Path Cover Problem Research Articles

Solving the path cover problem on circular-arc graphs by using an approximation algorithm

A path cover of a graph G = ( V , E ) is a family of vertex-disjoint paths that covers all vertices in V. Given a graph G, the path cover problem is to find a path cover of minimum cardinality. This paper presents a simple O ( n ) -time approximation algorithm for the path cover problem on circular-arc graphs given a set of n arcs with endpoints sorted. The cardinality of the path cover found by the approximation algorithm is at most one more than the optimal one. By using the result, we reduce the path cover problem on circular-arc graphs to the Hamiltonian cycle and Hamiltonian path problems on the same class of graphs in O ( n ) time. Hence the complexity of the path cover problem on circular-arc graphs is the same as those of the Hamiltonian cycle and Hamiltonian path problems on circular-arc graphs.

A time-optimal solution for the path cover problem on cographs

We show that the notoriously difficult problem of finding and reporting the smallest number of vertex-disjoint paths that cover the vertices of a graph can be solved time- and work-optimally for cographs. Our result implies that for this class of graphs the task of finding a Hamiltonian path can be solved time- and work-optimally in parallel.It was open for more than 10 years to find a time- and work-optimal parallel solution for this important problem. Our contribution is to offer an optimal solution to this important problem. We begin by showing that any algorithm that solves an instance of size n of the problem must take Ω(logn) time on the CREW, even if an infinite number of processors are available. We then go on to show that this time lower bound is tight by devising an EREW algorithm that, given an n-vertex cograph G represented by its cotree, finds and reports all the paths in a minimum path cover in O(logn) time using n/logn processors.

Deferred‐query: An efficient approach for some problems on interval graphs

This paper introduces the idea of a deferred-query approach to design O(n) algorithms for the domatic partition, optimal path cover, Hamiltonian path, Hamiltonian circuit, and maximum matching problems on interval graphs given n endpoint-sorted intervals. The previous best-known algorithms run in O(n log log n) or O(n +m) time, where m is the number of edges in the corresponding interval graphs. © 1999 John Wiley & Sons, Inc. Networks 34: 1–10, 1999

Optimal path cover problem on block graphs

Let G = ( V, E) be a block graph. First we show that an algorithm for finding the path partition number p( G) by J.H. Yan and G.J. Chang gives wrong answers to some block graphs. Then we present an efficient algorithm for finding a minimum path partition of G (not just the path partition number p( G)). The complexity of this algorithm is O( m), where m = ¦E¦.

Deferred-query: An efficient approach for some problems on interval graphs

This paper introduces the idea of a deferred-query approach to design O(n) algorithms for the domatic partition, optimal path cover, Hamiltonian path, Hamiltonian circuit, and maximum matching problems on interval graphs given n endpoint-sorted intervals. The previous best-known algorithms run in O(n log log n) or O(n + m) time, where m is the number of edges in the corresponding interval graphs.

Mapping Clones with a Given Ordering or Interleaving

We study the problem of constructing a most compact physical map for a collection of clones whose ordering or interleaving on a DNA molecule are given. Each clone is a contiguous section of the DNA and is represented by its fingerprint obtained from biochemical experiments. In this paper we consider two kinds of mapping: single complete digest mapping, in which the fingerprint of a clone is a multiset containing the sizes of the restriction fragments occurring in the clone, and mapping by hybridization of probes, in which the fingerprint of a clone is a multiset consisting of the short oligonucleotide probes occurring in the clone. Our goal is to position the clones and restriction fragments (or probes) on the DNA consistently with the given ordering or interleaving so that the total number of restriction fragments (resp. probes) required on the DNA is minimized. We first formulate this as a constrained path cover problem on a multistage graph. Using this formulation, it is shown that finding a most compact map for clones with a given ordering is NP-hard. The approximability of the problem is then considered. We present a simple approximation algorithm with ratio 2 . This is in fact the best possible as the above NP-hardness proof actually shows that achieving ratio 2-e is impossible for any constant e > 0 , unless P = NP. We also give a polynomial-time approximation scheme when the multiplicity is bounded by one (i.e., when the multisets are actually sets). The exact complexity of the problem in this special case is presently unknown. Finally we consider the mapping problem when an interleaving is given which depicts how the clones overlap with each other on the DNA. In the case of restriction fragment data, it is shown that finding a consistent map is NP-complete even if the multiplicity is bounded by 3 . This may suggest that information about the interleaving of clones does not necessarily make the problem computationally easier in single complete digest mapping. On the other hand, in the case of hybridization data, there is an efficient algorithm to construct a most compact map when the interleaving of clones is given.

Optimal path cover problem on block graphs and bipartite permutation graphs

The optimal path cover problem is to find a minimum number of vertex disjoint paths which together cover all the vertices of the graph. In this paper, we present linear-time algorithms for the optimal path cover problem for the class of block graphs and bipartite permutation graphs.

An $O(n^2 \log n)$ Algorithm for the Hamiltonian Cycle Problem on Circular-Arc Graphs

A circular arc family F is a collection of arcs on a circle. A circular-arc graph is the intersection graph of an arc family. A Hamiltonian cycle (HC) in a graph is a cycle that passes through every vertex exactly once. This paper presents an $O(n^2 \log n)$ algorithm to determine whether a given circular-arc graph contains an HC. This algorithm is based on two subroutines for interval graphs: (i) a linear time greedy algorithm for the node disjoint path cover problem and (ii) a linear time HC algorithm. If the given graph does not contain an HC, this paper can produce a proof either through the deletion of an appropriate cutset or through the failure to obtain a specific type of HC.

Linear algorithm for optimal path cover problem on interval graphs

A path cover of a graph G is a set of vertex-disjoint paths that cover all the vertices of G. An optimal path cover of G is a path cover of minimum cardinality. This problem is known to be NP-complete for arbitrary graphs. We present a linear algorithm for this problem on interval graphs. Given the adjacency lists of an interval graph with n vertices and m edges, our algorithm runs in O( m+ n) time.

On the computational complexity of path cover problems

In this paper the complexity of a number of path cover problems in acyclic digraphs, acyclic structured digraphs, and rooted trees is considered. The problems deal with finding path covers of a certain size for (a) some elements of the digraph (e.g., required pairs, vertex subsets) or (b) the vertices of the digraph when restrictions are placed on the members of the path cover (e.g., impossible pairs, length-constrained covers). An attempt is made to characterize the complexity of path cover problems in general and a connection between the complexity of a path cover problem and the existence of a reachability relation on the elements that are to be covered in the digraph is pointed out.

On structured digraphs and program testing

Certain graph theoretic problems dealing with the testing of structured programs are treated. A structured digraph is a digraph that represents a structured program. A labelling procedure which characterizes structured digraphs is described. An efficient algorithm for finding a minimum path cover for the vertices of digraphs that belong to an important family of structured digraphs is given. To model interactions among code segments the notions of `required pairs' and `must pairs' are introduced and the corresponding constrained path cover problems are shown to be NP-complete even for acyclic structured digraphs.

On Path Cover Problems in Digraphs and Applications to Program Testing

In this paper various path cover problems, arising in program testing, are discussed. Dilworth's theorem for acyclic digraphs is generalized. Two methods for fmding a minimum set of paths (minimum path cover) that covers the vertices (or the edges) of a digraph are given. To model interactions among code segments, the notions of required pairs and required paths are introduced. It is shown that rmding a minimum path cover for a set of required pairs is NP-hard. An efficient algorithm is given for findng a minimum path cover for a set of required paths. Other constrained path problems are contsidered and their complexities are discussed.