Approximation schemes for Min-Sum [formula omitted]-Clustering

The minimum latency problem, also known as the traveling repairman problem, is a variant of the traveling salesman problem in which the starting node of the tour is given and the goal is to minimize the sum of the arrival times at the other nodes. We present a quasi-polynomial time approximation scheme (QPTAS) for this problem when the instance is a weighted tree, when the nodes lie in $\mathbb{R}^d$ for some fixed d, and for planar graphs. We also present a polynomial time constant factor approximation algorithm for the general metric case. The currently best polynomial time approximation algorithm for general metrics, due to Goemans and Kleinberg, computes a 3.59-approximation.

Recently, Adamaszek and Wiese [1, 2] presented a quasi-polynomial time approximation scheme (QPTAS) for the problem of computing a maximum weight independent set for certain families of planar objects. This major advance on the problem was based on their proof that a certain type of separator exists for any independent set. Subsequently, Har-Peled [22] simplified and generalized their result. Mustafa et al. [36] also described a simplification, and somewhat surprisingly, showed that QPTAS's can be obtained for certain, albeit special, type of covering problems.Building on these developments, we revisit two NP-hard geometric partitioning problems -- convex decomposition and surface approximation. Partitioning problems combine the features of packing and covering. In particular, since the optimal solution does form a packing, the separator theorems are potentially applicable. Nevertheless, the two partitioning problems we study bring up additional difficulties that are worth examining in the context of the wider applicability of the separator methodology. We show how these issues can be handled in presenting quasi-polynomial time algorithms for these two problems with improved approximation guarantees.

We study the problem of embedding arbitrary finite metrics into a line metric in a non-contracting fashion to approximate the minimum average distortion. Since a path metric (or a line metric) is quite restricted, these embeddings could have high average distortions (Ω(n), where n is the number of points in the original metric). Furthermore, we prove that finding best embedding of even a tree metric into a line to minimize average distortion is NP-hard. Hence, we focus on approximating the best possible embedding for given input metric.We give a constant-factor approximation for the problem of embedding general metrics into the line metric. For the case of the metrics which can be represented as trees, we provide improved approximation ratios in polynomial time as well as a QPTAS (Quasi-Polynomial Time Approximation Scheme).KeywordsApproximation AlgorithmSteiner TreeMinimum LatencyOptimal TourEuler TourThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider the problem of embedding finite metrics with slack: We seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler [in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, 2004], we show that provable guarantees of this type can in fact be achieved in general: Any finite metric space can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into $\ell_1$ which exhibit gracefully degrading distortion: There is a single embedding into $\ell_1$ that achieves distortion at most $O(\log\frac{1}{\epsilon})$ on all but at most-1.5pt an $\epsilon$ fraction of distances simultaneously for all $\epsilon>0$. We extend this with distortion1pt $O(\log\frac{1}{\epsilon})^{1/p}$ to maps into general $\ell_p$, $p\geq1$, for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight and give a general technique to obtain lower bounds for $\epsilon$-slack embeddings from lower bounds for low-distortion embeddings.

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of n items, each associated with a non-negative weight, and T time periods with non-decreasing capacities \(W_1 \le \dots \le W_T\). When item i is inserted at time t, we gain a profit of \(p_{it}\); however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time \((\frac{1}{2}-\epsilon )\)-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys–Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and new structural properties of nearly-optimal solutions, we turn our algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.

In this paper, general metrics of the energy-delay (E-D) tradeoff in digital VLSI circuits are discussed. More specifically, the general class of metrics E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sup> with arbitrary exponents is adopted and evaluated for various commercial microprocessors. Results indicate that practical circuits are designed by minimizing a wider range of metrics compared to the ED or ED <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> metrics usually assumed in the literature. Hence, the general metrics E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sup> describes the energy-delay tradeoff in a more realistic way. An interesting interpretation of the adopted metrics is provided to gain an insight into the relationship between energy and delay in energy-efficient designs. Various properties are also derived analytically by resorting to the logical effort method. Simulations on a 65-nm technology are performed to exemplify and validate the theoretical results.

A note on a QPTAS for maximum weight triangulation of planar point sets

Given an edge-weighted graph and a set of known seed vertices of interest, a network scientist often desires to understand the graph relationships to explain connections between the seed vertices. If the size of the seed set is 2, shortest path calculations are an attractive computational kernel to explore the connections between the two vertices. When the seed set is 3 or larger (say up to 1,000s) Steiner minimal tree – min-weight acyclic connected subgraph (of the input graph) that contains all the seed vertices – is an attractive generalization of shortest weighted paths. In general, computing a Steiner minimal tree is NP-hard, but decades ago several polynomial-time algorithms were designed and proven to yield Steiner trees whose total weight is bounded within 2 times the minimal Steiner tree. Despite its rich theoretical literature, works related to parallel Steiner minimal tree computation and their scalable implementations are rather scarce. In this paper, we present a parallel 2-approximation Steiner minimal tree algorithm (with theoretical guarantees) and its MPI-based distributed implementation. In place of distance computation between all pairs of seed vertices, an expensive phase in many approximation algorithms, the solution we employ, exploits Voronoi cell computation. Also, this approach has higher parallel efficiency than others that involve minimum spanning tree computation on the entire graph. Furthermore, our distributed design exploits asynchronous processing and a message prioritization scheme to accelerate convergence of distance computation, employs techniques to avoid inefficient distributed spanning tree computation on the entire graph, and harnesses a combination of vertex and edge centric processing to offer fast time-to-solution. We demonstrate scalability and performance of our solution using real-world graphs with up to 128 billion edges and 512 compute nodes (8K processes), show the ability to find Steiner trees with up to 10K seed vertices in under one minute, and present in-depth analyses that highlight the benefits of our design choices. Using four real-world graphs and three seed sets for each, we compare our solution with the state-of-the-art exact Steiner minimal tree solver, SCIP-Jack, and two sequential algorithms with the same approximation bound as our algorithm. Our distributed solution comfortably outperforms these related works on graphs with 10s million edges and offers decent strong scaling – up to 90% efficient. We empirically show that, on average, the total distance (sum of edge weights) of the Steiner tree identified by our solution is 1.0527 times greater than the Steiner minimal tree (i.e., the optimal solution) – well within the theoretical bound of less than equal to 2.

We consider two closely related fundamental clustering problems in this paper. In the Min-Sum k -Clustering problem, one is given a metric space and has to partition the points into k clusters while minimizing the total pairwise distances between the points assigned to the same cluster. In the Balanced k -Median problem, the instance is the same and one has to obtain a partitioning into k clusters \(C_1,\ldots ,C_k\), where each cluster \(C_i\) has a center \(c_i\), while minimizing the total assignment costs for the points in the metric; here the cost of assigning a point j to a cluster \(C_i\) is equal to \(|C_i|\) times the distance between j and \(c_i\) in the metric.In this paper, we present an \(O(\log n)\)-approximation for both these problems where n is the number of points in the metric that are to be served. This is an improvement over the \(O(\epsilon ^{-1}\log ^{1 + \epsilon } n)\)-approximation (for any constant \(\epsilon > 0\)) obtained by Bartal, Charikar, and Raz [STOC ’01]. We also obtain a quasi-PTAS for Balanced k-Median in metrics with constant doubling dimension.As in the work of Bartal et al., our approximation for general metrics uses embeddings into tree metrics. The main technical contribution in this paper is an O(1)-approximation for Balanced k-Median in hierarchically separated trees (HSTs). Our improvement comes from a more direct dynamic programming approach that heavily exploits properties of standard HSTs. In this way, we avoid the reduction to special types of HSTs that were considered by Bartal et al., thereby avoiding an additional \(O(\epsilon ^{-1} \log ^\epsilon n)\) loss.

We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a k-hypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hypermatching Assignment Problem (HAP) is to assign a set of pairwise disjoint bundles to clients so as to maximize the total profit while respecting the budgets. This problem has various applications in production planning and budget-constrained auctions and generalizes well-studied problems in combinatorial optimization: for example the weighted (unweighted) k-hypergraph matching problem is the special case of HAP with one client having unbounded budget and general (unit) profits; the Generalized Assignment Problem (GAP) is the special case of HAP with k = 1.Let e > 0 denote an arbitrarily small constant. In this paper we obtain the following main results:• We give a randomized (k + 1 + e) approximation algorithm for HAP, which is based on rounding the 1-round Lasserre strengthening of a novel LP. This is one of a few approximation results based on Lasserre hierarchies and our approach might be of independent interest. We remark that for weighted k-hypergraph matching no LP nor SDP relaxation is known to have integrality gap better than k − 1 + 1/k for general k [Chan and Lau, SODA'10].• For the relevant special case that one wants to maximize the total revenue (i.e., bi,e = wi,e), we present a local search based (k + O (√k))/2 approximation algorithm for k = O(1). This almost matches the best known (k + 1 + e)/2 approximation ratio by Berman [SWAT'00] for the (less general) weighted k-hypergraph matching problem.• For the unweighted k-hypergraph matching problem, we present a (k + 1 + e)/3 approximation in quasipolynomial time. This improves over the (k + 2)/3 approximation by Halldorsson [SODA'95] (also in quasipolynomial time). In particular this suggests that a 4/3 + e approximation for 3-dimensional matching might exist, whereas the currently best known polynomial-time approximation ratio is 3/2.

A Study in Pairwise Clustering for Bi-dimensional Irregular Strip Packing Using the Dotted Board Model

We propose a graph model for mutual information based clustering problem. This problem was originally formulated as a constrained optimization problem with respect to the conditional probability distribution of clusters. Based on the stationary distribution induced from the problem setting, we propose a function which measures the relevance among data objects under the problem setting. This function is utilized to capture the relation among data objects, and the entire objects are represented as an edge-weighted graph where pairs of objects are connected with edges with their relevance. We show that, in hard assignment, the clustering problem can be approximated as a combinatorial problem over the proposed graph model when data is uniformly distributed. By representing the data objects as a graph based on our graph model, various graph based algorithms can be utilized to solve the clustering problem over the graph. The proposed approach is evaluated on the text clustering problem over 20 Newsgroup and TREC datasets. The results are encouraging and indicate the effectiveness of our approach.

The General Scheduling Problem (GSP) generalizes scheduling problems with sum of cost objectives such as weighted flow time and weighted tardiness. Given a set of jobs with processing times, release dates, and job dependent cost functions, we seek to find a minimum cost preemptive schedule on a single machine. The best known algorithm for this problem and also for weighted flow time/tardiness is an O(loglog P)-approximation (where P denotes the range of the job processing times), while the best lower bound shows only strong NP-hardness. When release dates are identical there is also a gap: the problem remains strongly NP-hard and the best known approximation algorithm has a ratio of e+\epsilon (running in quasi-polynomial time). We reduce the latter gap by giving a QPTAS if the numbers in the input are quasi-polynomially bounded, ruling out the existence of an APX-hardness proof unless NP\subseteq DTIME(2^polylog(n)). Our techniques are based on the QPTAS known for the UFP-Cover problem, a particular case of GSP where we must pick a subset of intervals (jobs) on the real line with associated heights and costs. If an interval is selected, its height will help cover a given demand on any point contained within the interval. We reduce our problem to a generalization of UFP-Cover and use a sophisticated divide-and-conquer procedure with interdependent non-symmetric subproblems. We also present a pseudo-polynomial time approximation scheme for two variants of UFP-Cover. For the case of agreeable intervals we give an algorithm based on a new dynamic programming approach which might be useful for other problems of this type. The second one is a resource augmentation setting where we are allowed to slightly enlarge each interval.

We obtain the first online algorithms for the node-weighted Steiner tree, Steiner forest and group Steiner tree problems that achieve a poly-logarithmic competitive ratio. Our algorithm for the Steiner tree problem runs in polynomial time, while those for the other two problems take quasi-polynomial time. Our algorithms can be viewed as online LP rounding algorithms in the framework of Buchbinder and Naor (Foundations and Trends in Theoretical Computer Science, 2009); however, while the natural LP formulation of these problems do lead to fractional algorithms with a poly-logarithmic competitive ratio, we are unable to round these LPs online without losing a polynomial factor. Therefore, we design new LP formulations for these problems drawing on a combination of paradigms such as spider decompositions, low-depth Steiner trees, generalized group Steiner problems, etc. and use the additional structure provided by these to round the more sophisticated LPs losing only a poly-logarithmic factor in the competitive ratio. As further applications of our techniques, we also design polynomial-time online algorithms with poly-logarithmic competitive ratios for two fundamental network design problems in edge-weighted graphs: the group Steiner forest problem (thereby resolving an open question raised by Chekuri et. al. (SODA 2008)) and the single source ℓ-vertex connectivity problem (which complements similar results for the corresponding edge-connectivity problem due to Gupta et. al. (STOC 2009)).

Boykov and Kolmogorov showed that it is possible to find globally minimal contours and surfaces via graph cuts by embedding an appropriate metric approximation into the graph edge weights and derived the requisite formulas for Euclidean and Riemannian metrics [3]. In [9] we have proposed an improved Euclidean metric approximation that is invariant under (horizontal and vertical) mirroring, applicable to grids with anisotropic resolution and with a smaller approximation error. In this paper, we extend our method to general Riemannian metrics that are essential for graph cut based image segmentation or stereo matching. It is achieved by the introduction of a transformation reducing the Riemannian case to the Euclidean one and adjusting the formulas from [9] to be able to cope with non-orthogonal grids. We demonstrate that the proposed method yields smaller approximation errors than the previous approaches both in theory and practice.