Arc-disjoint Paths Research Articles

Almost disjoint paths and separating by forbidden pairs

By Menger's theorem, the maximum number of arc-disjoint paths from a vertex s to a vertex t in a directed graph equals the minimum number of arcs needed to disconnect s and t, i.e., the minimum size of an s-t-cut. The max-flow problem in a network with unit capacities is equivalent to the arc-disjoint paths problem. Moreover, the max-flow and min-cut problems form a strongly dual pair. We relax the disjointedness requirement on the paths, allowing them to be almost disjoint, meaning they may share up to one arc. The resulting almost disjoint paths problem (ADP) asks for k s-t-paths such that any two of them are almost disjoint. The separating by forbidden pairs problem (SFP) is the corresponding dual problem and calls for a set of k arc pairs such that every s-t-path contains both arcs of at least one such pair.In this paper, we explore these two problems, showing that they have an unbounded duality gap in general and analyzing their complexity. We prove that ADP is NP-complete when k is part of the input and that SFP is Σ2P-complete, even for acyclic graphs. Furthermore, we efficiently solve ADP when k≤2 is fixed and present a polynomial time algorithm based on dynamic programming for ADP when k is constant.

The directed 2-linkage problem with length constraints

The weak 2-linkage problem for digraphs asks for a given digraph and vertices s1,s2,t1,t2 whether D contains a pair of arc-disjoint paths P1,P2 such that Pi is an (si,ti)-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs [8]. Recently it was shown [3] that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair of constants k1,k2, there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P1,P2 such that Pi is an (si,ti)-path of length no more than d(si,ti)+ki, for i=1,2, where d(si,ti) denotes the length of the shortest (si,ti)-path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P1,P2 to the weak 2-linkage problem where each path Pi has length at most d(si,ti)+clog1+ϵ⁡n for some constant c.

Arc-Disjoint Paths in Decomposable Digraphs

We prove that the weak k-linkage problem is polynomial for every fixed k for totally i¾?-decomposable digraphs, under appropriate hypothesis on i¾?. We then apply this and recent results by Fradkin and Seymour on the weak k-linkage problem for digraphs of bounded independence number or bounded cut-width to get polynomial algorithms for some classes of digraphs like quasi-transitive digraphs, extended semicomplete digraphs, locally semicomplete digraphs all of which contain the class of semicomplete digraphs as a subclass and directed cographs.

Algorithms for finding a rooted [formula omitted]-edge-connected orientation

A digraph is called rooted (k,1)-edge-connected if it has a root node r0 such that there exist k arc-disjoint paths from r0 to every other node and there is a path from every node to r0. Here we give a simple algorithm for finding a (k,1)-edge-connected orientation of a graph. A slightly more complicated variation of this algorithm has running time O(n4+n2m) that is better than the time bound of the previously known algorithms. With the help of this algorithm one can check whether an undirected graph is highly k-tree-connected, that is, for each edge e of the graph G, there are k edge-disjoint spanning trees of G not containing e. High tree-connectivity plays an important role in the investigation of redundantly rigid body–bar graphs.

Data Transmission Reliability of a Two-source Single-sink Computer Network with a Common Arc

In a multi-state network (MSN), data can be transmitted through different paths. The previous literatures focus on the cases with arc-disjoint paths, but not the cases in which the paths have a common arc. In this work, we consider the case that two source nodes are sending data to the sink node through their own minimal paths (MP), where these two MPs possess a common arc in the computer network. Each source node generates a data sequence. The time intervals of the generated data are randomly distributed. Also, the amount of the data generated in different time instants are randomly distributed. These two variables can be obtained by the Monte-Carlo simulation method. As all the data are transmitted through a common arc, the data need to compete for the priority using the arc and conflictions may be caused. We consider the case that the two MPs are in connection and calculate the reliabilities of the data under their transmission time constraints. Simulation results reveal that the conflictions could prolong their transmission time and thus affect their reliabilities. This work may provide a cross-reference for managers to decide the amount of the data and the time intervals to get the ideal network reliabilities.

Arc-disjoint paths and trees in 2-regular digraphs

An out-branching (in-branching) Bs+(Bs−) in a digraph D is a connected spanning subdigraph of D in which every vertex x≠s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems: •Given a 2-regular digraph D, decide whether it contains two arc-disjoint branchings Bu+, Bv−.•Given a 2-regular digraph D, decide whether it contains an out-branching Bu+ such that D remains connected after removing the arcs of Bu+. Both problems are NP-complete for general digraphs (Bang-Jensen (1991) [1], Bang-Jensen and Yeo (2012) [7]). We prove that the first problem remains NP-complete for 2-regular digraphs, whereas the second problem turns out to be polynomial when we do not prescribe the root in advance. The complexity when the root is prescribed in advance is still open. We also prove that, for 2-regular digraphs, the second problem is in fact equivalent to deciding whether D contains two arc-disjoint out-branchings.

Spider Covers and Their Applications

We introduce two new combinatorial optimization problems: the Maximum Spider Problem and the Spider Cover Problem; we study their approximability and illustrate their applications. In these problems we are given a directed graph , a distinguished vertex , and a family D of subsets of vertices. A spider centered at vertex s is a collection of arc-disjoint paths all starting at s but ending into pairwise distinct vertices. We say that a spider covers a subset of vertices X if at least one of the endpoints of the paths constituting the spider other than s belongs to X. In the Maximum Spider Problem the goal is to find a spider centered at s that covers the maximum number of elements of the family D. Conversely, the Spider Cover Problem consists of finding the minimum number of spiders centered at s that covers all subsets in D. We motivate the study of the Maximum Spider and Spider Cover Problems by pointing out a variety of applications. We show that a natural greedy algorithm gives a 2-approximation algorithm for the Maximum Spider Problem and a -approximation algorithm for the Spider Cover Problem.

Fractional routing using pairs of failure-disjoint paths

Given a set of commodities and a network where some arcs can fail while others are reliable, we consider a routing problem with respect to a survivability requirement that each commodity can be split among pairs of failure-disjoint paths. Two paths p and p′ form a pair of failure-disjoint paths if they share only reliable arcs. The same flow is sent over p and p′, but the flow sent on a common reliable arc is not doubled.We present a compact linear formulation of the problem. Also three non-compact formulations solvable by column generation are introduced. In the first formulation, the generated columns correspond to pairs of failure-disjoint paths, while in the second formulation the generated columns correspond to simple paths. The third formulation is solved by generating pairs of arc-disjoint paths. All formulations are compared numerically. On top of that we study some generalizations and some special cases of the problem of computing a shortest pair of failure-disjoint paths. One of these generalizations is equivalent to a single-commodity capacitated network design problem.

The hardness of routing two pairs on one face

We prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two classes of parallel demand edges, both lying on the infinite face of the routing graph. This was one of the open challenges concerning disjoint paths, explicitly asked by Muller (Math Program 105 (2–3):275–288, 2006). It also strengthens Schwarzler’s recent proof of one of the open problems of Schrijver’s book (Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003), about the complexity of the edge-disjoint paths problem with terminals on the outer boundary of a planar graph. We also give a directed acyclic reduction. This proves that the arc-disjoint paths problem is NP-complete in directed acyclic graphs, even with only two classes of demand arcs.

The internet's unexploited path diversity

The connectivity of the Internet at the Autonomous System level is influenced by the network operator policies implemented. These in turn impose a direction to the announcement of address advertisements and, consequently, to the paths that can be used to reach back such destinations. We propose to use directed graphs to properly represent how destinations propagate through the Internet and the number of arc-disjoint paths to quantify this network¿s path diversity. Moreover, in order to understand the effects that policies have on the connectivity of the Internet, numerical analyses of the resulting directed graphs were conducted. Results demonstrate that, even after policies have been applied, there is still path diversity which the Border Gateway Protocol cannot currently exploit.

An effective algorithm for obtaining the whole set of minimal cost pairs of disjoint paths with dual arc costs

In telecommunication networks design the problem of obtaining optimal (arc or node) disjoint paths, for increasing network reliability, is extremely important. The problem of calculating k c disjoint paths from s to t (two distinct nodes), in a network with k c different (arbitrary) costs on every arc such that the total cost of the paths is minimised, is NP-complete even for k c =2. When k c =2 these networks are usually designated as dual arc cost networks. In this paper we propose an exact algorithm for finding the whole set of arc-disjoint path pairs, with minimal cost in a network with dual arc costs. The correctness of the algorithm is based on a condition which guarantees that the optimal path pair cost has been obtained and on a proposition which guarantees that at the end of the algorithm all the optimal pairs have been obtained. The optimality condition is based on the calculation of upper and lower bounds on the optimal cost. Extensive experimentation is presented to show the effectiveness of the algorithm.

Reserving resilient capacity for a single commodity with upper‐bound constraints

AbstractContinuing research begun in a previous study [SIAM J Discr Math 14 (2001), 524–539], we investigate problems of reserving capacity in the arcs of a network, subject to the constraint that, on the failure of any one arc, there is enough reserved capacity on the remaining arcs to support a flow of value T from a source s to a destination t. We also impose upper bounds on the amount of capacity we may reserve on the arcs: This alters the nature of the problem. In the case where each arc has the same upper bound, we investigate the strategy of finding the minimum‐cost reservation that is itself an acyclic (s, t) flow: We show that such a reservation is easy to find, always has a simple form, and has a cost at most twice that of the optimal solution. In the case where each arc has its own upper bound, we explain why no such results can hold, but we do give an efficient algorithm for the case where we are asked for a reservation on a fixed set of arc‐disjoint paths. We consider the case where we are free to reserve on each arc as much capacity as we want but only in bundles of fixed size. © 2003 Wiley Periodicals, Inc.

Arc-Disjoint Paths in Expander Digraphs

Given a digraph D=(V,A) and a set of $\kappa$ pairs of vertices in V, we are interested in finding, for each pair (xi, yi), a directed path connecting xi to yi such that the set of $\kappa$ paths so found is arc-disjoint. For arbitrary graphs the problem is ${\cal NP}$-complete, even for $\kappa=2$. We present a polynomial time randomized algorithm for finding arc-disjoint paths in an r-regular expander digraph D. We show that if D has sufficiently strong expansion properties and the degree r is sufficiently large, then all sets of $\kappa=\Omega(n/\log n)$ pairs of vertices can be joined. This is within a constant factor of best possible.

Wavelength rerouting in optical networks, or the Venetian Routing problem

Wavelength rerouting has been suggested as a viable and cost-effective method to improve the blocking performance of wavelength-routed wavelength-division multiplexing (WDM) networks. This method leads to the following combinatorial optimization problem, dubbed Venetian Routing. Given a directed multigraph G along with two vertices s and t and a collection of pairwise arc-disjoint paths, we wish to find an st-path which arc-intersects the smallest possible number of the given paths. In this paper we prove the computational hardness of this problem even in various special cases, and present several approximation algorithms for its solution. In particular we show a non-trivial connection between Venetian Routing and Label Cover.

Routing permutations and 2–1 routing requests in the hypercube

Let H n be the directed symmetric n-dimensional hypercube. Using the computer, we show that for any permutation of the vertices of H 4, there exists a system of pairwise arc-disjoint directed paths from each vertex to its target in the permutation. This verifies Szymanski's conjecture (Proceedings of the International Conference on Parallel Processing, 1989, pp. I-103–I-110) for n=4. We also consider the so-called 2–1 routing requests in H n , where any vertex can be used twice as a source but only once as a target; we construct for any n⩾3 a 2–1 request that cannot be routed in H n by arc-disjoint paths: in other words, for n⩾3, H n is not (2–1)-rearrangeable.

Reserving Resilient Capacity in a Network

We examine various problems concerning the reservation of capacity in a given network, where each arc has a per-unit cost, so as to be "resilient" against one or more arc failures. For a given pair (s,t) of nodes and demand T, we require that, on the failure of any k arcs of the network, there is sufficient reserved capacity in the remainder of the network to support an (s,t) flow of value T. This problem can be solved in polynomial time for any fixed k, but we show that it is NP-hard if we are required to reserve an integer capacity on each arc. We concentrate on the case where the reservation has to consist of a collection of arc-disjoint paths: here we give a very simple algorithm to find a minimum cost fractional solution, based on finding successive shortest paths in the network. Unlike traditional network flow problems, the integral version is NP-hard: we do, however, give a polynomial time $\frac{15}{14}$-approximation algorithm in the case k=1 and show that this bound is best possible unless P = NP.

Two Arc-Disjoint Paths in Eulerian Digraphs

Let G be an Eulerian digraph, and let {x1, x2}, {y1,y2} be two pairs of vertices in G. A directed path from a vertex s to a vertex t is called an st-path. An instance (G;{x1, x2}, {y1,y2}) is called feasible if there is a choice of h,i,j,k with {h,i} = {j,k} = {1,2} such that G has two arc-disjoint xhxi- and yjyk-paths. In this paper, we characterize the structure of minimal infeasible instances, based on which an O(m+nlog n) time algorithm is presented to decide whether a given instance is feasible, where n and m are the number of vertices and arcs in the instance, respectively. If the instance is feasible, the corresponding two arc-disjoint paths can be computed in O(m(m+nlog n)) time.

Cost-minimizing construction of a unidirectional SHR with diverse protection

The widespread use of SONET technology makes the self-healing ring (SHR) architecture the most basic building-block in designing a large fiber-optic network which is not only survivable but also cost-effective. We address the design problem of placing a single SONET unidirectional ring with a single gateway hub in a region administered by a community of interest. Introduced for the purpose of further cost-saving in our problem setting is the flexibility that some offices, instead of being included in the ring, can be homed to the ring via two arc-disjoint paths. Given the set of offices and potential arcs, the objective is then to determine at the minimum total cost both the ring location and the homing to the ring of its nonmember nodes. We formulate the problem as a mixed integer programming model and develop an efficient solution procedure by devising six improvement heuristics. Extensive computational experiments are conducted with input data instances selected from the data ranges of the real-world environments. The practical value of the solution procedure for network planners is well evidenced by its excellent and consistent performance of quickly generated good-quality solutions over various input data instances.

Minimum deadlock-free message routing restrictions in binary hypercubes

This article presents a minimum set of deadlock-free routing restrictions (referred to as the Restriction) for binary hypercube networks and a new deadlock-free multiple-path fixed routing algorithm which provides a total of k + 1 arc-disjoint paths between two nodes of distance k. It is known that deadlock can be prevented by routing restriction in some highly regular networks. However, since routing restrictions imply a limited utilization of the existing paths, there should be as few restrictions as possible. The Restriction presented here is a relaxed version of the restrictions implied by the normal fixed routing algorithm. Proofs are presented to show that any routing that complies with the Restriction is deadlock-free and that the Restriction is minimal in that any relaxation to the Restriction will cause deadlock. The multiple-path fixed routing algorithm presented in this article fully utilizes the arc-disjoint paths allowed by the Restriction. The algorithm can be obtained by a small modification to the normal fixed routing algorithm. Results from a simulation of the hypercube network using the new routing algorithm is presented, which shows that the new algorithm improves the communication capacity of the hypercube.

A TWO-PHASE NETWORK DESIGN HEURISTIC FOR MINIMUM COST WATER DISTRIBUTION SYSTEMS UNDER A RELIABILITY CONSTRAINT

This paper presents a two-phase design and optimization procedure for constructing a pipe network water distribution system having a built-in degree of reliability. The first phase is comprised of an algorithm called TREESEARCH which iteratively constructs a tree pipe network. Starting with a shortest-path based tree, the procedure employs a linear programming subproblem to systematically modify this tree by adding and deleting one link at a time, with the aim of reducing the cost of the network while satisfying the flow continuity, energy balance, and pressure head requirement constraints. The second phase of the algorithm, called REDUNDANCY, is concerned with the issue of reliability. In this phase, the tree network constructed by the algorithm TREESEARCH is augmented through the addition of links so that there are at least two arc-disjoint paths from each source node to every demand node it serves. This augmentation is performed through the use of a set covering problem which recommends the links to be...