Minimum-dilation tour (and path) is NP-hard

Journal of Computer and System Sciences, 2015

We consider a bi-criteria generalization of the pathwidth problem, where, for given integers k, l and a graph G, we ask whether there exists a path decomposition P of G such that the width of P is at most k and the number of bags in P, i.e., the length of P, is at most l. We provide a complete complexity classification of the problem in terms of k and l for general graphs. Contrary to the original pathwidth problem, which is fixed-parameter tractable with respect to k, we prove that the generalized problem is NP-complete for any fixed k ≥ 4, and is also NP-complete for any fixed l ≥ 2. On the other hand, we give a polynomial-time algorithm that, for any (possibly disconnected) graph G and integers k ≤ 3 and l > 0, constructs a path decomposition of width at most k and length at most l, if any exists. As a by-product, we obtain an almost complete classification of the problem in terms of k and l for connected graphs. Namely, the problem is NP-complete for any fixed k ≥ 5 and it is polynomial-time for any k ≤ 3. This leaves open the case k = 4 for connected graphs.

2011

Abstract Given a set T of n points in ℝ 2, a Manhattan network on T is a graph G with the property that for each pair of points in T, G contains a rectilinear path between them of length equal to their distance in the L 1-metric. The minimum Manhattan network problem is to find a Manhattan network of minimum length, ie, minimizing the total length of the line segments in the network.

Computing Research Repository, 2000

We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and random directed graphs, D. If n is finite, we assume that G or D contains a hamilton circuit. If G is an arbitrary graph containing a hamilton circuit, we conjecture that Algorithm G always obtains a hamilton circuit in polynomial time.

We consider a bi-criteria generalization of the pathwidth problem, where, for given integers k, l and a graph G, we ask whether there exists a path decomposition P of G such that the width of P is at most k and the number of bags in P, i.e., the length of P, is at most l.

We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two n × n matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense n-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time n 3

Computational Geometry, 2014

A path or a polygonal domain is C-oriented if the orientations of its edges belong to a set of C given orientations; this is a generalization of the notable rectilinear case (C = 2). We study exact and approximation algorithms for minimum-link C-oriented paths and paths with unrestricted orientations, both in C-oriented and in general domains. Our two main algorithms are as follows: A subquadratic-time algorithm with a non-trivial approximation guarantee for general (unrestricted-orientation) minimum-link paths in general domains. An algorithm to find a minimum-link C-oriented path in a C-oriented domain. Our algorithm is simpler and more time-space efficient than the prior algorithm. We also obtain several related results: • 3SUM-hardness of determining the link distance with unrestricted orientations (even in a rectilinear domain). • An optimal algorithm for finding a minimum-link rectilinear path in a rectilinear domain. The algorithm and its analysis are simpler than the existing ones. • An extension of our methods to find a C-oriented minimum-link path in a general (not necessarily C-oriented) domain. • A more efficient algorithm to compute a 2-approximate C-oriented minimum-link path. • A notion of "robust" paths. We show how minimum-link C-oriented paths approximate the robust paths with unrestricted orientations to within an additive error of 1.

2005

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the length of a shortest path connecting p and q in G divided by their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers

1997

We consider the problems of computing r-approximate traveling salesman tours and r-approximate minimum spanning trees for a set of n points in IR d , where d 1 is a constant. In the algebraic computation tree model, the complexities of both these problems are shown to be (n log n=r), for all n and r such that r < n and r is larger than some constant. In the more powerful model of computation that additionally uses the oor function and random access, both problems can be solved in O(n) time if r = (n 1?1=d).

Fundamenta Informaticae, 2012

The results presented in the paper are threefold. Firstly, a new class of reduced-by-matching directed graphs is defined and its properties studied. The graphs are output from the algorithm which, for a given 1-graph, removes arcs which are unnecessary from the point of view of searching for a Hamiltonian circuit. In the best case, the graph is reduced to a quasiadjoint graph, what results in polynomial-time solution of the Hamiltonian circuit problem. Secondly, the systematization of several classes of digraphs, known from the literature and referring to directed line graphs, is provided together with the proof of its correctness. Finally, computational experiments are presented in order to verify the effectiveness of the reduction algorithm.

Discrete Applied Mathematics, 1999

We survey results on the sequential and parallel complexity of hamiltonian path and cycle problems in various classes of digraphs which generalize tournaments. We give detailed informations on the difference in difficulties for these problems for the various classes as well as prove new results on hamiltonian paths starting in a specified vertex for a quite general class of digraphs.