Minimum Number Of Paths Research Articles

Width Helps and Hinders Splitting Flows

Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow/circulation X on a directed graph G into weighted source-to-sink paths whose weighted sum equals X . We show that, for acyclic graphs, considering the width of the graph (the minimum number of paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class of width-stable graphs, for which a popular heuristic is a O (log Val(X))-approximation (Val(X) being the total flow of X ), and strengthen its worst-case approximation ratio from \(\Omega (\sqrt {m}) \) to Ω ( m /log m ) for sparse graphs, where m is the number of edges in the graph. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a (⌈log ‖ X ‖⌉ + 1)-approximation (‖ X ‖ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations (‖ X ‖ ≤ 1), using a generalised notion of width for this problem. Finally, we disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by [17], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

The Steiner cycle and path cover problem on interval graphs

The Steiner path problem is a common generalization of the Steiner tree and the Hamiltonian path problem, in which we have to decide if for a given graph there exists a path visiting a fixed set of terminals. In the Steiner cycle problem we look for a cycle visiting all terminals instead of a path. The Steiner path cover problem is an optimization variant of the Steiner path problem generalizing the path cover problem, in which one has to cover all terminals with a minimum number of paths. We study those problems for the special class of interval graphs. We present linear time algorithms for both the Steiner path cover problem and the Steiner cycle problem on interval graphs given as endpoint sorted lists. The main contribution is a lemma showing that backward steps to non-Steiner intervals are never necessary. Furthermore, we show how to integrate this modification to the deferred-query technique of Chang et al. to obtain the linear running times.

Induced path factors of regular graphs

AbstractAn induced path factor of a graph is a set of induced paths in with the property that every vertex of is in exactly one of the paths. The induced path number of is the minimum number of paths in an induced path factor of . We show that if is a connected cubic graph on vertices, then . Fix an integer . For each , define to be the maximum value of over all connected ‐regular graphs on vertices. As with even, we show that exists. We prove that and and that for .

A Novel Convolution-Based Algorithm for the Acyclic Network Symbolic Reliability Function Problem

Acyclic (binary-state) networks are commonly implemented in many diverse disciplines and applications, including information systems, processes management, project management. These networks owe their popularity to the fact that they do not use directed cycles. Network reliability is the most commonly used tool for evaluating and managing systems modeled on acyclic networks, and minimal paths (MPs) play a significant role in evaluating this reliability. This study therefore proposes a new algorithm based on a novel convolution concept for evaluating acyclic network reliability. The proposed algorithm is able to find all convolution-based MP sets within polynomial time, and then obtain the symbolic function of the acyclic network reliability in terms of those convolution-based MP sets based on the pivotal decomposition. Its total time complexity is O(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ), which is the best among all existing MP algorithms which are at least O(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|P|</sup> ), where O(|P|) = O(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ), and n and |P| are the number of nodes and MPs, respectively. The correctness and time complexity of the proposed algorithm is proven and examined. An example is used to display the novel convolution-based algorithm is implemented to solve the acyclic network reliability problem.

An Exact Enumeration Method to Find d-MPs in Multistate Networks

The enumeration approaches are important topics in network reliability calculation. For general purposes, the explicit enumeration (EE) is the popular method to apply. However, the low-cost feature of EE sacrifices its efficiency. A great improvement of EE is the invention of fast enumeration (FE), which creates a low-cost way of general purpose enumeration method with very good efficiency for applications. But for the enumeration in network reliability calculation, the optimal number of enumerations can be shown to be [Formula: see text], where [Formula: see text] is the number of minimal paths and [Formula: see text] is the demand of flow. FE still has the complexity far greater than the optimal one. This paper proposes an exact enumeration method for network reliability calculation, which has the complexity no greater than [Formula: see text]. This method greatly improves the enumeration efficiency than FE. So, it is believed to be very valuable to the large real-life applications. Benchmarks are made to show the efficiency of the proposed method.

Minimum Path Cover in Quasi-claw-Free Graphs

A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths; a minimum path cover of G is a path cover of G consisting of minimum number of paths; a path cover number of G, denoted by p(G), is the number of paths in a minimum path cover of G, i.e., $$p(G)=\min \{|{\mathcal {P}}|:$$$${\mathcal {P}}$$ is a path cover of $$G\}.$$ A graph G is quasi-claw-free if for any two vertices x, y with $$d(x,y)=2$$ in G, there exists a vertex u with $$u\in N(x)\cap N(y)$$ and $$N(u)\subseteq N[x]\cup N[y].$$ In this paper, we prove that for any quasi-claw-free graph G of order n, if $$\sigma _{k+1}(G)\ge {n-k}$$ for a positive integer k, then $$p(G)\le k,$$ where $$\sigma _{k+1}(G)$$ is the minimum degree sum of an independent set with $$k+1$$ vertices in G. Our result is a generalization of Ore’s sufficient conditions of hamiltonicity for quasi-claw-free graphs.

Covering 2‐connected 3‐regular graphs with disjoint paths

AbstractA path cover of a graph is a set of disjoint paths so that every vertex in the graph is contained in one of the paths. The path cover number of graph G is the cardinality of a path cover with the minimum number of paths. Reed in 1996 conjectured that a 2‐connected 3‐regular graph has path cover number at most . In this article, we confirm this conjecture.

On the Tractability of Shortest Path Problems in Weighted Edge-Coloured Graphs

A weighted edge-coloured graph is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such a graph paths are compared by their total weight in each colour, resulting in a Pareto set of minimal paths from one vertex to another. This paper will give a tight upper bound on the cardinality of a minimal set of paths for any weighted edge-coloured graph. Additionally, a bound is presented on the expected number of minimal paths in weighted edge–bicoloured graphs. These bounds indicate that despite weighted edge-coloured graphs are theoretically intractable, amenability to computation is typically found in practice.

Paths and cycles identifying vertices in twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The twisted cube is an important variation of the hypercube and preserves many of its desirable properties. Karpovsky et al. introduced the concept of identifying codes to model fault-detection in multiprocessor systems and Honkala et al. developed an identifying code by using cycles to identify the faulty processors in the hypercube. In this paper, we study the vertex identification problem on the twisted cube. We first propose an interesting construction scheme to build paths and cycles, and furthermore apply a minimum number of paths and cycles to identify the faulty processors of the twisted cube.

Path covering number and [formula omitted]-labeling number of graphs

A path covering of a graph G is a set of vertex disjoint paths of G containing all the vertices of G. The path covering number of G, denoted by P(G), is the minimum number of paths in a path covering of G. A k-L(2,1)-labeling of a graph G is a mapping f from V(G) to the set {0,1,…,k} such that |f(u)−f(v)|≥2 if dG(u,v)=1 and |f(u)−f(v)|≥1 if dG(u,v)=2. The L(2,1)-labeling numberλ(G) of G is the smallest number k such that G has a k-L(2,1)-labeling. The purpose of this paper is to study path covering numbers and L(2,1)-labeling numbers of graphs. Our main work extends most of the results in [S.S. Adams, A. Trazkovich, D.S. Troxell, B. Westgate, On island sequences of labelings with a condition at distance two, Discrete Appl. Math. 158 (2010) 1–7] and can answer an open problem in [J. Georges, D.W. Mauro, On the structure of graphs with non-surjective L(2, 1)-labelings, SIAM J. Discrete Math. 19 (2005) 208–223].

Equiparity Induced Path Decomposition in Trees

A decomposition of a graph G is a collection = {H1,H2,...,Hk} of subgraphs of G such that every edge ofG belongs to exactly oneHi. The decomposition is called a path decomposition ofG if each Hi is a path in G. Several studies have been undertaken on path decompositions by imposing certain conditions on the paths considered in the decomposition of the graph where the primary objective is to obtain the minimum number of paths required for a certain type of decomposition for a given graph. A path decomposition such that the paths in are induced as well as of same parity is defined as an equiparity induced path decomposition . The minimum number of paths in such a decomposition of a graph G is called the equiparity induced path decomposition number ofG and is denoted by pi(G). In this paper we determine the value of pi for trees of even size.

MASPTA-O: Multiagent System Planning to Avoid Threats Optimally

ABSTRACT Extensive use of modern technologies with expanding connectivity has created high risk threats requiring innovative strategies to counter them. In the literature, multiagent system planning has been used for threat avoidance where agents are employed in all the identified attack paths to avert the threat. This paper presents the two-level optimal approach MASPTA-O, or Multiagent System Planning to Avoid Threats Optimally. The first level identifies optimum number of threats to be mitigated using the Layered Threat Elimination model. At the second level, each identified threat is optimally mitigated by generating various mitigation plans using Mitigation Plan Generation algorithm and selecting minimum number of paths to avoid the threat. Agents are then inducted in these minimum number of attacker's paths to mitigate the threats. In the proposed work, a Master Agent has been introduced to continuously monitor the performance of the active agents for any abnormality and provides an alert signal to save the system from being compromised. Experimental study using the Java Agent Development Environment (JADE) has been conducted to test the proposed approach.

Finding a minimum path cover of a distance-hereditary graph in polynomial time

A path cover of a graph G = ( V , E ) is a set of pairwise vertex-disjoint paths such that the disjoint union of the vertices of these paths equals the vertex set V of G . The path cover problem is, given a graph, to find a path cover having the minimum number of paths. The path cover problem contains the Hamiltonian path problem as a special case since finding a path cover, consisting of a single path, corresponds directly to the Hamiltonian path problem. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. The complexity of the path cover problem on distance-hereditary graphs has remained unknown. In this paper, we propose the first polynomial-time algorithm, which runs in O( | V | 9 ) time, to solve the path cover problem on distance-hereditary graphs.

Vertex covering by paths on trees with its applications in machine translation

Given a tree and a set of paths in the tree, the problem of finding a minimum number of paths from the given path set to cover all the vertices in the tree is investigated in the paper. To distinguish from the classical path cover problem, such an optimization problem is referred to as vertex covering by paths. The problem and its edge variant, edge covering by paths, find applications in machine translation. Complexity and algorithmic results are presented for the problem and its edge variant.

The multiple TSP with time windows: vehicle bounds based on precedence graphs

Vehicle bounds for the multiple traveling salesman problem with time windows are found by covering two precedence graphs with the minimum number of paths. Instances with tight bounds are presented, as well as instances for which the bounds are loose. The similarity of these instances is discussed.

A Novel Approach to Determine Minimal Tie-Sets of Complex Network

Network reliability analysis is usually based on minimal path or cut enumeration from which the associated reliability expressions are deduced. The cut-set method is a popular approach in the reliability analysis of many systems from simple to complex configurations. The computational requirements necessary to determine the minimal cut-sets of a network depend mainly on the number of minimal paths between the source and the sink. A technique designated as the Path Tracing Algorithm is presented in this paper, which can handle both simple and complex networks, and considers both unidirectional and bi-directional branches. A step by step procedure is explained using a bridge-network. The algorithm is easy to program, and does not require limits on the size of the network. The applicability of the proposed technique is illustrated by application to a more complicated system.

On the k-path partition of graphs

The k-path partition problem is to partition a graph into the minimum number of paths, so that none of them has length more than k, for a given positive integer k. The problem is a generalization of the Hamiltonian path problem and the problem of partitioning a graph into the minimum number of paths. The k-path partition problem remains NP-complete on the class of chordal bipartite graphs if k is part of the input, and we show that it is NP-complete on the class of comparability graphs even for k=3. On the positive side, we present a polynomial-time solution for the problem, with any k, on bipartite permutation graphs, which form a subclass of chordal bipartite graphs.

Conduction in Granular Metals by Virtual Tunneling on the Fractal Percolation Cluster

We discuss the calculation of the dc conductivity of granular metals on the insulating side of the metal-insulator transition. The granular structure is represented by a fractal percolation cluster and the problem of virtual tunneling-assisted conductivity is shown to be related to estimating the number of minimal paths for the problem of chemical distance metric on the fractal. The resulting conductivity temperature dependence has the form ln σ const - (T 0 /T) x , where x = ζ/(D B + ζ), D B is the fractal dimensionality of the backbone cluster and ζ is the superlocalization exponent. This gives the value of x close to 0.4 both in two and three dimensions that agrees fairly well with the experimental value x 1/2 for many granular conductors.

Gallai's conjecture for disconnected graphs

The path number p( G) of a graph G is the minimum number of paths needed to partition the edge set of G. Gallai conjectured that p( G)⩽⌊( n+1)/2⌋ for every connected graph G of order n. Because the graph consisted of disjoint triangles, the best one could hope for in the disconnected case is p(G)⩽⌊ 2 3 n⌋ . We prove the sharper result that p(G)⩽ 1 2 u+⌊ 2 3 g⌋ where u is the number of odd vertices and g is the number of nonisolated even vertices.

Efficient delay yield estimate of digital circuits

A theoretical approach is presented for accurately estimating the delay yield of CMOS digital circuits when inter-die and intra-die variations are considered. Where this estimate is required to be of greater accuracy, a criterion is presented to determine the minimum number of paths that should be analysed by Monte Carlo simulation to achieve the required accuracy.