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In this paper the minimum spanning tree problem with uncertain edge costs is discussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within $O(\log^{1-\epsilon} n)$ for any $\epsilon>0$ unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within $O(\log n)$ unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance ratio of $O(\log^2 n)$ for min-max and 2-stage min-max problems are also proposed.

In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: Given a set of t topics and a collection of n users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is NP-hard and approximable within O(log t) in polynomial time. In this paper, we further investigate the problem and some of its special instances. We give various hardness results for instances where the number of topics in which an user is interested in is bounded by a constant, and also for the instances where the number of users interested in a common topic is constant. For the latter case, we present a first constant approximation algorithm. We also present some polynomial-time algorithms for very restricted instances of Min-TCO., 20 pages

We consider a problem which arises in optical routing. WDM/SONET rings are a network architecture used by telecommunications carriers for traffic streams. The dominant cost factor in such networks is the total number of add-drop multiplexers (ADMs) used. A list of traffic streams to be routed between pairs of nodes is given. In this paper we consider the problem where we need to assign a route and a wavelength to each traffic stream, minimizing the total number of used SONET ADMs. This is called the chord version of the SONET ADMs minimization problem, to denote the fact that the routing is not given a priori. The best previously known approximation algorithms for this problem have the performance guarantee of 32. We present an improved algorithm with performance guarantee of exactly 107≈1.42857. Moreover, we study some natural heuristics for this problem, and give tight analysis of their approximation ratios.

The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most $(1+\eps)$ times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless $P = NP$) nor to approximate it (unless $NP = BPP$). Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given $\eps,\delta>0$ it calculates a solution with contribution at most $(1+\eps)$ times the minimal contribution with probability at least $(1-\delta)$. Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds., Comment: 22 pages, to appear in Theoretical Computer Science

A wireless ad-hoc network consists of a collection of transceivers positioned in the plane. Each transceiver is equipped with a limited battery charge. The battery charge is reduced after each transmission, depending on the transmission distance. One of the major problems in wireless network design is to route network traffic efficiently, so as to maximize the network lifetime, i.e., the number of successful transmission rounds. In this paper, we consider Rooted Maximum Lifetime Broadcast/Convergecast problems in wireless settings. The instance consists of a directed graph G=(V,E) with edge-weight w(e) (the power needed to transmit a message along e) for every [email protected]?E, node capacity b(v) (the battery charge of v) for every [email protected]?V, and a root r. The goal is to find a maximum size collection {T"1,...,T"k} of Broadcast/Convergecast trees rooted at r such that @?"i"="1^kw(@d"T"""i(v))@?b(v), where @d"T(v) is the set of edges leaving v in T. In the Single Topology version, the same tree is used to transmit all the messages, namely, all the Broadcast/Convergecast trees T"i are identical. Using recent work on degree constrained network design problems (Nutov, 2008) [26], we give constant ratio approximation algorithms for various broadcast and convergecast problems, improving the previously best known approximation @W(@?1/[email protected]?) by Elkin et al. (2011) [12]. Similar results are shown for the more general Rooted Maximum Lifetime Mixedcast problem, where in addition we are given an integer @c>=0, and the goal is to find the maximum integer k so that k Broadcast and @ck Convergecast rounds can be performed. We also consider the model with partial level aggregation.

This paper studies α-coagnostic learnability of classes of Boolean formulas. To α-coagnostic learn C from H, the learner seeks a hypothesis h∈H whose probability of agreement (rather than disagreement as in agnostic learning) with a labeled example is within a factor α of the best agreement probability achieved by any f∈C. Although 1-coagnostic learning is equivalent to agnostic learning, this is not true for α-coagnostic learning for 12

Given a graph G=(V,E) in which a fugitive hides on vertices or along edges, graph searching problems are usually to find the minimum number of searchers required to capture the fugitive. In this paper, we consider the problem of finding the minimum number of steps to capture the fugitive. We introduce the fast edge searching problem in the edge search model, which is the problem of finding the minimum number of steps (called the fast edge-search time) to capture the fugitive. We establish relations between the fast edge searching and the fast searching that is the problem of finding the minimum number of searchers to capture the fugitive in the fast search model. While the family of graphs whose fast search number is at most k is not minor-closed for any positive integer k≥2, we show that the family of graphs whose fast edge-search time is at most k is minor-closed. We establish relations between the fast (fast edge) searching and the node searching. These relations allow us to transform the problem of computing node search numbers to the problem of computing fast edge-search numbers or fast search numbers. Using these relations, we prove that the problem of deciding, given a graph G and an integer k, whether the fast (edge-)search number of G is less than or equal to k is NP-complete; and it remains NP-complete for Eulerian graphs. We also prove that the problem of determining whether the fast (edge-)search number of G is half of the number of odd vertices in G is NP-complete; and it remains NP-complete for planar graphs with maximum degree 4. We present a linear time approximation algorithm for the fast edge-search time that always delivers solutions of at most (1+|V|−1|E|+1) times the optimal value. This algorithm also gives us a tight upper bound on the fast search number of graphs. We also show a lower bound on the fast search number using the minimum degree and the number of odd vertices.

In ad hoc wireless networks, a connected dominating set can be used as a virtual backbone to improve the performance. Many constructions for approximating the minimum connected dominating set are based on the construction of a maximal independent set. The relation between the size mis(G) of a maximum independent set and the size cds(G) of a minimum connected dominating set in the same graph G plays an important role in establishing the performance ratio of those approximation algorithms. Previously, it is known that mis(G) ≤ 4ċcds(G) + 1 for all unit disk graphs G. In this paper, we improve it by showing mis(G) ≤ 3.8ċcds(G) + 1.2.

We consider the problem of splitting an order for R goods, R≥1, among a set of sellers, each having bounded amounts of the goods, so as to minimize the total cost of the deal. In deal splitting with packages (DSP), the sellers offer packages containing combinations of the goods; in deal splitting with price tables (DST), the buyer can generate such combinations using price tables. Our problems, which often occur in online reverse auctions, generalize covering integer programs with multiplicity constraints (CIP), where we must fill up an R-dimensional bin by selecting (with a bounded number of repetitions) from a set of R-dimensional items, such that the overall cost is minimized. Thus, both DSP and DST are NP-hard, already for a single good, and hard to approximate for arbitrary number of goods.In this paper we focus on finding efficient approximations for DSP and DST instances where the number of goods is some fixed constant. In particular, we develop polynomial time approximation schemes (PTAS) for several subclasses of instances of practical interest. Our results include a PTAS for CIP in fixed dimension, and a more efficient (combinatorial) scheme for CIP∞, where the multiplicity constraints are omitted. Our approximation scheme for CIP∞ is based on a non-trivial application of the fast scheme for the fractional covering problem, proposed by Fleischer [L. Fleischer, A fast approximation scheme for fractional covering problems with variable upper bounds, in: Proc. of the 15th ACM-SIAM Symposium on Discrete Algorithm, 2004, pp. 994–1003].

Given a set V of elements, S a family of subsets of V, and G a connected graph on vertex set S,a connected set cover (CSC) is a subfamily R of S such that every element in V is covered by at least one set of R, and the subgraph G[R] of G induced by R is connected. If furthermore G[R] is k-connected and every element in V is covered by at least m sets in R, then R is a (k,m)-CSC. In this paper, we present two approximation algorithms for the minimum CSC problem, and one approximation algorithm for the minimum (2,m)-CSC problem. Performance ratios are analyzed. These are the first approximation algorithms for CSC problems in general graphs with guaranteed performance ratios.

In contrast to the minimization of deterministic finite automata (DFA's), the task of constructing a minimal nondeterministic finite automaton (NFA) for a given NFA is PSPACE-complete. Moreover, there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal NFA's. Since one is unable to efficiently estimate the size of a minimal NFA in an efficient way, one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal NFA for a given regular language. Here we consider the robust and most successful lower bound proof technique that is based on communication complexity. In this paper it is proved that even a strong generalization of this method fails for some concrete regular languages. ''To fail'' is considered here in a very strong sense. There is an exponential gap between the size of a minimal NFA and the achievable lower bound for a specific sequence of regular languages. The generalization of the concept of communication protocols is also strong here. It is shown that cutting the input word into 2^O^(^n^^^1^^^/^^^4^) pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata. It seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head.

Minimizing total energy to keep an ad hoc wireless network symmetrically connected is an NP-hard problem. Recently, several greedy approximations have been proposed, based on k-restricted decompositions of the network. Their performance ratios are established through estimations of the least upper bound ρk for the ratio between total powers of best possible k-restricted decomposition and the optimal solution. In this paper, we determine the exact value of ρk for all k.

Given a graph, the Hamiltonian path completion problem is to find an augmenting edge set such that the augmented graph has a Hamiltonian path. In this paper, we show that the Hamiltonian path completion problem will unlikely have any constant ratio approximation algorithm unless NP = P. This problem remains hard to approximate even when the given subgraph is a tree. Moreover, if the edge weights are restricted to be either 1 or 2, the Hamiltonian path completion problem on a tree is still NP-hard. Then it is observed that this problem is strongly NP-hard, so it does not have any fully polynomial-time approximation scheme (FPTAS) unless NP=P. When the given tree is a k-tree, we give an approximation algorithm with performance ratio 1.5.

The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The main result of the paper is a polynomial-time approximation scheme for minimum sum multicoloring the edges of trees. We also show that the problem is strongly NP-hard for trees, even if every demand is at most 2.

ϵ-points were introduced by the authors (see [S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic curves by lines, Theoret. Comput. Sci. 315(2–3) (2004) 627–650 (Special issue); S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic surfaces by lines, Comput. Aided Geom. Design 22(2) (2005) 147–181; S. Pérez-Díaz, J.R. Sendra, J. Sendra, Distance properties of ϵ-points on algebraic curves, in: Series Mathematics and Visualization, Computational Methods for Algebraic Spline Surfaces, Springer, Berlin, 2005, pp. 45–61]) as a generalization of the notion of approximate root of a univariate polynomial. The notion of ϵ-point of an algebraic hypersurface is quite intuitive. It essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module. Intuition says that an ϵ-point of a hypersurface is a point close to it. In this paper, we formally analyze this assertion giving bounds of the distance of the ϵ-point to the hypersurface. For this purpose, we introduce the notions of height, depth and weight of an ϵ-point. The height and the depth control when the distance bounds are valid, while the weight is involved in the bounds., Ministerio de Educación y Ciencia, Comunidad de Madrid, Universidad de Alcalá

While the original stable marriage problem requires all participants to rank all members of the opposite sex in a strict order, two natural variations are to allow for incomplete preference lists and ties in the preferences. Either variation is polynomially solvable, but it has recently been shown to be NP-hard to find a maximum cardinality stable matching when both of the variations are allowed. It is easy to see that the size of any two stable matchings differ by at most a factor of two, and so, an approximation algorithm with a factor two is trivial. In this paper, we give a randomized approximation algorithm RANDBRK and show that its expected approximation ratio is at most 10/7(1.3913) for RANDBRK. Furthermore, we show that these restrictions except for the last one can be removed without increasing the approximation ratio too much.

Given a single machine and a set of jobs with due dates, the classical NP-hard problem of scheduling to minimize total tardiness is a well-understood one. Lawler gave a fully polynomial-time approximation scheme (FPTAS) for it some 20 years ago. If the jobs have positive weights the problem of minimizing total weighted tardiness seems to be considerably more intricate. it. In this paper, we give some of the first approximation algorithms for it. We examine first the weighted problem with a fixed number of due dates and we design a pseudopolynomial algorithm for it. We show how to transform the pseudopolynomial algorithm to an FPTAS for the case where the weights are polynomially bounded.For the case with an arbitrary number of due dates and polynomially bounded processing times, we provide a quasipolynomial algorithm which produces a schedule whose value has an additive error proportional to the weighted sum of the due dates. We also investigate the performance of algorithms for minimizing the related total weighted late work objective.

In the multicast k-tree routing problem, a data copy is sent from the source node to at most k destination nodes in every transmission. The goal is to minimize the total cost of sending data to all destination nodes, which is measured as the sum of the costs of all routing trees. This problem was formulated out of optical networking and has applications in general multicasting. Several approximation algorithms, with increasing performance, have been proposed in the last several years; the most recent ones rely heavily on a tree partitioning technique. In this paper, we present a further improved approximation algorithm along the line. The algorithm has a worst-case performance ratio of 54ρ+32, where ρ denotes the best approximation ratio for the Steiner minimum tree problem. The proofs of the technical routing lemmas also provide some insights into why such a performance ratio could be the best possible that one can get using this tree partitioning technique.

In this paper, we study two interesting variants of the classical bin packing problem, called Lazy Bin Covering (LBC) and Cardinality Constrained Maximum Resource Bin Packing (CCMRBP) problems. For the offline LBC problem, we first prove the approximation ratio of the First-Fit-Decreasing and First-Fit-Increasing algorithms, then present an APTAS. For the online LBC problem, we give a competitive analysis for the algorithms of Next-Fit, Worst-Fit, First-Fit, and a modified HARMONICM algorithm. The CCMRBP problem is a generalization of the Maximum Resource Bin Packing (MRBP) problem Boyar et al. (2006) [1]. For this problem, we prove that its offline version is no harder to approximate than the offline MRBP problem.

Given an input string S and a target string T when S is a permutation of T , the interchange rearrangement problem is to apply on S a sequence of interchanges, such that S is transformed into T . The interchange operation exchanges the position of the two elements on which it is applied. The goal is to transform S into T at the minimum cost possible, referred to as the distance between S and T . The distance can be defined by several cost models that determine the cost of every operation. There are two known models: The Unit-cost model and the Length-cost model. In this paper, we suggest a natural cost model: The Element-cost model. In this model, the cost of an operation is determined by the elements that participate in it. Though this model has been studied in other fields, it has never been considered in the context of rearrangement problems. We consider both the special case where all elements in S and T are distinct, referred to as a permutation string, and the general case, referred to as a general string. An efficient optimal algorithm for the permutation string case and efficient approximation algorithms for the general string case, which is N P -hard, are presented. The study is broadened to include the transposition rearrangement problem under the Element-cost model and under the other known models, in order to provide additional perspective on the new model.