O(log⁡2k/log⁡log⁡k)O(\log^2k/\log\log{k})-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm

In the Directed Steiner Tree (DST) problem we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost arborescence that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial-time. By adjusting the parameters in the hardness result of Halperin and Krauthgamer, we show the matching lower bound of $\Omega(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial-time algorithms. This is the first improvement on the DST problem since the classical quasi-polynomial-time $O(\log^3 k)$ approximation algorithm by Charikar et al. (The paper erroneously claims an $O(\log^2k)$ approximation due to a mistake in prior work.) Our approach is based on two main ingredients. First, we derive an approximation preserving reduction to the Label-Consistent Subtree (LCST) problem. The LCST instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all prior work on DST achieves a reduction to a tree instance of the related Group Steiner Tree (GST) problem of similar height, however losing a logarithmic factor in the approximation ratio. Our second ingredient is an LP-rounding algorithm to approximately solve LCST instances, which is inspired by the framework developed by Rothvo{\ss}. We consider a Sherali-Adams lifting of a proper LP relaxation of LCST. Our rounding algorithm proceeds level by level from the root to the leaves, rounding and conditioning each time on a proper subset of label variables. A small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves

APPROXIMATION ALGORITHMS FOR POINT PATTERN MATCHING AND SEARCHI NG

Point pattern matching is a fundamental problem in computational geometry. For given a reference set and pattern set, the problem is to find a geometric transformation applied to the pattern set that minimizes some given distance measure with respect to the reference set. This problem has been heavily researched under various distance measures and error models. Point set similarity searching is variation of this problem in which a large database of point sets is given, and the task is to preprocess this database into a data structure so that, given a query point set, it is possible to rapidly find the nearest point set among elements of the database. Here, the term nearest is understood in above sense of pattern matching, where the elements of the database may be transformed to match the given query set. The approach presented here is to compute a low distortion embedding of the pattern matching problem into an (ideally) low dimensional metric space and then apply any standard algorithm for nearest neighbor searching over this metric space. This main focus of this dissertation is on two problems in the area of point pattern matching and searching algorithms: (i) improving the accuracy of alignment-based point pattern matching and (ii) computing low-distortion embeddings of point sets into vector spaces. For the first problem, new methods are presented for matching point sets based on alignments of small subsets of points. It is shown that these methods lead to better approximation bounds for alignment-based planar point pattern matching algorithms under the Hausdorff distance. Furthermore, it is shown that these approximation bounds are nearly the best achievable by alignment-based methods. For the second problem, results are presented for two different distance measures. First, point pattern similarity search under translation for point sets in multidimensional integer space is considered, where the distance function is the symmetric difference. A randomized embedding into real space under the L1 metric is given. The algorithm achieves an expected distortion of O(log2 n). Second, an algorithm is given for embedding Rd under the Earth Mover's Distance (EMD) into multidimensional integer space under the symmetric difference distance. This embedding achieves a distortion of O(log D), where D is the diameter of the point set. Combining this with the above result implies that point pattern similarity search with translation under the EMD can be embedded in to real space in the L1 metric with an expected distortion of O(log2 n log D)

On rooted kk-connectivity problems in quasi-bipartite digraphs

We consider the directed Rooted Subset $k$-Edge-Connectivity problem: given a set $T \subseteq V$ of terminals in a digraph $G=(V+r,E)$ with edge costs and an integer $k$, find a min-cost subgraph of $G$ that contains $k$ edge disjoint $rt$-paths for all $t \in T$. The case when every edge of positive cost has head in $T$ admits a polynomial time algorithm due to Frank, and the case when all positive cost edges are incident to $r$ is equivalent to the $k$-Multicover problem. Recently, [Chan et al. APPROX20] obtained ratio $O(\ln k \ln |T|)$ for quasi-bipartite instances, when every edge in $G$ has an end in $T+r$. We give a simple proof for the same ratio for a more general problem of covering an arbitrary $T$-intersecting supermodular set function by a minimum cost edge set, and for the case when only every positive cost edge has an end in $T+r$

A Unified Framework for Hopsets

Given an undirected graph G = (V,E), an (?,?)-hopset is a graph H = (V,E\u27), so that adding its edges to G guarantees every pair has an ?-approximate shortest path that has at most ? edges (hops), that is, d_G(u,v) ? d_{G?H}^(?)(u,v) ? ?? d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter ?. In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^?),O(k^{1-?}))-hopset of size O?(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ? < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (?,?)-hopset of size O(n^{1+1/k}) must have ??? ? ?(k)

Minimum Scan Cover and Variants - Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph G that is embedded in Euclidean space. The edges of G need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances

Algorithms for Unit-Disk Graphs and Related Problems

In this dissertation, we study algorithms for several problems on unit-disk graphs and related problems. The unit-disk graph can be viewed as an intersection graph of a set of congruent disks. Unit-disk graphs have been extensively studied due to many of their applications, e.g., modeling the topology of wireless sensor networks. Lots of problems on unit-disk graphs have been considered in the literature, such as shortest paths, clique, independent set, distance oracle, diameter, etc. Specifically, we study the following problems in this dissertation: L1 shortest paths in unit-disk graphs, reverse shortest paths in unit-disk graphs, minimum bottleneck moving spanning tree, unit-disk range reporting, distance selection, etc. We develop efficient algorithms for these problems and our results are either first-known solutions or somehow improve the previous work. Given a set P of n points in the plane and a parameter r \u3e 0, a unit-disk graph G(P) can be defined using P as its vertex set and two points of P are connected by an edge if the distance between these two points is at most r. The weight of an edge is one in the unweighted case and is equal to the distance between the two endpoints in the weighted case. Note that the distance between two points can be measured by different metrics, e.g., L1 or L2 metric. In the first problem of L1 shortest paths in unit-disk graphs, we are given a point set P and a source point s ∈ P, the problem is to find all shortest paths from s to all other vertices in the L1 weighted unit-disk graph defined on set P. We present an O(n log n) time algorithm, which matches the Ω(n log n)-time lower bound. In the second problem, we are given a set P of n points, parameters r, λ \u3e 0, and two points s and t of P, the goal is to compute the smallest r such that the shortest path length between s and t in the unit-disk graph with respect to set P and parameter r is at most λ. This problem can be defined in both unweighted and weighted cases. We propose an algorithm of O(⌊λ⌋ · n log n) time and another algorithm of O(n5/4 log7/4 n) time for the unweighted case. We also given an O(n5/4 log5/2 n) time algorithm for the weighted case. In the third problem, we are given a set P of n points that are moving in the plane, the problem is to compute a spanning tree for these moving points that does not change its combinatorial structure during the point movement such that the bottleneck weight of the spanning tree (i.e., the largest Euclidean length of all edges) during the whole movement is minimized. We present an algorithm that runs in O(n4/3 log3 n) time. The fourth problem is unit-disk range reporting in which we are given a set P of n points in the plane and a value r, we need to construct a data structure so that given any query disk of radius r, all points of P in the disk can be reported efficiently. We build a data structure of O(n) space in O(n log n) time that can answer each query in O(k + log n) time, where k is the output size. The time complexity of our algorithm is the same as the previous result but our approach is much simpler. Finally, for the problem of distance selection, we are given a set P of n points in the plane and an integer 1 ≤ k ≤ (n2), the distance selection problem is to find the k-th smallest interpoint distance among all pairs of points of p. We propose an algorithm that runs in O(n4/3 log n) time. Our techniques yield two algorithmic frameworks for solving geometric optimization problems. Many algorithms and techniques developed in this dissertation are quite general and fundamental, and we believe they will find other applications in future