Hamilton Paths in Grid Graphs

Itai, Alon; Papadimitriou, Christos H.; Szwarcfiter, Jayme Luiz

doi:10.1137/0211056

Cited by 444 publications

(296 citation statements)

References 4 publications

Supporting

Mentioning

294

Contrasting

Unclassified

Order By: Relevance

“…A grid graph is a finite node-induced subgraph of G ∞ . It is shown in [13] that the Hamiltonian cycle problem in a grid graph is NP-hard. Given a grid graph on n nodes, we will construct an n × n relaxed Supnick matrix with non-negative entries, such that there exists a Hamiltonian cycle in the graph, if and only if the optimal TSP tour on the corresponding relaxed Supnick matrix has zero length.…”

Section: Theorem 1 the Tsp On A Relaxed Supnick Matrix Is Np-hardmentioning

confidence: 99%

“…The proof is by reduction from the Hamiltonian cycle problem in grid graphs [13]. We follow the definitions from [13].…”

Section: Theorem 1 the Tsp On A Relaxed Supnick Matrix Is Np-hardmentioning

confidence: 99%

“…The proof is by reduction from the Hamiltonian cycle problem in grid graphs [13]. We follow the definitions from [13]. Let G ∞ be the infinite graph, whose vertex set consists of all the integer points in the plane, and in which two vertices are connected, if and only if the Euclidean distance between them is equal to one.…”

Section: Theorem 1 the Tsp On A Relaxed Supnick Matrix Is Np-hardmentioning

confidence: 99%

See 2 more Smart Citations

One-Sided Monge TSP Is NP-Hard

Deı̆neko

Tiskin

2006

Computational Science and Its Applications - ICCSA 2006

View full text Add to dashboard Cite

Abstract. The Travelling Salesman Problem (TSP) is a classical NPhard optimisation problem. There exist, however, special cases of the TSP that can be solved in polynomial time. Many of the well-known TSP special cases have been characterized by imposing special four-point conditions on the underlying distance matrix. Probably the most famous of these special cases is the TSP on a Monge matrix, which is known to be polynomially solvable (as are some other generally NP-hard problems restricted to this class of matrices). By relaxing the four-point conditions corresponding to Monge matrices in different ways, one can define other interesting special cases of the TSP, some of which turn out to be polynomially solvable, and some NP-hard. However, the complexity status of one such relaxation, which we call one-sided Monge TSP (also known as the TSP on a relaxed Supnick matrix ), has remained unresolved. In this note, we show that this version of the TSP problem is NP-hard. This completes the full classification of all possible four-point conditions for symmetric TSP.

show abstract

Section: Theorem 1 the Tsp On A Relaxed Supnick Matrix Is Np-hardmentioning

confidence: 99%

“…The proof is by reduction from the Hamiltonian cycle problem in grid graphs [13]. We follow the definitions from [13].…”

Section: Theorem 1 the Tsp On A Relaxed Supnick Matrix Is Np-hardmentioning

confidence: 99%

Section: Theorem 1 the Tsp On A Relaxed Supnick Matrix Is Np-hardmentioning

confidence: 99%

See 1 more Smart Citation

One-Sided Monge TSP Is NP-Hard

Deı̆neko

Tiskin

2006

Computational Science and Its Applications - ICCSA 2006

View full text Add to dashboard Cite

show abstract

“…The Hamiltonian path problem is known to be NP-complete in general graphs [10,11], and remains NP-complete even when restricted to some small classes of graphs such as split graphs [13], chordal bipartite graphs, split strongly chordal graphs [17], circle graphs [5], planar graphs [11], and grid graphs [14]. However, it makes sense to investigate the tractability of the longest path problem on the classes of graphs for which the Hamiltonian path problem admits polynomial time solutions.…”

Section: Introductionmentioning

confidence: 99%

The Longest Path Problem Is Polynomial on Interval Graphs

Ioannidou

Mertzios

Nikolopoulos

2009

Mathematical Foundations of Computer Science 2009

View full text Add to dashboard Cite

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.

show abstract

“…It is clear that the longest path problem is NP-hard on every class of graphs on which the Hamiltonian path problem is NP-complete; note that, the Hamiltonian path problem is known to be NP-complete on general graphs [12,13], and remains NP-complete even when restricted to some small classes of graphs such as split graphs [15], chordal bipartite graphs, split strongly chordal graphs [19], directed path graphs [20], circle graphs [7], planar graphs [13], and grid graphs [16]. On the other hand, there are several classes of graphs on which the Hamiltonian path problem admits polynomial time solutions; these classes include proper interval graphs [3], interval graphs [1,5,8], circular-arc graphs [8], biconvex graphs [2], and cocomparability graphs [6].…”

mentioning

confidence: 99%

The Longest Path Problem has a Polynomial Solution on Interval Graphs

2010

View full text Add to dashboard Cite

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.

show abstract

Hamilton Paths in Grid Graphs

Cited by 444 publications

References 4 publications

One-Sided Monge TSP Is NP-Hard

One-Sided Monge TSP Is NP-Hard

The Longest Path Problem Is Polynomial on Interval Graphs

The Longest Path Problem has a Polynomial Solution on Interval Graphs

Contact Info

Product

Resources

About