A linear-time algorithm for thek-fixed-endpoint path cover problem on cographs

Networks ◽  
2007 ◽  
Vol 50 (4) ◽  
pp. 231-240 ◽  
Author(s):  
Katerina Asdre ◽  
Stavros D. Nikolopoulos
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Path Cover ◽  
Cover Problem

scholarly journals Computing directed Steiner path covers

2021 ◽  
Author(s):  
Frank Gurski ◽  
Dominique Komander ◽  
Carolin Rehs ◽  
Jochen Rethmann ◽  
Egon Wanke
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Time Algorithm ◽  
Integer Programs ◽  
Path Cover ◽  
Hamiltonian Path Problem ◽  
Directed Paths ◽  
Minimum Number ◽  
Cover Problem ◽  
Vertex Disjoint

AbstractIn this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph $$G=(V,E)$$ G = ( V , E ) and a set $$T \subseteq V$$ T ⊆ V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

scholarly journals The reverse total weighted distance problem on networks with variable edge lengths

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1333-1342
Author(s):  
Kien Nguyen ◽  
Nguyen Hung
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Time Algorithm ◽  
Set Cover ◽  
Linear Time Algorithm ◽  
Iterative Approach ◽  
Weighted Distance ◽  
Set Cover Problem ◽  
Cover Problem ◽  
Distance Problem

We address the problem of reducing the edge lengths of a network within a given budget so that the sum of weighted distances from each vertex to others is minimized. We call this problem the reverse total weighted distance problem on networks. We first show that the problem is NP-hard by reducing the set cover problem to it in polynomial time. Particularly, we develop a linear time algorithm to solve the problem on a tree. For the problem on cycles, we devise an iterative approach without mentioning the exact complexity. Additionally, if the cycle has uniform edge lengths, we can prove that the specified approach runs in O(n3) time as each edge of the cycle can be reduced at most once, where n is the number of vertices in the underlying cycle.

scholarly journals Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow

2002 ◽  
Vol 44 (2) ◽  
pp. 193-204 ◽  
Author(s):  
D. S. Franzblau ◽  
A. Raychaudhuri
Keyword(s):  
Linear Time ◽  
Optimal Path ◽  
Hamiltonian Path ◽  
Maximum Flow ◽  
Time Algorithm ◽  
Frequency Assignment ◽  
Minimum Size ◽  
Linear Time Algorithm ◽  
Path Cover ◽  
Vertex Disjoint

A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

A Linear time algorithm for deciding security

1976 ◽  
Author(s):  
A. K. Jones ◽  
R. J. Lipton ◽  
L. Snyder
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm
Sign in / Sign up

Export Citation Format

Share Document