Sayan Bhattacharya

Budget constrained auctions with heterogeneous items

In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders have arbitrary demand and budget constraints (and additive valuations). Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with a discrete type space for each bidder, where her valuations for different items can be correlated. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles.

Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of both time and space, not much was known about achieving such type of algorithm. In this paper we study this issue for a problem which lies at the core of many graph mining applications called densest subgraph problem. We develop an algorithm that achieves time- and space-efficiency for this problem simultaneously. It is one of the first of its kind for graph problems to the best of our knowledge. Given an input graph, the densest subgraph is the subgraph that maximizes the ratio between the number of edges and the number of nodes. For any ε>0, our algorithm can, with high probability, maintain a (4+ε)-approximate solution under edge insertions and deletions using ~O(n) space and ~O(1) amortized time per update; here,

Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

n

Coordination mechanisms from (almost) all scheduling policies

is the number of nodes in the graph and ~O hides the O(polylog_{1+ε} n) term. The approximation ratio can be improved to (2+ε) with more time. It can be extended to a (2+ε)-approximation sublinear-time algorithm and a distributed-streaming algorithm. Our algorithm is the first streaming algorithm that can maintain the densest subgraph in one pass. Prior to this, no algorithm could do so even in the special case of an incremental stream and even when there is no time restriction. The previously best algorithm in this setting required O(log n) passes [BahmaniKV12]. The space required by our algorithm is tight up to a polylogarithmic factor.

New deterministic approximation algorithms for fully dynamic matching

We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph in o([EQUATION]m) time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a (2 + e) approximation in O(log n/e2) amortized time per update. For maximum matching, we show how to maintain a (3 + e) approximation in O(m1/3/e2) amortized time per update, and a (4 + e) approximation in O(m1/3/e2) worst-case time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld [13].

Consideration set generation in commerce search

We study the price of anarchy of coordination mechanisms for a scheduling problem where each job j has a weight wj, processing time pij, assignment cost hij, and communication delay (or release date) rij, on machine i. Each machine is free to declare its own scheduling policy. Each job is a selfish agent and selects a machine that minimizes its own disutility, which is equal to its weighted completion time plus its assignment cost. The goal is to minimize the total disutility incurred by all the jobs. Our model is general enough to capture scheduling jobs in a distributed environment with heterogeneous machines (or data centers) that are situated across different locations. Our main result is a characterization of scheduling policies that give a small (robust) Price of Anarchy. More precisely, we show that whenever each machine independently declares any scheduling policy that satisfies a certain bounded stretch condition introduced in this paper, the game induced between the jobs has a small Price of Anarchy. Our characterization is powerful enough to test almost all popular scheduling policies. On the technical side, to derive our results, we use a potential function whose derivative leads to an instantaneous smoothness condition, and linear programming and dual fitting. To the best of our knowledge, this is a novel application of these techniques in the context of coordination mechanisms, and we believe these tools will find more applications in analyzing PoA of games. We also extend our results to the lk-norms and l∞ norm (makespan) objectives.

Fully dynamic approximate maximum matching and minimum vertex cover in O (log 3 n ) worst case update time

We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2+є)-approximate maximum matching in general graphs with O(poly(logn, 1/є)) update time. (2) An algorithm that maintains an αK approximation of the value of the maximum matching with O(n2/K) update time in bipartite graphs, for every sufficiently large constant positive integer K. Here, 1≤ αK < 2 is a constant determined by the value of K. Result (1) is the first deterministic algorithm that can maintain an o(logn)-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best randomized polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with arbitrarily small polynomial update time on bipartite graphs. Previously the best update time for this problem was O(m1/4) [Bernstein et al. ICALP 2015], where m is the current number of edges in the graph.

Approximation algorithm for security games with costly resources

In commerce search, the set of products returned by a search engine often forms the basis for all user interactions leading up to a potential transaction on the web. Such a set of products is known as the consideration set. In this study, we consider the problem of generating consideration set of products in commerce search so as to maximize user satisfaction. One of the key features of commerce search that we exploit in our study is the association of a set of important attributes with the products and a set of specified attributes with the user queries. Those important attributes not used in the query are treated as unspecified. The attribute space admits a natural definition of user satisfaction via user preferences on the attributes and their values, viz. require that the surfaced products be close to the specified attribute values in the query, and diverse with respect to the unspecified attributes. We model this as a general Max-Sum Dispersion problem wherein we are given a set of n nodes in a metric space and the objective is to select a subset of nodes with total cost at most a given budget, and maximize the sum of the pairwise distances between the selected nodes. In our setting, each node denotes a product, the cost of a node being inversely proportional to its relevance with respect to specified attributes. The distance between two nodes quantifies the diversity with respect to the unspecified attributes. The problem is NP-hard and a 2-approximation was previously known only when all the nodes have unit cost. In our setting, we do not make any assumptions on the cost. We label this problem as the General Max-Sum Dispersion problem. We give the first constant factor approximation algorithm for this problem, achieving an approximation ratio of 2. Further, we perform extensive empirical analysis on real-world data to show the effectiveness of our algorithm.

Deterministic Fully Dynamic Approximate Vertex Cover and Fractional Matching in O(1) Amortized Update Time

We consider the problem of maintaining an approximately maximum (fractional) matching and an approximately minimum vertex cover in a dynamic graph. Starting with the seminal paper by Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. There remains, however, a polynomial gap between the best known worst case update time and the best known amortised update time for this problem, even after allowing for randomisation. Specifically, Bernstein and Stein [ICALP 2015, SODA 2016] have the best known worst case update time. They present a deterministic data structure with approximation ratio (3/2 + ϵ) and worst case update time O(m1/4 / ϵ2), where m is the number of edges in the graph. In recent past, Gupta and Peng [FOCS 2013] gave a deterministic data structure with approximation ratio (1 + ϵ) and worst case update time [EQUATION]. No known randomised data structure beats the worst case update times of these two results. In contrast, the paper by Onak and Rubinfeld [STOC 2010] gave a randomised data structure with approximation ratio O(1) and amortised update time O(log2 n), where n is the number of nodes in the graph. This was later improved by Baswana, Gupta and Sen [FOCS 2011] and Solomon [FOCS 2016], leading to a randomised date structure with approximation ratio 2 and amortised update time O(1). We bridge the polynomial gap between the worst case and amortised update times for this problem, without using any randomisation. We present a deterministic data structure with approximation ratio (2 + ϵ) and worst case update time O(log3 n), for all sufficiently small constants ϵ.

Design of Dynamic Algorithms via Primal-Dual Method

In recent years, algorithms for computing game-theoretic solutions have been developed for real-world security domains. These games are between a defender, who must allocate her resources to defend potential targets, and an attacker, who chooses a target to attack. Existing work has assumed the set of defenders resources to be fixed. This assumption precludes the effective use of approximation algorithms, since a slight change in the defenders allocation strategy can result in a massive change in her utility. In contrast, we consider a model where resources are obtained at a cost, initiating the study of the following optimization problem: Minimize the total cost of the purchased resources, given that every target has to be defended with at least a certain probability. We give an efficient logarithmic approximation algorithm for this problem.