Ali Vakilian

On Streaming and Communication Complexity of the Set Cover Problem

We develop the first streaming algorithm and the first two-party communication protocol that uses a constant number of passes/rounds and sublinear space/communication for logarithmic approximation to the classic Set Cover problem. Specifically, for n elements and m sets, our algorithm/protocol achieves a space bound of O(m ·n δ log2 n logm) using O(41/δ ) passes/rounds while achieving an approximation factor of O(41/δ logn) in polynomial time (for δ = Ω(1/logn)). If we allow the algorithm/protocol to spend exponential time per pass/round, we achieve an approximation factor of O(41/δ ). Our approach uses randomization, which we show is necessary: no deterministic constant approximation is possible (even given exponential time) using o(m n) space. These results are some of the first on streaming algorithms and efficient two-party communication protocols for approximation algorithms. Moreover, we show that our algorithm can be applied to multi-party communication model.

Towards Tight Bounds for the Streaming Set Cover Problem

We consider the classic Set Cover problem in the data stream model. For n elements and m sets (m ≥ n) we give a O(1/δ)-pass algorithm with a strongly sub-linear ~O(mnδ) space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [10] that uses exponentially larger number of passes. We complement this result by showing that the tradeoff between the number of passes and space exhibited by our algorithm is tight, at least when the approximation factor is equal to 1. Specifically, we show that any algorithm that computes set cover exactly using ({1 over 2δ}-1) passes must use ~Ω(mnδ) space in the regime of m=O(n). Furthermore, we consider the problem in the geometric setting where the elements are points in R2 and sets are either discs, axis-parallel rectangles, or fat triangles in the plane, and show that our algorithm (with a slight modification) uses the optimal ~O(n) space to find a logarithmic approximation in O(1/δ) passes. Finally, we show that any randomized one-pass algorithm that distinguishes between covers of size 2 and 3 must use a linear (i.e., Ω(mn)) amount of space. This is the first result showing that a randomized, approximate algorithm cannot achieve a space bound that is sublinear in the input size. This indicates that using multiple passes might be necessary in order to achieve sub-linear space bounds for this problem while guaranteeing small approximation factors.

Prize-Collecting Survivable Network Design in Node-Weighted Graphs

We consider node-weighted network design problems, in particular the survivable network design problem (SNDP) and its prize-collecting version (PC-SNDP). The input consists of a node-weighted undirected graph G = (V,E) and integral connectivity requirements r(st) for each pair of nodes st. The goal is to find a minimum node-weighted subgraph H of G such that, for each pair st, H contains r(st) edge-disjoint paths between s and t. PC-SNDP is a generalization in which the input also includes a penalty π(st) for each pair, and the goal is to find a subgraph H to minimize the sum of the weight of H and the sum of the penalties for all pairs whose connectivity requirements are not fully satisfied by H. Let k = max st r(st) be the maximum requirement. There has been no non-trivial approximation for node-weighted PC-SNDP for k > 1, the main reason being the lack of an LP relaxation based approach for node-weighted SNDP. In this paper we describe multiroute-flow based relaxations for the two problems and obtain approximation algorithms for PC-SNDP through them. The approximation ratios we obtain for PC-SNDP are similar to those that were previously known for SNDP via combinatorial algorithms. Specifically, we obtain an O(k 2 logn)-approximation in general graphs and an O(k 2)-approximation in graphs that exclude a fixed minor. The approximation ratios can be improved by a factor of k but the running times of the algorithms depend polynomially on n k .

Node-weighted network design in planar and minor-closed families of graphs

We consider node-weighted network design in planar and minor-closed families of graphs. In particular we focus on the edge-connectivity survivable network design problem (EC-SNDP). The input consists of a node-weighted undirected graph G=(V,E) and integral connectivity requirements r(uv) for each pair of nodes uv. The goal is to find a minimum node-weighted subgraph H of G such that, for each pair uv, H contains r(uv) edge-disjoint paths between u and v. Our main result is an O(k)-approximation algorithm for EC-SNDP where k= max uvr(uv) is the maximum requirement. This improves the O(k logn)-approximation known for node-weighted EC-SNDP in general graphs [15]. Our algorithm and analysis applies to the more general problem of covering a proper function with maximum requirement k. Our result is inspired by, and generalizes, the work of Demaine, Hajiaghayi and Klein [5] who gave constant factor approximation algorithms for node-weighted Steiner tree and Steiner forest problems (and more generally covering 0-1 proper functions) in planar and minor-closed families of graphs.

Which concepts are worth extracting

It is well established that extracting and annotating occurrences of entities in a collection of unstructured text documents with their concepts improve the effectiveness of answering queries over the collection. However, it is very resource intensive to create and maintain large annotated collections. Since the available resources of an enterprise are limited and/or its users may have urgent information needs, it may have to select only a subset of relevant concepts for extraction and annotation. We call this subset a conceptual design for the annotated collection. In this paper, we introduce the problem of cost effective conceptual design, where given a collection, a set of relevant concepts, and a fixed budget, one likes to find a conceptual design that improves the effectiveness of answering queries over the collection the most. We prove that the problem is generally NP-hard in the number of relevant concepts and propose two efficient approximation algorithms to solve the problem: Approximate Popularity Maximization (APM for short) and Approximate Annotation-benefit Maximization (AAM for short). We show that if there is not any constraints regrading the overlap of concepts, APM is a fully polynomial time approximation scheme. We also prove that if the relevant concepts are mutually exclusive, APM has a constant approximation ratio and AAM is a fully polynomial time approximation scheme. Our empirical results using Wikipedia collection and a search engine query log validate the proposed formalization of the problem and show that APM and AAM efficiently compute conceptual designs. They also indicate that in general APM delivers the optimal conceptual designs if the relevant concepts are not mutually exclusive. Also, if the relevant concepts are mutually exclusive, the conceptual designs delivered by AAM improve the effectiveness of answering queries over the collection more than the solutions provided by APM.

Cost-effective conceptual design using taxonomies

It is known that annotating entities in unstructured and semi-structured datasets by their concepts improves the effectiveness of answering queries over these datasets. Ideally, one would like to annotate entities of all relevant concepts in a dataset. However, it takes substantial time and computational resources to annotate concepts in large datasets, and an organization may have sufficient resources to annotate only a subset of relevant concepts. Clearly, it would like to annotate a subset of concepts that provides the most effective answers to queries over the dataset. We propose a formal framework that quantifies the amount by which annotating entities of concepts from a taxonomy in a dataset improves the effectiveness of answering queries over the dataset. Because the problem is

Fractional Set Cover in the Streaming Model

\mathbf {NP}

Node-weighted prize-collecting survivable network design problems

NP-hard, we propose efficient approximation and pseudo-polynomial time algorithms for several cases of the problem. Our extensive empirical studies validate our framework and show accuracy and efficiency of our algorithms.