Covering problems in edge- and node-weighted graphs

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a natural linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal-dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem, respectively. These results match the currently known best results for purely edge-weighted graphs.Comment: To appear in SWAT 201

Lagrangian Relaxation and Partial Cover

Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover.Comment: 20 pages, extended abstract appeared in STACS 200

Online Network Design Algorithms via Hierarchical Decompositions

We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design. In a sense, our work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings and results in randomized $O(\log n)$-competitive algorithms, where $n$ is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get $O(\log k)$ competitive ratios, where $k$ is the number of terminals. In this paper, we apply our approach to obtain deterministic $O(\log k)$-competitive online algorithms for the following problems. - Steiner network with edge duplication. Previously, only a randomized $O(\log n)$-competitive algorithm was known. - Rent-or-buy. Previously, only deterministic $O(\log^2 k)$-competitive and randomized $O(\log k)$-competitive algorithms by Awerbuch, Azar and Bartal (2004) were known. - Connected facility location. Previously, only a randomized $O(\log^2 k)$-competitive algorithm by San Felice, Williamson and Lee (2014) was known. - Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (2011) and give a simpler analysis.Comment: Accepted to SODA 201

You Manage What You Measure: Using Mobile Phones to Strengthen Outcome Monitoring in Rural Sanitation

This paper addresses the sanitation challenge in India, where it is home to the majority of people defecating in the open in the world and also one of the top rapidly growing emerging economies. The paper focuses on the need for a reliable and timely monitoring system to ensure investments in sanitation lead to commensurate outcomes

From Cost Sharing Mechanisms to Online Selection Problems

We consider a general class of online optimization problems, called online selection problems, where customers arrive sequentially, and one has to decide upon arrival whether to accept or reject each customer. If a customer is rejected, then a rejection cost is incurred. The accepted customers are served with minimum possible cost, either online or after all customers have arrived. The goal is to minimize the total production costs for the accepted customers plus the rejection costs for the rejected customers. These selection problems are related to online variants of offline prize collecting combinatorial optimization problems that have been widely studied in the computer science literature. In this paper, we provide a general framework to develop online algorithms for this class of selection problems. In essence, the algorithmic framework leverages any cost sharing mechanism with certain properties into a poly-logarithmic competitive online algorithm for the respective problem; the competitive ratios are shown to be near-optimal. We believe that the general and transparent connection we establish between cost sharing mechanisms and online algorithms could lead to additional online algorithms for problems beyond the ones studied in this paper.National Science Foundation (U.S.) (CAREER Award CMMI-0846554)United States. Air Force Office of Scientific Research (FA9550-11-1-0150)United States. Air Force Office of Scientific Research (FA9550-08-1-0369)Solomon Buchsbaum AT&T Research Fun

Frameworks for Nonclairvoyant Network Design with Deadlines or Delay

Clairvoyant network design with deadlines or delay has been studied extensively, culminating in an O(log n)-competitive general framework, where n is the number of possible request types (Azar and Touitou, FOCS 2020). In the nonclairvoyant setting, the problem becomes much harder, as ?(?n) lower bounds are known for certain problems (Azar et al., STOC 2017). However, no frameworks are known for the nonclairvoyant setting, and previous work focuses only on specific problems, e.g., multilevel aggregation (Le et al., SODA 2023). In this paper, we present the first nonclairvoyant frameworks for network design with deadlines or delay. These frameworks are nearly optimal: their competitive ratio is O?(?n), which matches known lower bounds up to logarithmic factors