path (Karger et al. 1993). Max Linear Satisfy: Given a set of linear equations, find a largest possible subset that are simultaneously satisfiable (Arora et al. 1993) Nearest Codeword: Given a linear error correcting code specified by a matrix, and given a vector, find the codeword closest in Hamming distance to the ....

Karger, D., Motwani, R., and Ramkumar, G. D. S. (1993). On approximating the longest path in a graph. In Workshop on Algorithms and Data Structures, volume 709 of Lecture Notes in Computer Science, pages 421--430. Springer Verlag.

....of leaves. It is easy to see that this problem is NP complete by a simple reduction from the Hamiltonian path problem. It is natural to ask if techniques similar to ours can be used to approximate this problem as well. We can use recent results on hardness of approximating the longest path problem [81] to infer that the minimum leaf spanning tree problem is hard to approximate. The key observation is that a spanning tree on n nodes with at most leaves contains a path with at least 2n= edges. This can be shown by considering an Euler tour of the tree and estimating the longest length of a ....

....polynomial time approximation algorithm with a multiplicative performance guarantee of f can be used to obtain a spanning tree of a Hamiltonian graph with at most 2f leaves. By our observation above, this yields a path in the graph of size at least n=f . The results of Karger, Motwani and Ramkumar [81] imply that f is at least 2 Omega Gamma1 4 1 Gammaffl n) for any ffl 1 unless P = NP 3 . 3 Recall that P stands for the complexity class Deterministic Quasi polynomial time, or DTIME[n polylog n ] 132 Chapter 8 Conclusions and Open Issues We have investigated several ....

D. Karger, R. Motwani, and G. D. S. Ramkumar, "On approximating the longest path in a graph," to appear in Proceedings, Workshop on Algorithms and Data Structures 1993.

....hereditary graph properties appears in [LY2] The set cover reduction of [LY1] as well as some of the reductions of [LY2] use 2P1R proofs. Recently it was shown that the longest path problem (of which our LONGEST CR PATH problem is a variant) is hard to approximate in the sense of Theorem 1. 1 [KMR]. Efficient four prover proofs were constructed in [BGLR] leading to improvement of many approximation results including those for set cover, clique, chromatic number, and MAX 3SAT. 2 Proofs, Optimization and Approximation It is convenient to view a 2P1R proof as an optimization problem; the ....

D. Karger, R. Motwani and G.D.S. Ramkumar. On Approximating the Longest Path in a Graph. Manuscript.

....Arora, Babai, Stern, and Sweedyk [3] prove hardness results for a collection of problems involving integral lattices, codes, or linear equations inequations. These include Nearest Lattice Vector, Nearest Codeword, and the Shortest Lattice Vector under the 1 norm. Karger, Motwani, and Ramkumar [70] prove the hardness of approximating the longest path in a graph to within a 2 log 1 n factor, for any 0. There are many other results which we haven t mentioned here; see the compendium [37] or the survey [4] Improved analysis of outer veri ers. Our construction of an ecient outer veri ....

D. Karger, R. Motwani, and G. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18(1):82-98, May 1997.

....Arora, Babai, Stern, and Sweedyk [3] prove hardness results for a collection of problems involving integral lattices, codes, or linear equations inequations. These include Nearest Lattice Vector, Nearest Codeword, and the Shortest Lattice Vector under the ## norm. Karger, Motwani, and Ramkumar [70] prove the hardness of approximating the longest path in a graph to within a 2 log 1 # n factor, for any # 0. There are many other results which we haven t mentioned here; see the compendium [37] or the survey [4] Improved analysis of outer verifiers. Our construction of an e#cient outer ....

D. Karger, R. Motwani, and G. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18(1):82-98, May 1997.

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D. Karger, R. Motwani, and G. Ramkumar, "On approximating the longest path in a graph," in Proc. Third Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science 709 (Springer-Verlag, Berlin, 1993) pp. 421--432.

....known instances of hardness results seem to be shown for problems which are additive. In particular, this is true for all MAX SNP problems, MAX CLIQUE, CHROMATIC NUMBER, and SET COVER. One case where a hardness result does not seem to directly apply to an additive problem is that of LONGEST PATH [17]. However in this case, the closely related LONGEST S T PATH problem is easily seen to be additive and the hardness result essentially stems from this problem. Lastly, the most interesting optimization problems which do not seem to be additive are problems related to GRAPH BISECTION or PARTITION, ....

D. Karger, R. Motwani and G.D.S. Ramkumar. On approximating the longest path in a graph. In Proceedings of the Third Workshop on Algorithms and Data Structures (1993), pp. 421--432.

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D. R. Karger, R. Motwani, and G. D. S. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18(1):82--98, 1997.

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D. Karger, R. Motwani, and G.D.S. Ramkumar. On approximating the longest path in a graph. In Proceedings of the Workshop on algorithms and data structures, Montreal, Canada, pages 421-432, 1993.

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D. Karger, R. Motwani and G. D, S. Ramkumar, On approximating the longest path in a graph, Proceedings of Workshop on Algorithms and Data Structures, 1993, LNCS v. 709, pp 421-430.

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D. Karger, R. Motwani, and G. Ramkumar. On approximating the longest path in a graph. Algorithmica, 18(1):82--98, 1997.

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D. Karger,R. Motwani,andG. D. S. Ramkumar, On approximating the longest path in a graph, Algorithmica, 18 (1997), pp. 82--98.

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D. Karger, R. Motwani, G. Ramkumar, On Approximating the Longest Path in a Graph, Algoritmica 18 (1997), 82--98.

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D. Karger, R. Motwani, and G.D.S. Ramkumar. On approximating the longest path in a graph. In Proceedings of Workshop on Algorithms and Data Structures, pages 421--430. LNCS (SpringerVerlag) , v. 709, 1993.

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D. Karger, R. Motwani and G. D. S. Ramkumar. On Approximating the Longest Path in a Graph. Manuscript, 1992.

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