Ramakrishna Gummadi

Optimal bidding strategies in dynamic auctions with budget constraints

We consider the problem of a bidder with limited budget competing in a series of second-price auctions. A motivating example is that of sponsored search auctions, where advertisers bid in a sequence of repeated generalized second price auctions. To characterize the optimal bidding strategy, we formulate the problem as a discounted Markov Decision Process, and provide explicit solutions when the bidder is involved in a large number of auctions.

Feasible Rate Allocation in Wireless Networks

Rate allocation is a fundamental problem in the operation of a wireless network because of the necessity to schedule the operation of mutually interfering links between the nodes. Among the many reasons behind the importance of efficiently determining the membership of an arbitrary rate vector in the feasibility region, is its high relevance in optimal cross layer design. A key feature in a wireless network is that links without common nodes can also conflict (secondary interference constraints). While the node exclusive model problem has efficient algorithms, it has long been known that this is a hard problem with these additional secondary constraints. However, wireless networks are usually deployed in geographic areas that do not span the most general class of all graphs possible. This is the underlying theme of this paper, where we provide algorithms for two restricted instances of wireless network topologies. In the first tractable instance, we consider nodes placed arbitrarily in a region such that (a) the node density is bounded, and (b) a node can only transmit or interfere with other nodes that are within a certain limited radius. We obtain a simple (1 - epsi) polynomial-time approximation scheme for checking feasibility (for any epsi > 0). The second instance considers the membership problem of an arbitrary rate-vector in the feasible set, where the nodes are distributed within a slab of fixed width (there are no density assumptions). Specifically, the results in [13] are shown to extend to a much more general class of graphs, which we call the (dmin,dmax) class of graphs, and this generalization is used to obtain a strongly polynomial time algorithm that decides membership of a rate-vector where the hosts are distributed within an infinite corridor with fixed cross-section.

Optimal Bidding Strategies and Equilibria in Dynamic Auctions with Budget Constraints

How should agents bid in repeated sequential auctions when they are budget constrained? A motivating example is that of sponsored search auctions, where advertisers bid in a sequence of generalized second price (GSP) auctions. These auctions, specifically in the context of sponsored search, have many idiosyncratic features that distinguish them from other models of sequential auctions: First, each bidder competes in a large number of auctions, where each auction is worth very little. Second, the total bidder population is often large, which means it is unrealistic to assume that the bidders could possibly optimize their strategy by modeling specific opponents. Third, the presence of a virtually unlimited supply of these auctions means bidders are necessarily expense constrained. Motivated by these three factors, we first frame the generic problem as a discounted Markov Decision Process for which the environment is independent and identically distributed over time. We also allow the agents to receive income to augment their budget at a constant rate. We first provide a structural characterization of the associated value function and the optimal bidding strategy, which specifies the extent to which agents underbid from their true valuation due to long term budget constraints. We then provide an explicit characterization of the optimal bid shading factor in the limiting regime where the discount rate tends to zero, by identifying the limit of the value function in terms of the solution to a differential equation that can be solved efficiently. Finally, we proved the existence of Mean Field Equilibria for both the repeated second price and GSP auctions with a large number of bidders.

Optimal control of a broadcasting server

A stochastic control problem motivated by broadcast applications is considered in this paper. A natural queueing model abstraction in which each service to a queue clears all the customers at once is adopted, which can also be considered as a batch processing queueing model with infinite batch size. Each broadcast can be charged a non-negative cost. In addition, there is a cost whose rate is given as a function of the number of customers waiting in the system at any point. For any cost rate which is a convex function in the number of customers, it is shown that the optimal control is of the threshold type in order to minimize the infinite horizon discounted cost. This result complements the existing literature on batch processing queueing models that have typically only considered monotone costs. For a system with two classes of customers where each service can clear all customers of any given class, with monotone waiting costs and zero service costs, we show that the optimal control can be represented as a double-switch curve in the two dimensional state space. The structure of the optimal policy for multiple queues is a natural next question, and an interesting future direction is to explore the performance of simple index policies.

The Role of Feedback in the Choice between Routing and Coding for Wireless Unicast

We consider the benefits of coding in wireless networks, specifically its role in exploiting the local broadcast property of the wireless medium. We first argue that for unicast, the throughput achieved with network coding is the same as that achieved without any coding. This argument highlights the role of a general max-flow min-cut duality and is more explicit than previous proofs. The maximum throughput can be achieved in multiple ways without any coding, for example, using backpressure routing, or using some centralized flow scheduler that is aware of the network topology. However, all such schemes, in order to take advantage of the local broadcast property, require dynamic routing decisions for choosing the next hop for each packet from among the nodes where it is successfully received. This choice seems to depend critically on feedback signalling information like queue lengths, or ARQ. In contrast, note that the use of network coding can achieve the same without such feedback, in exchange for decoding overhead. A key issue to be resolved in making a comparison between routing and coding would be how critical feedback signalling is, for the throughput of routing policies. With this motivation, we first explore how feedback at a given node affects its throughput, with arbitrary rates of its one hop neighbors to the destination. \emph{Static} routing policies which are essentially \emph{feedback independent}, are considered. An explicit characterization of the optimal policies under such a feedback constraint is obtained, which turns out to be a natural generalization of both flooding and traditional routing (which does not exploit local broadcast, because the next hop is fixed prior to the transmission). When losses at the receivers are independent (still allowing for dependencies on transmissions by two different nodes, to model interference), the reduction in capacity due to constraining the feedback is limited to a constant fraction (

On Tractable Instances of Modular Supervisory Control

1-e^{-1} = 63\%

The role of coding in the choice between routing and coding for wireless unicast

) of the coding capacity, and gets arbitrarily close to optimal as the capacity itself is low. This result also extends to a more general version on feedforward networks without any assigned rates of the one hop neighbors to the destination. However, if there are dependencies in the losses seen by receivers from a single broadcast, the reduction could be arbitrarily bad, even with just two hops.

Broadcasting with side information

An instance of a modular supervisory control problem involves a plant automaton, described either as a monolithic, finite-state automaton (SUP1M), or as the synchronous product of several finite-state automata (SUPMM), along with a set of finite state, specification automata on a common alphabet. The marked language of the synchronous product of these automata represents the desired specification. A supervisory policy that solves the instance selectively disables certain events, based on the past history of event-occurrences, such that the marked behavior of the supervised system is a non-empty subset of the desired specification. Testing the existence of a supervisory policy for a variety of in stances of modular supervisory control is PSPACE-complete [1]. This problem remains intractable even when the plant is a monolithic finite state automaton and the specification automata are restricted to have only two states with a specific structure [2]. We refer to this intractable class as SU P1Ω in this paper. After introducing complement sets for events in a plant automaton, we identify a subclass of SUP1Ω that can be solved in polynomial time. Using this class as the base, inspired by a family of subclasses of SAT (cf. section 4.2, [3]) that can be solved in polynomial time [4], we develop a family of subclasses of SUP1Ω that can be solved in polynomial time. The results of this paper are also used to identify a polynomial time hierarchy for certain intractable subclasses of SUPMM identified in this paper.

Mean field equilibria of multi armed bandit games

We consider the benefits of coding in wireless networks, specifically its role in exploiting the local broadcast property of the wireless medium. We first argue that for unicast, the throughput achieved with network coding is the same as that achieved without any coding. This argument highlights the role of a general max-flow min-cut duality and is more explicit than previous proofs. The maximum throughput can be achieved in multiple ways without any coding, for example, using backpressure routing, or using some centralized flow scheduler that is aware of the network topology. However, all such schemes, in order to take advantage of the local broadcast property, require dynamic routing decisions for choosing the next hop for each packet from among the nodes where it is successfully received. This choice seems to depend critically on feedback signaling information like queue lengths, or ARQ. In contrast, note that the use of network coding can achieve the same without such feedback, in exchange for decoding overhead. A key issue to be resolved in making a comparison between routing and coding would be how critical feedback signaling is, for the throughput of routing policies. With this motivation, we first explore how feedback at a given node affects its throughput, with arbitrary rates of its one-hop neighbors to the destination. Static routing policies which are essentially feedback independent, are considered. An explicit characterization of the optimal policies under such a feedback constraint is obtained, which turns out to be a natural generalization of both flooding and traditional routing (which does not exploit local broadcast, because the next hop is fixed prior to the transmission). When losses at the receivers are independent (still allowing for dependencies on transmissions by two different nodes, to model interference), the reduction in capacity due to constraining the feedback is limited to a constant fraction (e−1=37%) of the coding capacity, and gets arbitrarily close to optimal as the unconstrained capacity goes to zero. We also extend this analysis to a layered multihop network and also compare the throughput of flooding to backpressure via simulations for a layered network assuming independent losses. Finally, if there are dependencies in the losses seen by receivers from a single broadcast, the reduction could be arbitrarily bad, even with just two hops.

Computing the Capacity Region of a Wireless Network

We consider the problem of multicasting data from a source to receivers that possess arbitrary subsets of the data apriori as side information. Fountain codes, which are an ideal solution to the standard multicasting problem without any side information, have also been proposed as a potential approach for the side information problem in multiple independent studies recently. Relevant to such a context, we formulate and study an optimization problem over degree distributions to minimize the overhead necessary for complete decoding, and prove that: (i) Degree distributions converging to the standard soliton distribution cannot exploit side information in terms of the overhead necessary for complete decoding. (ii) An asymptotic shifted soliton distribution achieves an overhead which is within a constant factor (<; 2) of the optimal overhead (iii) There exist no degree distributions which achieve asymptotically optimal overhead for any non trivial constant fraction of the data as side information. While (iii) is discouraging, this limitation can be sidestepped by using systematic versions, where intermediate symbols are generated from the source symbols, to which the fountain code is then applied. One important implication of this is that the systematic versions are in a sense indispensable to achieve asymptotic rate optimality for the side information problem.