Approximation Schemes for Minimum Latency Problems

We give a polynomial time, (1 + ∊)-approximation algorithm for the traveling repairman problem (TRP) in the Euclidean plane, on weighted planar graphs, and on weighted trees. This improves on the known quasi-polynomial time approximation schemes for these problems. The algorithm is based on a simple technique that reduces the TRP to what we call the segmented TSP. Here, we are given numbers l1, …, lK and n1, …, nK and we need to find a path that visits at least nh points within path distance lh from the starting point for all h ∊ {1, …, K}. A solution is α-approximate if at least nh points are visited within distance αlh. It is shown that any algorithm that is α-approximate for every constant K in some metric space, gives an α(1 + ∊)-approximation for the TRP in the same metric space. Subsequently, approximation schemes are given for this segmented TSP problem in different metric spaces. The segmented TSP with only one segment (K = 1) is equivalent to the k-TSP for which a (2 + ∊)-approximation is known for a general metric space. Hence, this approach through the segmented TSP gives new impulse for improving on the 3.59-approximation for TRP in a general metric space. A similar reduction applies to many other minimum latency problems. To illustrate the strength of this approach we apply it to the well-studied scheduling problem of minimizing total weighted completion time under precedence constraints, 1|prec|Σ wjCj, and present a polynomial time approximation scheme for the case of interval order precedence constraints. This improves on the known 3/2-approximation for this problem. Both approximation schemes apply as well if release dates are added to the problem.

We investigate the hardness of approximating a transshipment problem with a permutable transit vector. We have already known that the transshipment problem of minimizing the total shipping cost is NP-hard. In this paper, we prove that even if the unit shipping cost associated with every arc is given as a positive integer, there is no polynomial time constant factor approximation algorithm for the transshipment problem under the hypothesis that PWe obtain the inapproximability result by a gap-introducing reduction from an NP-complete problem by the name of 3DM. On the other hand, we propose a preparatory heuristic algorithm to the transshipment problem, which is based on a relaxation into the minimum cost flow problem and the minimum weight perfect matching on a bipartite graph. We also provide an upper bound of the total shipping cost obtained by the proposed heuristic.

We consider the k-median clustering with outliers problem: Given a finite point set in a metric space and parameters k and m, we want to remove m points (called outliers), such that the cost of the optimal k-median clustering of the remaining points is minimized. We present the first polynomial time constant factor approximation algorithm for this problem.

The crossing number of a projective graph is quadratic in the face–width

In this paper, we present a constant-factor approximation algorithm for the unsplittable flow problem on a path. This improves on the previous best known approximation factor of O(log n). The approximation ratio of our algorithm is 7+e for any e>;0. In the unsplittable flow problem on a path, we are given a capacitated path P and n tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge e of P, the total demand of selected tasks that use e does not exceed the capacity of e. This is a well-studied problem that occurs naturally in various settings, and therefore it has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. Polynomial time constant factor approximation algorithms for the problem were previously known only under the no-bottleneck assumption (in which the maximum task demand must be no greater than the minimum edge capacity). We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either 1, 2, or 3.

Approximate min–max theorems for Steiner rooted-orientations of graphs and hypergraphs

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

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 give a polynomial time approximation scheme for the weighted traveling repairman problem (TRP) in the Euclidean plane, on trees, and on planar graphs. This improves upon the quasi-polynomial tim...

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.

<p>In combinatorial optimization, covering problems are those problems where given a ground set and a family of subsets of the ground set, the objective is to find a minimum cost set of subsets whose union contains the ground set. We consider covering problems in the context of Computational Geometry, which is a subfield of Computer Science that deals with problems associated with geometric objects such as points, lines, disks, polygons etc. In particular, geometric covering is an active research area, where the ground set and the family of subsets are induced by geometric objects. Covering problems in combinatorial optimizations often have a geometric analogue that arises naturally, and though such problems remain NP-hard, it is often possible to exploit the geometric properties of the set system to obtain better approximation algorithms.</p> <p>In this work, the fundamental problem that we consider is a generalization of geometric covering, where each element in the ground set may need to covered by more than one subset. To be precise, the problem is defined as follows: given two sets of points X, Y and a coverage function κ : X → Z+ ∪ {0}, construct balls centered on the points in Y such that each point in X is covered by at least κ(x) distinct balls. The objective in this problem is to minimize the total cost, which is a function of the radii of the balls. This problem is termed as the metric multi-cover (MMC) problem.</p> <p>We first consider version of the MMC problem where κ(x) = 1 for all clients, i.e. the 1-covering case. The known results that give a constant factor approximation for this problem are variations of LP-based primal-dual algorithm. We use a modified local search technique, motivated by geometric idea, to derive a simple, constant- factor approximation for this problem in Chapter 2.</p> <p>We then look at the MMC problem where the point sets X,Y are in the Euclidean plane, and each client x ∈ X needs to be covered by at least κ(x) distinct disks centered on the points in Y . In Chapter 4, we give the first polynomial time constant factor approximation for this problem, in which the constant is independent of the coverage function κ. Our solution also has an incremental property, which allows the algorithm to handle increases in the coverage requirement by increasing the radii of the current server disks, without affecting the approximation factor.</p> <p>In the next problem, we move from the Euclidean plane to arbitrary metric spaces where we consider the uniform MMC problem. In this problem, each client x has the demand κ(x) = k, where k > 0 is an integer. We give the first constant factor approximation (independent of k) for this problem. The key contribution that led to this result is the formulation of a partitioning scheme of the servers in the uniform MMC problem, that reduces the uniform MMC problem to k instances of 1-covering problem, while preserving the optimality of the solution to a constant multiplicative factor. We present the partitioning scheme as an independent result in Chapter 5, which we then use to solve the uniform MMC problem in Chapter 6.</p> <p>The MMC problem with arbitrary coverage function κ is then considered in Chapter 7. The key challenge that the non-uniform version presents is that the symmetry of the server partitioning scheme breaks down as the coverage demands of clients are independent of each other. We present a constant factor algorithm for this problem in Chapter 7.</p> <p>The last problem that we consider is the t-MMC problem, which is a restricted version of the uniform MMC problem. The objective is to compute a cover in which each client is covered by at least k distinct server disks, using atmost t server disks in total. This problem is a generalization of the clustering problem (where k = 1), and to our knowledge this is the first time this generalization has been considered. We give a constant factor approximation for this problem in Chapter 8.</p>

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.

The two-flight low risk helicopter transportation problem under the split pickup scenario

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

In this paper, we study the following vehicle routing problem: given n vertices in a metric space, a specified root vertex r (the depot), and a length bound D, find a minimum cardinality set of r-paths that covers all vertices, such that each path has length at most D. This problem is $\mathcal{NP}$-complete, even when the underlying metric is induced by a weighted star. We present a 4-approximation for this problem on tree metrics. On general metrics, we obtain an O(logD) approximation algorithm, and also an $(O(\log \frac{1}{\epsilon}),1+\epsilon)$ bicriteria approximation. All these algorithms have running times that are almost linear in the input size. On instances that have an optimal solution with one r-path, we show how to obtain in polynomial time, a solution using at most 14 r-paths. We also consider a linear relaxation for this problem that can be solved approximately using techniques of Carr & Vempala [7]. We obtain upper bounds on the integrality gap of this relaxation both in tree metrics and in general.