On approximating the longest path in a graph

Karger, David R.; Motwani, Rajeev; Ramkumar, G.

doi:10.1007/3-540-57155-8_267

Cited by 81 publications

(107 citation statements)

References 5 publications

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“…· n · m) where n and m are respectively the number of vertices and the number of edges of the graph. Karger, Motwani and Ramkumar [8] gave a polynomial time algorithm which finds a path of length Ω(log n) in any 1-tough graph. A similar result was obtained also by Fürer and Raghavachari [4].…”

Section: Introductionmentioning

confidence: 99%

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On the Approximation of Finding A(nother) Hamiltonian Cycle in Cubic Hamiltonian Graphs

Bazgan

Sántha

Tuza

1999

Journal of Algorithms

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Section: Introductionmentioning

confidence: 99%

“…On the negative side, Karger, Motwani and Ramkumar [8] have proved that unless P =N P , Longest Path is not constant approximable in polynomial time. Their proof consists of two parts.…”

Section: Introductionmentioning

confidence: 99%

On the Approximation of Finding A(nother) Hamiltonian Cycle in Cubic Hamiltonian Graphs

Bazgan

Sántha

Tuza

1999

Journal of Algorithms

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“…However, finding a longest path seems to be more difficult than deciding whether or not a graph admits a Hamiltonian path. Indeed, it has been proved that even if a graph has a Hamiltonian path, the problem of finding a path of length n − n ε for any ε < 1 is NP-hard, where n is the number of vertices of the graph [15]. Moreover, there is no polynomial-time constant-factor approximation algorithm for the longest path problem unless P=NP [15].…”

Section: Introductionmentioning

confidence: 99%

The Longest Path Problem Is Polynomial on Interval Graphs

Ioannidou

Mertzios

Nikolopoulos

2009

Mathematical Foundations of Computer Science 2009

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Abstract. The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno in [20], where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm runs in O(n 4 ) time, where n is the number of vertices of the input graph, and bases on a dynamic programming approach.

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“…Although the two problems are similar, finding a longest path in a graph seems to be more difficult than deciding whether or not the graph admits a Hamiltonian path. Indeed, it has been proved that even if a graph has a Hamiltonian path, the problem of finding a path of length n − n ε for any ε < 1 is NP-hard, where n is the number of vertices of the graph [17]. Moreover, there is no polynomial-time constant-factor approximation algorithm for the longest path problem unless P = NP [17].…”

mentioning

confidence: 99%

“…Indeed, it has been proved that even if a graph has a Hamiltonian path, the problem of finding a path of length n − n ε for any ε < 1 is NP-hard, where n is the number of vertices of the graph [17]. Moreover, there is no polynomial-time constant-factor approximation algorithm for the longest path problem unless P = NP [17]. For related results see also [9-11, 25, 26].…”

mentioning

confidence: 99%

The Longest Path Problem has a Polynomial Solution on Interval Graphs

2010

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The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno (Proc. of the 15th Annual International Symp. on Algorithms and Computation (ISAAC), LNCS, vol. 3341, pp. 871-883, 2004), where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm uses a dynamic programming approach and runs in O(n 4 ) time, where n is the number of vertices of the input graph.

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On approximating the longest path in a graph

Cited by 81 publications

References 5 publications

On the Approximation of Finding A(nother) Hamiltonian Cycle in Cubic Hamiltonian Graphs

On the Approximation of Finding A(nother) Hamiltonian Cycle in Cubic Hamiltonian Graphs

The Longest Path Problem Is Polynomial on Interval Graphs

The Longest Path Problem has a Polynomial Solution on Interval Graphs

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