Connecting colored point sets

We study the following Ramsey-type problem. Let S=B∪R be a two-colored set of n points in the plane. We show how to construct, in time, a crossing-free spanning tree T(B) for B, and a crossing-free spanning tree T(R) for R, such that both the number of crossings between T(B) and T(R) and the diameters of T(B) and T(R) are kept small. The algorithm is conceptually simple and is implementable without using any non-trivial data structure. This improves over a previous method in Tokunaga [Intersection number of two connected geometric graphs, Inform. Process. Lett. 59 (1996) 331-333] that is less efficient in implementation and does not guarantee a diameter bound. Implicit to our approach is a new proof for the result in the reference above on the minimum number of crossings between T(B) and T(R).     

Efficient algorithms for a family of matroid intersection problems

Consider a matroid where each element has a real-valued cost and a color, red or green; a base is sought that contains q red elements and has smallest possible cost. An algorithm for the problem on general matroids is presented, along with a number of variations. Its efficiency is demonstrated by implementations on specific matroids. In all cases but one, the running time matches the best-known algorithm for the problem without the red element constraint: On graphic matroids, a smallest spanning tree with q red edges can be found in time O(n log n) more than what is needed to find a minimum spanning tree. A special case is finding a smallest spanning tree with a degree constraint; here the time is only O(m + n) more than that needed to find one minimum spanning tree. On transversal and matching matroids, the time is the same as the best-known algorithms for a minimum cost base. This also holds for transversal matroids for convex graphs, which model a scheduling problem on unit-length jobs with release times and deadlines. On partition matroids, a linear-time algorithm is presented. Finally an algorithm related to our general approach finds a smallest spanning tree on a directed graph, where the given root has a degree constraint. Again the time matches the best-known algorithm for the problem without the red element (i.e., degree) constraint.     

Uniform election in trees and polyominoids

Election is a classical paradigm in distributed algorithms. This paper aims to design and analyze a distributed algorithm choosing a node in a graph which models a network. In case the graph is a tree, a simple schema of algorithm acts as follows: it removes leaves until the graph is reduced to a single vertex; the elected one. In Métivier et al. (2003) [7], the authors studied a randomized variant of this schema which gives the same probability of being elected to each node of the tree. They conjectured that the expected election duration of this algorithm is O(ln(n)) where n denotes the size of the tree, and asked whether it is possible to use the same algorithm to obtain a fair election in other classes of graphs.In this paper, we prove their conjecture. We then introduce a new structure called polyominoid graphs. We show how a spanning tree for these graphs can be computed locally so that our algorithm, applied to this spanning tree, gives a uniform election algorithm on polyominoids.     

The complexity of a minimum reload cost diameter problem

We consider the minimum diameter spanning tree problem under the reload cost model which has been introduced by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: Minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85]. In this model an undirected edge-coloured graph G is given, together with a nonnegative symmetrical integer matrix R specifying the costs of changing from a colour to another one. The reload cost of a path in G arises at its internal nodes, when passing from the colour of one incident edge to the colour of the other. We prove that, unless P=NP, the problem of finding a spanning tree of G having a minimum diameter with respect to reload costs, when restricted to graphs with maximum degree 4, cannot be approximated within any constant α<2 if the reload costs are unrestricted, and cannot be approximated within any constant β<5/3 if the reload costs satisfy the triangle inequality. This solves a problem left open by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85].     

Obligation rules for minimum cost spanning tree situations and their monotonicity properties

We consider the class of Obligation rules for minimum cost spanning tree situations. The main result of this paper is that such rules are cost monotonic and induce also population monotonic allocation schemes. Another characteristic of Obligation rules is that they assign to a minimum cost spanning tree situation a vector of cost contributions which can be obtained as product of a double stochastic matrix with the cost vector of edges in the optimal tree provided by the Kruskal algorithm. It turns out that the Potters value (P-value) is an element of this class.     

An optimal pram algorithm for a spanning tree on trapezoid graphs

LetG be a graph withn vertices andm edges. The problem of constructing a spanning tree is to find a connected subgraph ofG withn vertices andn?1 edges. In this paper, we propose anO(logn) time parallel algorithm withO(n/logn), processors on an EREW PRAM for constructing a spanning tree on trapezoid graphs.     

Np-hardness proof and an approximation algorithm for the minimum vertex ranking spanning tree problem

The minimum vertex ranking spanning tree problem (MVRST) is to find a spanning tree of G whose vertex ranking is minimum. In this paper, we show that MVRST is NP-hard. To prove this, we polynomially reduce the 3-dimensional matching problem to MVRST. Moreover, we present a (⌈D_s/2⌉+1)/(⌊log₂(D_s+1)⌋+1)-approximation algorithm for MVRST where D_s is the minimum diameter of spanning trees of G.     

Reformulations and solution algorithms for the maximum leaf spanning tree problem

Given a graph G = (V, E), the maximum leaf spanning tree problem (MLSTP) is to find a spanning tree of G with as many leaves as possible. The problem is easy to solve when G is complete. However, for the general case, when the graph is sparse, it is proven to be NP-hard. In this paper, two reformulations are proposed for the problem. The first one is a reinforced directed graph version of a formulation found in the literature. The second recasts the problem as a Steiner arborescence problem over an associated directed graph. Branch-and-Cut algorithms are implemented for these two reformulations. Additionally, we also implemented an improved version of a MLSTP Branch-and-Bound algorithm, suggested in the literature. All of these algorithms benefit from pre-processing tests and a heuristic suggested in this paper. Computational comparisons between the three algorithms indicate that the one associated with the first reformulation is the overall best. It was shown to be faster than the other two algorithms and is capable of solving much larger MLSTP instances than previously attempted in the literature.     

An Inverse-Ackermann Type Lower Bound For Online Minimum Spanning Tree Verification*

Given a spanning tree T of some graph G, the problem of minimum spanning tree verification is to decide whether T = MST(G). A celebrated result of Komlós shows that this problem can be solved with a linear number of comparisons. Somewhat unexpectedly, MST verification turns out to be useful in actually computing minimum spanning trees from scratch. It is this application that has led some to wonder whether a more flexible version of MST verification could be used to derive a faster deterministic minimum spanning tree algorithm. In this paper we consider the online MST verification problem in which we are given a sequence of queries of the form “Is e in the MST of T ∪{e}?”, where the tree T is fixed. We prove that there are no linear-time solutions to the online MST verification problem, and in particular, that answering m queries requires Ω(mα(m,n)) time, where α(m,n) is the inverse-Ackermann function and n is the size of the tree. On the other hand, we show that if the weights of T are permuted randomly there is a simple data structure that preprocesses the tree in expected linear time and answers queries in constant time. * A preliminary version of this paper appeared in the proceedings of the 43rd IEEE Symposium on Foundations of Computer Science (FOCS 2002), pages 155–163. † This work was supported by Texas Advanced Research Program Grant 003658-0029-1999, NSF Grant CCR-9988160, and an MCD Graduate Fellowship.     

David G. Luenberger,Linear and nonlinear programming, 2nd edition,Addison-Wesley,Reading (1984), xv + 492 pages, $35.50

This paper considers a stochastic version of the linear continuous type knapsack problem in which the cost coefficients are random variables. The problem is to find an optimal solution and an optimal probability level of the chance constraint. This problem P₀ is first transformed into a deterministic equivalent problem P. Then a subproblem with a positive parameter is introduced and a close relation between P and its subproblem is shown. Further, an auxiliary problem of the subproblem is introduced and a direct relation between P and the auxiliary problem is derived through a relation connecting the subproblem and its auxiliary problem. Fully utilizing these relations, an efficient algorithm is proposed that finds an optimal solution of P in at most O(n⁴) computational time where n is the number of decision variables. Finally, further research problems are discussed.     

An O(n) algorithm for discrete n-point convex approximation with applications to continuous case

Given a bounded real function ? defined on a closed bounded real interval I, the problem is to find a convex function g so as to minimize the supremum of $\text{¦f(t) ? g(t)¦}$ for all t in I, over the class of all convex functions on I. The usual approach is to consider a discrete version of the problem on a grid of (n + 1) points in I, apply a conventional linear program to obtain an optimal solution, and let the grid size go to zero. This paper presents an alternative algorithm of complexity O(n), which is based on the concept of the greatest convex minorant of a function, for computation of a special “maximal” optimal solution to the discrete problem. It establishes the rate of convergence of this optimal solution to a solution of the original problem as the grid size goes to zero. It presents an alternative efficient linear program that generates the maximal optimal solution to the discrete problem. It also gives an O(n) algorithm for the discrete n-point monotone approximation problem.     

Rectilinear line segment intersection,layered segment trees,and dynamization

The aim of the present paper is to provide an efficient solution to the following problem: “Given a family of n rectilinear line segments in two-space report all intersections in the family with a query consisting of an arbitrary rectilinear line segment.” We provide an algorithm which takes O(nlog₂n) preprocessing time, o(nlog₂n) space and O(log₂n + k) query time, where k is the number of reported intersections. This solution serves to introduce a powerful new data structure, the layered segment tree, which is of independent interest. Second it yields, by way of recent dynamization techniques, a solution to the on-line version of the above problem, that is the operations INSERT and DELETE and QUERY with a line segment are allowed. Third it also yields a new nonscanning solution to the batched version of the above problem. Finally we apply these techniques to the problem obtained by replacing “line segment” by “rectangle” in the above problem, giving an efficient solution in this case also.     

Solving the deterministic and stochastic uncapacitated facility location problem: from a heuristic to a simheuristic

The uncapacitated facility location problem (UFLP) is a popular combinatorial optimization problem with practical applications in different areas, from logistics to telecommunication networks. While most of the existing work in the literature focuses on minimizing total cost for the deterministic version of the problem, some degree of uncertainty (e.g., in the customers’ demands or in the service costs) should be expected in real-life applications. Accordingly, this paper proposes a simheuristic algorithm for solving the stochastic UFLP (SUFLP), where optimization goals other than the minimum expected cost can be considered. The development of this simheuristic is structured in three stages: (i) first, an extremely fast savings-based heuristic is introduced; (ii) next, the heuristic is integrated into a metaheuristic framework, and the resulting algorithm is tested against the optimal values for the UFLP; and (iii) finally, the algorithm is extended by integrating it with simulation techniques, and the resulting simheuristic is employed to solve the SUFLP. Some numerical experiments contribute to illustrate the potential uses of each of these solving methods, depending on the version of the problem (deterministic or stochastic) as well as on whether or not a real-time solution is required.     

Finding minimal spanning trees in a euclidean coordinate space

The minimal spanning tree problem of a point set in ak-dimensional Euclidean space is considered and a new version of the multifragmentMST-algorithm of Bentley and Friedman is given. The minimal spanning tree is found by repeatedly joining the minimal subtree with the closest subtree. Ak-d tree is used for choosing the connecting edges. Computation time of the algorithm depends on the configuration of the point set: for normally distributed random points the algorithm is very fast. Two extreme cases demandingO(n logn) andO(n ²) operations,n being the cardinality of the point set, are also given.     

A multi-population hybrid biased random key genetic algorithm for hop-constrained trees in nonlinear cost flow networks

Genetic algorithms and other evolutionary algorithms have been successfully applied to solve constrained minimum spanning tree problems in a variety of communication network design problems. In this paper, we enlarge the application of these types of algorithms by presenting a multi-population hybrid genetic algorithm to another communication design problem. This new problem is modeled through a hop-constrained minimum spanning tree also exhibiting the characteristic of flows. All nodes, except for the root node, have a nonnegative flow requirement. In addition to the fixed charge costs, nonlinear flow dependent costs are also considered. This problem is an extension of the well know NP-hard hop-constrained Minimum Spanning Tree problem and we have termed it hop-constrained minimum cost flow spanning tree problem. The efficiency and effectiveness of the proposed method can be seen from the computational results reported.     

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).     

Projection problems for symmetric polytopes

Let K⊂Rn be a convex body (a compact, convex subset with non-empty interior), ΠK its projection body. Finding the least upper bound, as K ranges over the class of origin-symmetric convex bodies, of the affine-invariant ratio V(ΠK)/V(K)ⁿ⁻¹, being called Schneider's projection problem, is a well-known open problem in the convex geometry. To study this problem, Lutwak, Yang and Zhang recently introduced a new affine invariant functional for convex polytopes in Rn. For origin-symmetric convex polytopes, they posed a conjecture for the new functional U(P). In this paper, we give an affirmative answer to the conjecture in Rn, thereby, obtain a modified version of Schneider's projection problem.     

A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low-weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning treeTusingadoptionsto meet the degree constraints is considered. A novel network-flow-based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previous algorithm. If the degree constraintd(v) for eachvis at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 − min{(d(v) − 2)/(deg_T(v) − 2) : deg_T(v) > 2}, where deg_T(v) is the initial degree ofv. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds foranyalgorithm at all, then it also holds for the given algorithm. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. ChoosingTto be a minimum spanning tree yields approximation algorithms with factors less than 2 for the general problem on geometric graphs with distances induced by variousL_pnorms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.     

Lower semicontinuity property of multiparameter optimal stopping value and its application to multiparameter prophet inequalities

This paper is concerned with the optimal stopping problem for discrete time multiparameter stochastic processes with the index set Nd. The optimal stopping value of a discrete time multiparameter integrable stochastic process whose negative part is uniformly integrable, is lower semicontinuous for the topology of convergence in distribution. The multiparameter version of prophet inequality for the one-parameter optimal stopping problem is formulated and the lower semicontinuity property of the optimal stopping value is applied to the multiparameter prophet inequality.     

On generalizations of network design problems with degree bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely laminar crossing spanning tree), and (2) by incorporating ‘degree bounds’ in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems. Our main result is a (1, b + O(log n))-approximation algorithm for the minimum crossing spanning tree (MCST) problem with laminar degree constraints. The laminar MCST problem is a natural generalization of the well-studied bounded-degree MST, and is a special case of general crossing spanning tree. We give an additive Ω(log ^c m) hardness of approximation for general MCST, even in the absence of costs (c > 0 is a fixed constant, and m is the number of degree constraints). This also leads to a multiplicative Ω(log ^c m) hardness of approximation for the robust k-median problem (Anthony et al. in Math Oper Res 35:79–101, 2010), improving over the previously known factor 2 hardness. We then consider the crossing contra-polymatroid intersection problem and obtain a (2, 2b + Δ ? 1)-approximation algorithm, where Δ is the maximum element frequency. This models for example the degree-bounded spanning-set intersection in two matroids. Finally, we introduce the crossing latticep olyhedron problem, and obtain a (1, b + 2Δ ? 1) approximation algorithm under certain condition. This result provides a unified framework and common generalization of various problems studied previously, such as degree bounded matroids.