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This paper examines a number of variants of the sparse k-spanner problem and presents hardness results concerning their approximability. Previously, it was known that most k-spanner problems are weakly inapproximable (namely, they are NP-hard to approximate with ratio O(log n), for every k ≥ 2) and that the unit-length k-spanner problem for constant stretch requirement k ≥ 5 is strongly inapproximable (namely, it is NP-hard to approximate with ratio $O(2^{{\log}^{1 - \epsilon} n})$ ). The results of this paper significantly expand the ranges of hardness for k-spanner problems. In general, strong hardness is shown for a number of k-spanner problems, for certain ranges of the stretch requirement k depending on the particular variant at hand. The problems studied differ by the types of edge weights and lengths used, and they include directed, augmentation and client-server variants. The paper also considers k-spanner problems in which the stretch requirement k is relaxed (e.g., $k = \Omega(\log n))$ . For these cases, no inapproximability results were known (even for a constant approximation ratio) for any spanner problem. Moreover, some versions of the k-spanner problem are known to enjoy the ratio-degradation property; namely, their complexity decreases exponentially with the inverse of the stretch requirement. So far, no hardness result existed precluding any k-spanner problem from enjoying this property. This paper establishes strong inapproximability results for the case of relaxed stretch requirement (up to $k = O(n^{{1 - \delta}})$ , for any $0 < \delta < 1$ ), for a large variety of k-spanner problems. It is also shown that these problems do not enjoy the ratio-degradation property. [ABSTRACT FROM AUTHOR]

This paper studies the problem of maximizing the number of correct results of dependent tasks computed unreliably. We consider a distributed system composed of a reliable server that coordinates the computation of a massive number of unreliable workers. Any worker computes correctly with probability p < 1. Any incorrectly computed task corrupts all dependent tasks. The goal is to determine which tasks should be computed by the (reliable) server and which by the (unreliable) workers, and when, so as to maximize the expected number of correct results, under a constraint d on the computation time. This problem is motivated by distributed computing applications that persist partial results of computations for future use in other computations and that want to ensure that the persisted results are of high quality. We show that this optimization problem is NP-hard. Then we study optimal scheduling solutions for the mesh with the tightest deadline. We present combinatorial arguments that describe all optimal solutions for two ranges of values of worker reliability p, when p is close to zero and when p is close to one. [ABSTRACT FROM AUTHOR]

We introduce new data structures for compressed suffix trees whose size are linear in the text size. The size is measured in bits; thus they occupy only O(n log|A|) bits for a text of length n on an alphabet A. This is a remarkable improvement on current suffix trees which require O(n log n) bits. Though some components of suffix trees have been compressed, there is no linear-size data structure for suffix trees with full functionality such as computing suffix links, string-depths and lowest common ancestors. The data structure proposed in this paper is the first one that has linear size and supports all operations efficiently. Any algorithm running on a suffix tree can also be executed on our compressed suffix trees with a slight slowdown of a factor of polylog(n). [ABSTRACT FROM AUTHOR]

Any Boolean function can be defined by a Boolean circuit, provided we may use sufficiently strong functions in its gates. On the other hand, what Boolean functions can be defined depends on these gate functions: Each set B of gate functions defines the class of Boolean functions that can be defined by circuits over B. Although these classes have been known since the 1920s, their computational complexity was never investigated. In this paper we will study how difficult it is to decide for a Boolean function f and a class B, whether f is in B. Moreover, we will provide such a decision algorithm with additional information: How difficult is it to decide whether or not f is in B, provided we already know a circuit for f, but with gates from another class A? Given such a circuit, we know that f is in A. Is the problem harder if we do not have a concrete representation for f, but still know that it is from A? For nearly all possible combinations, we show that this is not the case, and that the problem is either in P or coNP-complete. [ABSTRACT FROM AUTHOR]

For t>0 and g≥0, a vertex-weighted graph of total weight W is ( t, g) -trimmable if it contains a vertex-induced subgraph of total weight at least (1−1/ t) W and with no simple path of more than g edges. A family of graphs is trimmable if for every constant t>0, there is a constant g≥0 such that every vertex-weighted graph in the family is ( t, g)-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. We also show that every family of directed graphs of bounded layer bandwidth (a less restrictive condition than bounded directed bandwidth) is trimmable. As an application of these results, we derive polynomial-time approximation schemes for various forms of the problem of labeling a subset of given weighted point features with nonoverlapping sliding axes-parallel rectangular labels so as to maximize the total weight of the labeled features, provided that the ratios of label heights or the ratios of label lengths are bounded by a constant. This settles one of the last major open questions in the theory of map labeling. [ABSTRACT FROM AUTHOR]

We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an n-element universe U and a family ℱ of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of U with k sets from ℱ in time within a polynomial factor (in n) of the number of supersets of the members of ℱ. Relying on an projection theorem of Chung et al. (J. Comb. Theory Ser. A 43:23–37, ) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs with maximum degree Δ. In particular, we show how to compute the domatic number in time within a polynomial factor of (2Δ+1−2) n/(Δ+1) and the chromatic number in time within a polynomial factor of (2Δ+1−Δ−1) n/(Δ+1). For any constant Δ, these bounds are O((2− ε) n) for ε>0 independent of the number of vertices n. [ABSTRACT FROM AUTHOR]

We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless W[ P]-hard problems are fixed parameter tractable by one-side error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razborov (SIAM J. Comput. 38(4):1347–1363, ). [ABSTRACT FROM AUTHOR]

We give a general model of partially asynchronous, distributed load-balancing algorithms for the discrete load model in parallel computers, where the processor loads are treated as non-negative integers. We prove that all load-balancing algorithms in this model are finite. This means that all load-balancing algorithms based on this model are guaranteed to reach a stable situation at a certain time (which depends on the particular algorithm) at which no load will be sent from one processor to another. With an additional assumption, we prove that the largest load difference between any two processors, in the final stable situation of the load-balancing algorithms in this model, is upper-bounded by the diameter of the topology. [ABSTRACT FROM AUTHOR]

We present approximation algorithms for two closely related bicriteria problems. Given a graph with two weight functions on the edges, length and cost, we consider the Bounded-Diameter Minimum-Cost Steiner Tree (BDMST) problem and the Bounded-Diameter Minimum-Cardinality Edge Addition (BDMC) problem. We present a parameterized approximation algorithm for the BDMST problem with a bicriteria approximation ratio of (O(p log s/log p),O(log s/log p)) where the first factor gives the relaxation on the diameter bound, the second factor is the cost-approximation factor, s is the number of required nodes and p, 1 ≤ p < s, is an input parameter. The parameter p allows a trade-off between the two approximation factors. This is the first improvement of the cost-approximation factor since the scheme proposed by Marathe et al. [9]. For example, p can be set to sα to obtain an (O(sα),O(1)) approximation for a constant α. The algorithm is very simple and is suitable for distributed implementations. We also present the first algorithm for Bounded-Hops Minimum-Cost Steiner Tree for complete graphs with triangle inequality. This model includes graphs defined by points in a Euclidean space of any dimension. The algorithm guarantees an approximation ratio of (O(logds),O(logds)) where d is the bound on the diameter. This is an improvement over the general-case approximation when d is comparable with s. For example, the ratio is (O(1),O(1)) for any d = sα where α is a constant between 0 and 1. For the case where the number of terminals is a constant and all edge lengths are unit, we have a polynomial-time algorithm. This can be extended to any length function providing a (1 + ε) in the approximation with ε showing up in the time complexity of the algorithm. For another special case, where the cost of any edge is either 1 or 0 and the length of each edge is positive, an algorithm with approximation ratio of (O(log(c log s)), O(log(c log s))) is presented, where c is the cost of the optimal solution. This approximation is a significant improvement over (O(log s),O(log s)) when the cost of the solution c is much smaller than the number of terminals s. This is useful when an existing multicast network is to be modified to accommodate new terminals with the addition of as few new edges as possible. We also propose an approximation algorithm for the Bounded-Diameter Minimum-Cardinality Edge Addition problem, which achieves an O(log n) approximation while relaxing the diameter bound by 2. While this ratio is the same as the one claimed in [3], this algorithm is simple and combinatorial rather than based on the Linear Program solution and can be readily modified for a distributed implementation. [ABSTRACT FROM AUTHOR]