An O(m log n) algorithm for the max + sum spanning tree problem

In a graph in which each edge has two weights, the max + sum spanning tree problem seeks a spanning tree that has the minimum value for the combined total of the maximum of one of the edge weights and the sum of the other weights among all the spanning trees in the graph. Exploiting an efficient data structure, an O(m log n) algorithm is presented for solving this problem. This improves the currently known bound of O(mn).     

Enumeration of Pareto optimal multi-criteria spanning trees – a proof of the incorrectness of Zhou and Gen's proposed algorithm

The minimum spanning tree (MST) problem is a well-known optimization problem of major significance in operational research. In the multi-criteria MST (mc-MST) problem, the scalar edge weights of the MST problem are replaced by vectors, and the aim is to find the complete set of Pareto optimal minimum-weight spanning trees. This problem is NP-hard and so approximate methods must be used if one is to tackle it efficiently. In an article previously published in this journal, a genetic algorithm (GA) was put forward for the mc-MST. To evaluate the GA, the solution sets generated by it were compared with solution sets from a proposed (exponential time) algorithm for enumerating all Pareto optimal spanning trees. However, the proposed enumeration algorithm that was used is not correct for two reasons: (1) It does not guarantee that all Pareto optimal minimum-weight spanning trees are returned. (2) It does not guarantee that those trees that are returned are Pareto optimal. In this short paper we prove these two theorems.     

On the uniform edge-partition of a tree

We study the problem of uniformly partitioning the edge set of a tree with n edges into k connected components, where k?n. The objective is to minimize the ratio of the maximum to the minimum number of edges of the subgraphs in the partition. We show that, for any tree and k?4, there exists a k-split with ratio at most two. For general k, we propose a simple algorithm that finds a k-split with ratio at most three in O(nlogk) time. Experimental results on random trees are also shown.     

Approximation algorithms for constrained generalized tree alignment problem

In generalized tree alignment problem, we are given a set S of k biologically related sequences and we are interested in a minimum cost evolutionary tree for S. In many instances of this problem partial phylogenetic tree for S is known. In such instances, we would like to make use of this knowledge to restrict the tree topologies that we consider and construct a biologically relevant minimum cost evolutionary tree. So, we propose the following natural generalization of the generalized tree alignment problem, a problem known to be MAX-SNP Hard, stated as follows:

Constrained Generalized Tree Alignment Problem [S. Divakaran, Algorithms and heuristics for constrained generalized alignment problem, DIMACS Technical Report 2007-21, 2007]: Given a set S of k related sequences and a phylogenetic forest comprising of node-disjoint phylogenetic trees that specify the topological constraints that an evolutionary tree of S needs to satisfy, construct a minimum cost evolutionary tree for S.

In this paper, we present constant approximation algorithms for the constrained generalized tree alignment problem. For the generalized tree alignment problem, a special case of this problem, our algorithms provide a guaranteed error bound of 2−2/k.     

New branch-and-bound algorithms for k-cardinality tree problems

Given an undirected graph, the k-cardinality tree problem (KCTP) is the problem of finding a subtree with exactly k edges whose sum of weights is minimum. In this paper we present a lower bound for KCTP based on the work by Kataoka et al. [Kataoka, S., N. Araki and T. Yamada, Upper and lower bounding procedures for the minimum rooted k-subtree problem, European Journal of Operational Research, 122 (2000), 561–569]. This new bound is the basis for the development of a branch-and-bound algorithm for the problem. Experiments carried out on instances from KCTLib revealed that the new exact algorithm largely outperforms the previous approach.     

Approximation hardness of min-max tree covers

We prove the first inapproximability bounds to study approximation hardness for a min-max k-tree cover problem and its variants. The problem is to find a set of k trees to cover vertices of a given graph with metric edge weights, so as to minimize the maximum total edge weight of any of the k trees. Our technique can also be applied to improve inapproximability bounds for min-max problems that use other covering objectives, such as stars, paths, and tours.     

On Minimum Edge Ranking Spanning Trees

In this paper, we introduce the problem of computing a minimum edge ranking spanning tree (MERST); i.e., find a spanning tree of a given graph G whose edge ranking is minimum. Although the minimum edge ranking of a given tree can be computed in polynomial time, we show that problem MERST is NP-hard. Furthermore, we present an approximation algorithm for MERST, which realizes its worst case performance ratio where n is the number of vertices in G and Δ* is the maximum degree of a spanning tree whose maximum degree is minimum. Although the approximation algorithm is a combination of two existing algorithms for the restricted spanning tree problem and for the minimum edge ranking problem of trees, the analysis is based on novel properties of the edge ranking of trees.     

On minimum power connectivity problems

Given a (directed or undirected) graph with edge costs, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Motivated by applications for wireless networks, we present polynomial and improved approximation algorithms, as well as inapproximability results, for some classic network design problems under the power minimization criteria. Our main result is for the problem of finding a min-power subgraph that contains k internally-disjoint vs-paths from every node v to a given node s: we give a polynomial algorithm for directed graphs and a logarithmic approximation algorithm for undirected graphs. In contrast, we will show that the corresponding edge-connectivity problems are unlikely to admit even a polylogarithmic approximation.     

Packing Steiner trees: polyhedral investigations

LetG=(V, E) be a graph andT⊆V be a node set. We call an edge setS a Steiner tree forT ifS connects all pairs of nodes inT. In this paper we address the following problem, which we call the weighted Steiner tree packing problem. Given a graphG=(V, E) with edge weightsw _e , edge capacitiesc _e ,e∈E, and node setT ₁,…,T _N , find edge setsS ₁,…,S _N such that eachS _k is a Steiner tree forT _k , at mostc _e of these edge sets use edgee for eache∈E, and the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from a routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the Steiner tree packing problem from a polyhedral point of view and define an associated polyhedron, called the Steiner tree packing polyhedron. The goal of this paper is to (partially) describe this polyhedron by means of inequalities. It turns out that, under mild assumptions, each inequality that defines a facet for the (single) Steiner tree polyhedron can be lifted to a facet-defining inequality for the Steiner tree packing polyhedron. The main emphasis of this paper lies on the presentation of so-called joint inequalities that are valid and facet-defining for this polyhedron. Inequalities of this kind involve at least two Steiner trees. The classes of inequalities we have found form the basis of a branch & cut algorithm. This algorithm is described in our companion paper (in this issue).     

The complete optimal stars-clustering-tree problem

We consider the following complete optimal stars-clustering-tree problem: Given a complete graph G=(V,E) with a weight on every edge and a collection of subsets of V, we want to find a minimum weight spanning tree T such that each subset of the vertices in the collection induces a complete star in T. One motivation for this problem is to construct a minimum cost (weight) communication tree network for a collection of (not necessarily disjoint) groups of customers such that each group induces a complete star. As a result the network will provide a “group broadcast” property, “group fault tolerance” and “group privacy”. We present another motivation from database systems with replications. For the case where the intersection graph of the subsets is connected we present a structure theorem that describes all feasible solutions. Based on it we provide a polynomial algorithm for finding an optimal solution. For the case where each subset induces a complete star minus at most k leaves we prove that the problem is NP-hard.     

An O(n2) simplex algorithm for a class of linear programs with tree structure

In connection with the optimal design of centralized circuit-free networks linear 0–1 programming problems arise which are related to rooted trees. For each problem the variables correspond to the edges of a given rooted tree T. Each path from a leaf to the root of T, together with edge weights, defines a linear constraint, and a global linear objective is to be maximized. We consider relaxations of such problems where the variables are not restricted to 0 or 1 but are allowed to vary continouosly between these bounds. The values of the optimal solutions of such relaxations may be used in a branch and bound procedure for the original 0–1 problem. While in [10] a primal algorithm for these relaxations is discussed, in this paper, we deal with the dual linear program and present a version of the simplex algorithm for its solution which can be implemented to run in time O(n²). For balanced trees T this time can be reduced to O(n log n).     

Minimum cut bases in undirected networks

Given an undirected, connected network G=(V,E) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as another graph theoretic problem closely related to it, namely, the cycle basis problem. We consider two versions of the problem: the unconstrained and the fundamental cut basis problem.For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem, and is thus solvable in strongly polynomial time. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each from a spanning tree T, is shown to be NP-hard. In this proof, we also show that a tree which induces the minimum fundamental cycle basis is also an optimal solution for the minimum fundamental cut basis problem in unweighted graphs.We present heuristics, integer programming formulations and summarize first experiences with numerical tests.     

Exact arborescences,matchings and cycles

Suppose we are given a graph in which edge has an integral weight. An ‘exact’ problem is to determine whether a desired structure exists for which the sum of the edge weights is exactly k for some prescribed k.We consider the special case of the problem in which all costs are zero or one for arborescences and show that a ‘continuity’ property is prossessed similar to that possessed by matroids. This enables us to determine in polynomial time the complete set of values of k for which a solution exists. We also give a minmax theorem for the maximum possible value of k, in terms of a packing of certain directed cuts in the graph.We also show how enumerative techniques can be used to solve the general exact problem for arborescences (implying spanning trees), perfect matchings in planar graphs and sets of disjoint cycles in a class of planar directed graphs which includes those of degree three. For these problems, we thereby obtain polynomial algorithms provided that the weights are bounded by a constant or encoded in unary.     

A contribution to the stochastic flow shop scheduling problem

This paper deals with performance evaluation and scheduling problems in m machine stochastic flow shop with unlimited buffers. The processing time of each job on each machine is a random variable exponentially distributed with a known rate. We consider permutation flow shop. The objective is to find a job schedule which minimizes the expected makespan. A classification of works about stochastic flow shop with random processing times is first given. In order to solve the performance evaluation problem, we propose a recursive algorithm based on a Markov chain to compute the expected makespan and a discrete event simulation model to evaluate the expected makespan. The recursive algorithm is a generalization of a method proposed in the literature for the two machine flow shop problem to the m machine flow shop problem with unlimited buffers. In deterministic context, heuristics (like CDS [Management Science 16 (10) (1970) B630] and Rapid Access [Management Science 23 (11) (1977) 1174]) and metaheuristics (like simulated annealing) provide good results. We propose to adapt and to test this kind of methods for the stochastic scheduling problem. Combinations between heuristics or metaheuristics and the performance evaluation models are proposed. One of the objectives of this paper is to compare the methods together. Our methods are tested on problems from the OR-Library and give good results: for the two machine problems, we obtain the optimal solution and for the m machine problems, the methods are mutually validated.     

On the approximability of the minimum strictly fundamental cycle basis problem

We consider the problem of finding a strictly fundamental cycle basis of minimum weight in the cycle space associated with an undirected connected graph G, where a nonnegative weight is assigned to each edge of G and the total weight of a basis is defined as the sum of the weights of all the cycles in the basis. Several heuristics have been proposed to tackle this NP-hard problem, which has some interesting applications. In this paper we show that this problem is APX-hard, even when restricted to unweighted graphs, and hence does not admit a polynomial-time approximation scheme, unless P=NP. Using a recent result on the approximability of lower-stretch spanning trees (Elkin et al. (2005) [7]), we obtain that the problem is approximable within O(log²nloglogn) for arbitrary graphs. We obtain tighter approximability bounds for dense graphs. In particular, the problem restricted to complete graphs admits a polynomial-time approximation scheme.     

A heuristic for the p-center problems in graphs

Generalizing a result of Hochbaum and Shmoys, a polynomial algorithm with a worst-case error ratio of 2 is described for the p-center problem is connected graphs with edge lengths and vertex weights. A slight modification of this algorithm provides ratio 2 also for the absolute p-center problem. Both these heuristics are best possible in the sense that any smaller ratio would imply that P = NP.     

On a labeling problem in graphs

Motivated by applications in software programming, we consider the problem of covering a graph by a feasible labeling. Given an undirected graph G=(V,E), two positive integers k and t, and an alphabet Σ, a feasible labeling is defined as an assignment of a set L_v⊆Σ to each vertex v∈V, such that (i) |L_v|≤k for all v∈V and (ii) each label α∈Σ is used no more than t times. An edge e={i,j} is said to be covered by a feasible labeling if L_i∩L_j≠0?. G is said to be covered if there exists a feasible labeling that covers each edge e∈E.In general, we show that the problem of deciding whether or not a tree can be covered is strongly NP-complete. For k=2, t=3, we characterize the trees that can be covered and provide a linear time algorithm for solving the decision problem. For fixed t, we present a strongly polynomial algorithm that solves the decision problem; if a tree can be covered, then a corresponding feasible labeling can be obtained in time polynomial in k and the size of the tree. For general graphs, we give a strongly polynomial algorithm to resolve the covering problem for k=2, t=3.     

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0.     

Variable neighborhood search for the vertex weighted k-cardinality tree problem

This paper presents some new heuristics based on variable neighborhood search to solve the vertex weighted k-cardinality tree problem. An efficient local search procedure is also developed for use within these heuristics. Our computational results demonstrate that the new heuristics substantially outperform the state-of-the-art methodologies, including a tabu search and genetic algorithm recently proposed in the literature. We also show that a decomposition approach is best for larger problem sizes than previously investigated. Thus, our findings advance in a significant way the capacity to solve this important class of problems.     

Efficient algorithms for merging

Efficient algorithms are given to find the maximum lengthn of an ordered list in which 4 elements can be merged using exactlyk comparisons. A top down algorithm for the (2,n) merge problem is discussed and is shown to obtain the optimal merge length first reported by Hwang and Lin. Our algorithms combine this top down approach and strong heuristics, some of which derived from Hwang's optimal algorithm for the (3,n) problem, and produce a lengthn which is close to the optimal lengthf ₄(k).