On the K shortest path trees problem

We address the problem of finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k-1) shortest path trees. Consequently, we design an O(f(n,m,C_max)+Km) time and O(K+m) space algorithm to determine the K shortest path trees, in a directed network with n nodes, m arcs and maximum absolute length C_max, where O(f(n,m,C_max)) is the best time needed to solve the shortest simple paths connecting a source node with any other non-source node.     

A comprehensive survey on the quickest path problem

This work is a survey on a special minsum-maxmin bicriteria problem, known as the quickest path problem, that can model the transmission of data between two nodes of a network. Moreover, the authors review the problems of ranking the K quickest paths, and the K quickest loopless paths, and compare them in terms of the worst-case complexity order. The classification presented led to the proposal of a new variant of a known K quickest loopless paths algorithm. Finally, applications of quickest path algorithms are mentioned, as well as some comparative empirical results. This work was partially supported by FEDER and OE, under the projects POCTI/MAT/139/2001, POCTI/ISFL-1/152, POSI/SRI/37346/2001, and POCTI/MAT/37707/2001.     

Some optimal path problems subject to improvements

In this paper, the functional equation technique of dynamic programming is applied to solve the problems of a) determining an optimal path from a given origin to a fixed destination when the path is subject to a given number of improvements, b) finding an optimal path from a given origin to an assigned destination by passing at least once through each node of a set of specified nodes when the path is subject to a given number of improvements, c) obtaining an optimal path from a given origin to a fixed destination by passing at least once through at least one node of each ofK sets of specified nodes when the path is subject to a given number of improvements.

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Zusammenfassung In dieser Arbeit wird die Funktionalgleichung des dynamischen Programmierens verwendet, um folgende drei Netzwerkprobleme zu lösen: a) Bestimmung eines optimalen Pfades von einem gegebenen Anfangs- zu einem gegebenen Endknoten, wenn längs einem Pfad eine gewisse Anzahl von Verbesserungen möglich sind. b) Wie a), wobei zusätzlich der Pfad mindestens einmal durch jeden Knoten einer spezifizierten Knotenmenge gehen soll. c) Wie a), wobei der Pfad durch mindestens einen Knoten in jeder vonK spezifizierten Knotenmenge gehen soll.

     

Algorithms for shortest paths and d-cycle problems

Let G be a weighted graph with n vertices and m edges. We address the d-cycle problem, i.e., the problem of finding a subgraph of minimum weight with given cyclomatic number d. Hartvigsen [Minimum path bases, J. Algorithms 15 (1993) 125–142] presented an algorithm with running time O(n²m) and O(n^2d−1m²) for the cyclomatic numbers d=1 and d2, respectively. Using a (d+1)-shortest-paths algorithm, we develop a new more efficient algorithm for the d-cycle problem with running time O(n^2d−1+n²m+n³logn).     

Improved algorithms for path partition and related problems

We study the L_∞ path partition problem: given a path of n weighted vertices and an integer k, remove k−1 edges from the path so that the maximum absolute deviation of the weights of the resulting k sub-paths from their mean is minimized. Previously, the best algorithm solves this problem in O(nklogk) time. We present an O(nk) time algorithm. We also give improved solutions for two related problems: the L_d path partition problem and the web proxies placement problem.     

New scaling algorithms for the assignment and minimum mean cycle problems

In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recentauction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing the optimality conditions, and the amount of relaxation is successively reduced to zero. On a network with 2n nodes,m arcs, and integer arc costs bounded byC, the algorithm runs in O( m log(nC)) time and uses very simple data structures. This time bound is comparable to the time taken by Gabow and Tarjan's scaling algorithm, and is better than all other time bounds under thesimilarity assumption, i.e.,C = O(n ^k ) for somek. We next consider the minimum mean cycle problem. Themean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains. Theminimum mean cycle problem is to identify a cycle whose mean cost is minimum. We show that by using ideas of the assignment algorithm in an approximate binary search procedure, the minimum mean cycle problem can also be solved in O( m lognC) time. Under the similarity assumption, this is the best available time bound to solve the minimum mean cycle problem.     

A new pivot selection rule for the network simplex algorithm

We present a new network simplex pivot selection rule, which we call theminimum ratio pivot rule, and analyze the worst-case complexity of the resulting network simplex algorithm. We consider networks withn nodes,m arcs, integral arc capacities and integral supplies/demands of nodes. We define a {0, 1}-valued penalty for each arc of the network. The minimum ratio pivot rule is to select that eligible arc as the entering arc whose addition to the basis creates a cycle with the minimum cost-to-penalty ratio. We show that the so-defined primal network simplex algorithm solves minimum cost flow problem within O(nΔ) pivots and in O(Δ(m + n logn)) time, whereΔ is any upper bound on the sum of all arc flows in every feasible flow. For assignment and shortest path problems, our algorithm runs in O(n ²) pivots and O(nm +n ² logn) time.     

Bipartite anti‐Ramsey numbers of cycles

We determine the maximum number of colors in a coloring of the edges of K_m,n such that every cycle of length 2k contains at least two edges of the same color. One of our main tools is a result on generalized path covers in balanced bipartite graphs. For positive integers q≤ a, let g(a,q) be the maximum number of edges in a spanning subgraph G of K_a,a such that the minimum number of vertex‐disjoint even paths and pairs of vertices from distinct partite sets needed to cover V(G) is q. We prove that g(a,q) = a² ? aq + max {a, 2q ? 2}. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 9–28, 2004     

Efficient reduction for path problems on circular-arc graphs

An efficient reduction process for path problems on circular-arc graphs is introduced. For the parity path problem, this reduction gives anO(n+m) algorithm for proper circular-arc graphs, and anO(n+m) algorithm for general circular-arc graphs. This reduction also gives anO(n+m) algorithm for the two path problem on circular-arc graphs.     

The tricriterion shortest path problem with at least two bottleneck objective functions

The focus of this paper is on the tricriterion shortest path problem where two objective functions are of the bottleneck type, for example MinMax or MaxMin. The third objective function may be of the same kind or we may consider, for example, MinSum or MaxProd. Let p(n) be the complexity of a classical single objective algorithm responsible for this third function, where n is the number of nodes and m be the number of arcs of the graph. An O(m²p(n)) algorithm is presented that can generate the minimal complete set of Pareto-optimal solutions. Finding the maximal complete set is also possible. Optimality proofs are given and extensions for several special cases are presented. Computational experience for a set of randomly generated problems is reported.     

Polynomial algorithms for a class of minimum rank-two cost path problems

In this paper, we develop two algorithms for finding a directed path of minimum rank-two monotonic cost between two specified nodes in a network with n nodes and m arcs. Under the condition that one of the vectors characterizing the cost function f is binary, one yields an optimal solution in O(n³) or O(nm log n) time if f is quasiconcave; the other solves any problem in O(nm + n ² log n) time.     

On vertex symmetric digraphs

The problem of how “near” we can come to a n-coloring of a given graph is investigated. I.e., what is the minimum possible number of edges joining equicolored vertices if we color the vertices of a given graph with n colors. In its generality the problem of finding such an optimal color assignment to the vertices (given the graph and the number of colors) is NP-complete. For each graph G, however, colors can be assigned to the vertices in such a way that the number of offending edges is less than the total number of edges divided by the number of colors. Furthermore, an Ω(epn) deterministic algorithm for finding such an n-color assignment is exhibited where e is the number of edges and p is the number of vertices of the graph (e?p?n). A priori solutions for the minimal number of offending edges are given for complete graphs; similarly for equicolored K_m in K_p and equicolored graphs in K_p.     

Computational experiments with a lazy version of a <Emphasis Type="Italic">K</Emphasis> quickest simple path ranking algorithm

The quickest path problem is related to the classical shortest path problem, but its objective function concerns the transmission time of a given amount of data throughout a path, which involves both cost and capacity. The K-quickest simple paths problem generalises the latter, by looking for a given number K of simple paths in non-decreasing order of transmission time. Two categories of algorithms are known for ranking simple paths according to the transmission time. One is the adaptation of deviation algorithms for ranking shortest simple paths (Pascoal et al. in Comput. Oper. Res. 32(3):509–520, 2005; Rosen et al. in Comput. Oper. Res. 18(6):571–584, 1991), and another is based on ranking shortest simple paths in a sequence of networks with fixed capacity lower bounds (Chen in Inf. Process. Lett. 50:89–92, 1994), and afterwards selecting the K quickest ones. After reviewing the quickest path and the K-quickest simple paths problems we describe a recent algorithm for ranking quickest simple paths (Pascoal et al. in Ann. Oper. Res. 147(1):5–21, 2006). This is a lazy version of Chen’s algorithm, able to interchange the calculation of new simple paths and the output of each k-quickest simple path. Finally, the described algorithm is computationally compared to its former version, as well as to deviation algorithms.      

Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints.We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge.Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.     

On the complexity of some arborescences finding problems on a multihop radio network

An arborescence of a multihop radio network is a directed spanning tree (with rootx) such that the edges are directed away from the root. Based upon an arborescence,x canbroadcast a message to other nodes according to the directed edges of the spanning tree. The minimum transmission power arborescence problem is to find an arborescence such that the message can be broadcasted to other nodes by using a minimal amount of transmission power. The minimum delay arborescence problem is to find an arborescence such that a message can be broadcasted to other nodes by using a minimal number of broadcast transmission. In this paper we show that both these problems areNP-complete. The reductions are from the maximum leaf spanning tree problem.Areverse arborescence is similar to an arborescence except that the edges are directed toward the root. Based upon a reverse arborescence, the root node cancollect information from other nodes. In this paper we also show that the reverse minimum transmission power arborescence problem can be solved with the same computational complexity as that of finding a minimum cost spanning tree, and the reverse minimum delay arborescence problem can be solved with the same computational complexity as that of finding a spanning tree.     

On dual minimum cost flow algorithms

We describe a new dual algorithm for the minimum cost flow problem. It can be regarded as a variation of the best known strongly polynomial minimum cost flow algorithm, due to Orlin. Indeed we obtain the same running time of O(m log m(m+n log n)), where n and m denote the number of vertices and the number of edges. However, in contrast to Orlin's algorithm we work directly with the capacitated network (rather than transforming it to a transshipment problem). Thus our algorithm is applicable to more general problems (like submodular flow) and is likely to be more efficient in practice.  Our algorithm can be interpreted as a cut cancelling algorithm, improving the best known strongly polynomial bound for this important class of algorithms by a factor of m. On the other hand, our algorithm can be considered as a variant of the dual network simplex algorithm. Although dual network simplex algorithms are reportedly quite efficient in practice, the best worst-case running time known so far exceeds the running time of our algorithm by a factor of n.     

Comparing different metaheuristic approaches for the median path problem with bounded length

In this paper we consider the Bounded Length Median Path Problem which can be defined as the problem of locating a path-shaped facility that departures from a given origin and arrives at a given destination in a network. The length of the path is assumed to be bounded by a given maximum length. At each vertex of the network (customer-point) the demand for the service is given and the cost to reach the closest service-point is computed. The objective is to minimize the sum of these costs over all the customer-points in the network.     

Increasing the Weight of Minimum Spanning Trees

The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NP-hard and an Ω(1/log k)-approximation algorithm is presented for it, where (input parameter) k > 1 is the number of edges to be removed. The second problem is studied, assuming that the increase in the weight of an edge has an associated cost proportional to the magnitude of the change. An O(n³m² log(n²/m)) time algorithm is presented to solve it.     

On minimum degree in Hamiltonian path graphs

For integers p and s satisfying 2 ? s ? p ? 1, let m(p,s) denote the maximum number of edges in a graph G of order p such that the minimum degree in the hamiltonian path graph of G equals s. The values of m(p, s) are determined for 2 ? s ? p/2 and for (2p ? 2)/3 ? s ? p ? 1, and upper and lower bounds on m(p, s) are obtained for p/2 < s < (2p ? 2)/3.