arXiv subjects

Rahul Vaze

Publications and source records attributed to Rahul Vaze.

At least 55 records · Page 3Linked to original sources

arXiv2018

Submodular Maximization Under A Matroid Constraint: Asking more from an old friend, the Greedy Algorithm

The classical problem of maximizing a submodular function under a matroid constraint is considered. Defining a new measure for the increments made by the greedy algorithm at each step, called the discriminant, improved approximation ratio guarantees are derived for the greedy algorithm. At each step, discriminant measures the multiplicative gap in the incremental valuation between the item chosen by the greedy algorithm and the largest potential incremental valuation for eligible items not selected by it. The new guarantee subsumes all the previous known results for the greedy algorithm, including the curvature based ones, and the derived guarantees are shown to be tight via constructing specific instances. More refined approximation guarantee is derived for a special case called the submodular welfare maximization/partition problem that is also tight, for both the offline and the online case.

cs.DS↗

arXiv2017

Online Knapsack Problem under Expected Capacity Constraint

Online knapsack problem is considered, where items arrive in a sequential fashion that have two attributes; value and weight. Each arriving item has to be accepted or rejected on its arrival irrevocably. The objective is to maximize the sum of the value of the accepted items such that the sum of their weights is below a budget/capacity. Conventionally a hard budget/capacity constraint is considered, for which variety of results are available. In modern applications, e.g., in wireless networks, data centres, cloud computing, etc., enforcing the capacity constraint in expectation is sufficient. With this motivation, we consider the knapsack problem with an expected capacity constraint. For the special case of knapsack problem, called the secretary problem, where the weight of each item is unity, we propose an algorithm whose probability of selecting any one of the optimal items is equal to $1-1/e$ and provide a matching lower bound. For the general knapsack problem, we propose an algorithm whose competitive ratio is shown to be $1/4e$ that is significantly better than the best known competitive ratio of $1/10e$ for the knapsack problem with the hard capacity constraint.

cs.DS↗

arXiv2017

Robust Online Speed Scaling With Deadline Uncertainty

A speed scaling problem is considered, where time is divided into slots, and jobs with payoff $v$ arrive at the beginning of the slot with associated deadlines $d$. Each job takes one slot to be processed, and multiple jobs can be processed by the server in each slot with energy cost $g(k)$ for processing $k$ jobs in one slot. The payoff is accrued by the algorithm only if the job is processed by its deadline. We consider a robust version of this speed scaling problem, where a job on its arrival reveals its payoff $v$, however, the deadline is hidden to the online algorithm, which could potentially be chosen adversarially and known to the optimal offline algorithm. The objective is to derive a robust (to deadlines) and optimal online algorithm that achieves the best competitive ratio. We propose an algorithm (called min-LCR) and show that it is an optimal online algorithm for any convex energy cost function $g(.)$. We do so without actually evaluating the optimal competitive ratio, and give a general proof that works for any convex $g$, which is rather novel. For the popular choice of energy cost function $g(k) = k^α, α\ge 2$, we give concrete bounds on the competitive ratio of the algorithm, which ranges between $2.618$ and $3$ depending on the value of $α$. The best known online algorithm for the same problem, but where deadlines are revealed to the online algorithm has competitive ratio of $2$ and a lower bound of $\sqrt{2}$. Thus, importantly, lack of deadline knowledge does not make the problem degenerate, and the effect of deadline information on the optimal competitive ratio is limited.

cs.DS↗

arXiv2017

Energy-Delay-Distortion Problem

An energy-limited source trying to transmit multiple packets to a destination with possibly different sizes is considered. With limited energy, the source cannot potentially transmit all bits of all packets. In addition, there is a delay cost associated with each packet. Thus, the source has to choose, how many bits to transmit for each packet, and the order in which to transmit these bits, to minimize the cost of distortion (introduced by transmitting lower number of bits) and queueing plus transmission delay, across all packets. Assuming an exponential metric for distortion loss and linear delay cost, we show that the optimization problem is jointly convex. Hence, the problem can be exactly solved using convex solvers, however, because of the complicated expression derived from the KKT conditions, no closed form solution can be found even with the simplest cost function choice made in the paper, also the optimal order in which packets should be transmitted needs to be found via brute force. To facilitate a more structured solution, a discretized version of the problem is also considered, where time and energy are divided in discrete amounts. In any time slot (fixed length), bits belonging to any one packet can be transmitted, while any discrete number of energy quanta can be used in any slot corresponding to any one packet, such that the total energy constraint is satisfied. The discretized problem is a special case of a multi-partitioning problem, where each packet's utility is super-modular and the proposed greedy solution is shown to incur cost that is at most $2$-times of the optimal cost.

cs.IT↗

arXiv2017

On Distributed Power Control for Uncoordinated Dual Energy Harvesting Links: Performance Bounds and Near-Optimal Policies

In this paper, we consider a point-to-point link between an energy harvesting transmitter and receiver, where neither node has the information about the battery state or energy availability at the other node. We consider a model where data is successfully delivered only in slots where both nodes are active. Energy loss occurs whenever one node turns on while the other node is in sleep mode. In each slot, based on their own energy availability, the transmitter and receiver need to independently decide whether or not to turn on, with the aim of maximizing the long-term time-average throughput. We present an upper bound on the throughput achievable by analyzing a genie-aided system that has noncausal knowledge of the energy arrivals at both the nodes. Next, we propose an online policy requiring an occasional one-bit feedback whose throughput is within one bit of the upper bound, asymptotically in the battery size. In order to further reduce the feedback required, we propose a time-dilated version of the online policy. As the time dilation gets large, this policy does not require any feedback and achieves the upper bound asymptotically in the battery size. Inspired by this, we also propose a near-optimal fully uncoordinated policy. We use Monte Carlo simulations to validate our theoretical results and illustrate the performance of the proposed policies.

cs.IT↗

arXiv2017

Capacity of Cellular Wireless Network

Earlier definitions of capacity for wireless networks, e.g., transport or transmission capacity, for which exact theoretical results are known, are well suited for ad hoc networks but are not directly applicable for cellular wireless networks, where large-scale basestation (BS) coordination is not possible, and retransmissions/ARQ under the SINR model is a universal feature. In this paper, cellular wireless networks, where both BS locations and mobile user (MU) locations are distributed as independent Poisson point processes are considered, and each MU connects to its nearest BS. With ARQ, under the SINR model, the effective downlink rate of packet transmission is the reciprocal of the expected delay (number of retransmissions needed till success), which we use as our network capacity definition after scaling it with the BS density. Exact characterization of this natural capacity metric for cellular wireless networks is derived. The capacity is shown to first increase polynomially with the BS density in the low BS density regime and then scale inverse exponentially with the increasing BS density. Two distinct upper bounds are derived that are relevant for the low and the high BS density regimes. A single power control strategy is shown to achieve the upper bounds in both the regimes. This result is fundamentally different from the well known capacity results for ad hoc networks, such as transport and transmission capacity that scale as the square root of the (high) BS density. Our results show that the strong temporal correlations of SINRs with PPP distributed BS locations is limiting, and the realizable capacity in cellular wireless networks in high-BS density regime is much smaller than previously thought. A byproduct of our analysis shows that the capacity of the ALOHA strategy with retransmissions is zero.

cs.IT↗

arXiv2016

Online Knapsack Problem and Budgeted Truthful Bipartite Matching

Two related online problems: knapsack and truthful bipartite matching are considered. For these two problems, the common theme is how to `match' an arriving left vertex in an online fashion with any of the available right vertices, if at all, so as to maximize the sum of the value of the matched edges, subject to satisfying a sum-weight constraint on the matched left vertices. Assuming that the left vertices arrive in an uniformly random order (secretary model), two almost similar algorithms are proposed for the two problems, that are $2e$ competitive and $24$ competitive, respectively. The proposed online bipartite matching algorithm is also shown to be truthful: there is no incentive for any left vertex to misreport its bid/weight. Direct applications of these problems include job allocation with load balancing, generalized adwords, crowdsourcing auctions, and matching wireless users to cooperative relays in device-to-device communication enabled cellular network.

cs.DS↗

arXiv2016

Online Budgeted Truthful Matching

An online truthful budgeted matching problem is considered for a bipartite graph, where the right vertices are available ahead of time, and individual left vertices arrive sequentially. On arrival of a left vertex, its edge utilities (or weights) to all the right vertices and a corresponding cost (or bid) are revealed. If a left vertex is matched to any of the right vertices, then it has to be paid at least as much as its cost. The problem is to match each left vertex instantaneously and irrevocably to any one of the right vertices, if at all, to find the maximum weight matching that is truthful, under a payment budget constraint. Truthfulness condition requires that no left vertex has any incentive of misreporting its cost. Assuming that the vertices arrive in an uniformly random order (secretary model) with arbitrary utilities, a truthful algorithm is proposed that is $24β$-competitive (where $β$ is the ratio of the maximum and the minimum utility) and satisfies the payment budget constraint. Direct applications of this problem include crowdsourcing auctions, and matching wireless users to cooperative relays in device-to-device enabled cellular network.

cs.DS↗

arXiv2016

Paging with Multiple Caches

Modern content delivery networks consist of one or more back-end servers which store the entire content catalog, assisted by multiple front-end servers with limited storage and service capacities located near the end-users. Appropriate replication of content on the front-end servers is key to maximize the fraction of requests served by the front-end servers. Motivated by this, a multiple cache variant of the classical single cache paging problem is studied, which is referred to as the Multiple Cache Paging (MCP) problem. In each time-slot, a batch of content requests arrive that have to be served by a bank of caches, and each cache can serve exactly one request. If a content is not found in the bank, it is fetched from the back-end server, and one currently stored content is ejected, and counted as fault. As in the classical paging problem, the goal is to minimize the total number of faults. The competitive ratio of any online algorithm for the MCP problem is shown to be unbounded for arbitrary input, thus concluding that the MCP problem is fundamentally different from the classical paging problem. Consequently, stochastic arrivals setting is considered, where requests arrive according to a known/unknown stochastic process. It is shown that near optimal performance can be achieved with simple policies that require no co-ordination across the caches.

cs.DS↗

arXiv2016

Online energy efficient packet scheduling for a common deadline with and without energy harvesting

The problem of online packet scheduling to minimize the required conventional grid energy for transmitting a fixed number of packets given a common deadline is considered. The total number of packets arriving within the deadline is known, but the packet arrival times are unknown, and can be arbitrary. The proposed algorithm tries to finish the transmission of each packet assuming all future packets are going to arrive at equal time intervals within the left-over time. The proposed online algorithm is shown to have competitive ratio that is logarithmic in the number of packet arrivals. The hybrid energy paradigm is also considered, where in addition to grid energy, energy is also available via extraction from renewable sources. The objective here is to minimize the grid energy use. A suitably modified version of the previous algorithm is also shown to have competitive ratio that is logarithmic in the number of packet arrivals.

cs.IT↗

arXiv2015

Online Budgeted Repeated Matching

A basic combinatorial online resource allocation problem is considered, where multiple servers have individual capacity constraints, and at each time slot, a set of jobs arrives, that have potentially different weights to different servers. At each time slot, a one-to-one matching has to be found between jobs and servers, subject to individual capacity constraints, in an online manner. The objective is to maximize the aggregate weight of jobs allotted to servers, summed across time slots and servers, subject to individual capacity constraints. This problem generalizes the well known adwords problem, and is also relevant for various other modern applications. A simple greedy algorithm is shown to be 3-competitive, whenever the weight of any edge is at most half of the corresponding server capacity. Moreover, a randomized version of the greedy algorithm is shown to be 6-competitive for the unrestricted edge weights case. For parallel servers with small-weight jobs, we show that a load-balancing algorithm is near-optimal.

cs.DS↗

arXiv2015

Achieving Non-Zero Information Velocity in Wireless Networks

In wireless networks, where each node transmits independently of other nodes in the network (the ALOHA protocol), the expected delay experienced by a packet until it is successfully received at any other node is known to be infinite for signal-to-interference-plus-noise-ratio (SINR) model with node locations distributed according to a Poisson point process. Consequently, the information velocity, defined as the limit of the ratio of the distance to the destination and the time taken for a packet to successfully reach the destination over multiple hops, is zero, as the distance tends to infinity. A nearest neighbor distance based power control policy is proposed to show that the expected delay required for a packet to be successfully received at the nearest neighbor can be made finite. Moreover, the information velocity is also shown to be non-zero with the proposed power control policy. The condition under which these results hold does not depend on the intensity of the underlying Poisson point process.

cs.IT↗

arXiv2015

Optimal Offline and Competitive Online Strategies for Transmitter-Receiver Energy Harvesting

A joint transmitter-receiver energy harvesting model is considered, where both the transmitter and receiver are powered by (renewable) energy harvesting source. Given a fixed number of bits, the problem is to find the optimal transmission power profile at the transmitter and ON-OFF profile at the receiver to minimize the transmission time. With infinite capacity at both the transmitter and receiver, optimal offline and optimal online policies are derived. The optimal online policy is shown to be two-competitive in the arbitrary input case. With finite battery capacities at both ends, only random energy arrival sequence with given distribution are considered, for which an online policy with bounded expected competitive ratio is proposed.

cs.IT↗

arXiv2015

Long term Throughput and Approximate Capacity of Transmitter-Receiver Energy Harvesting Channel with Fading

We first consider an energy harvesting channel with fading, where only the transmitter harvests energy from natural sources. We bound the optimal long term throughput by a constant for a class of energy arrival distributions. The proposed method also gives a constant approximation to the capacity of the energy harvesting channel with fading. Next, we consider a more general system where both the transmitter and the receiver employ energy harvesting to power themselves. In this case, we show that finding an approximation to the optimal long term throughput is far more difficult, and identify a special case of unit battery capacity at both the transmitter and the receiver for which we obtain a universal bound on the ratio of the upper and lower bound on the long term throughput.

cs.IT↗

arXiv2015

Critical Database Size for Effective Caching

Replicating or caching popular content in memories distributed across the network is a technique to reduce peak network loads. Conventionally, the performance gain of caching was thought to result from making part of the requested data available closer to end users. Recently, it has been shown that by using a carefully designed technique to store the contents in the cache and coding across data streams a much more significant gain can be achieved in reducing the network load. Inner and outer bounds on the network load v/s cache memory tradeoff were obtained in (Maddah-Ali and Niesen, 2012). We give an improved outer bound on the network load v/s cache memory tradeoff. We address the question of to what extent caching is effective in reducing the server load when the number of files becomes large as compared to the number of users. We show that the effectiveness of caching become small when the number of files becomes comparable to the square of the number of users.

cs.IT↗

arXiv2014

The Online Disjoint Set Cover Problem and its Applications

Given a universe $U$ of $n$ elements and a collection of subsets $\mathcal{S}$ of $U$, the maximum disjoint set cover problem (DSCP) is to partition $\mathcal{S}$ into as many set covers as possible, where a set cover is defined as a collection of subsets whose union is $U$. We consider the online DSCP, in which the subsets arrive one by one (possibly in an order chosen by an adversary), and must be irrevocably assigned to some partition on arrival with the objective of minimizing the competitive ratio. The competitive ratio of an online DSCP algorithm $A$ is defined as the maximum ratio of the number of disjoint set covers obtained by the optimal offline algorithm to the number of disjoint set covers obtained by $A$ across all inputs. We propose an online algorithm for solving the DSCP with competitive ratio $\ln n$. We then show a lower bound of $Ω(\sqrt{\ln n})$ on the competitive ratio for any online DSCP algorithm. The online disjoint set cover problem has wide ranging applications in practice, including the online crowd-sourcing problem, the online coverage lifetime maximization problem in wireless sensor networks, and in online resource allocation problems.

cs.DS↗

arXiv2014

Autoregressive Cascades on Random Networks

This paper considers a model for cascades on random networks in which the cascade propagation at any node depends on the load at the failed neighbor, the degree of the neighbor as well as the load at that node. Each node in the network bears an initial load that is below the capacity of the node. The trigger for the cascade emanates at a single node or a small fraction of the nodes from some external shock. Upon failure, the load at the failed node gets divided randomly and added to the existing load at those neighboring nodes that have not yet failed. Subsequently, a neighboring node fails if its accumulated load exceeds its capacity. The failed node then plays no further part in the process. The cascade process stops as soon as the accumulated load at all nodes that have not yet failed is below their respective capacities. The model is shown to operate in two regimes, one in which the cascade terminates with only a finite number of node failures. In the other regime there is a positive probability that the cascade continues indefinitely. Bounds are obtained on the critical parameter where the phase transition occurs.

physics.soc-ph↗

arXiv2014

Optimal WiFi Sensing via Dynamic Programming

The problem of finding an optimal sensing schedule for a mobile device that encounters an intermittent WiFi access opportunity is considered. At any given time, the WiFi is in any of the two modes, ON or OFF, and the mobile's incentive is to connect to the WiFi in the ON mode as soon as possible, while spending as little sensing energy. We introduce a dynamic programming framework which enables the characterization of an explicit solution for several models, particularly when the OFF periods are exponentially distributed. While the problem for non-exponential OFF periods is ill-posed in general, a usual workaround in literature is to make the mobile device aware if one ON period is completely missed. In this restricted setting, using the DP framework, the deterministic nature of the optimal sensing policy is established, and value iterations are shown to converge to the optimal solution. Finally, we address the blind situation where the distributions of ON and OFF periods are unknown. A continuous bandit based learning algorithm that has vanishing regret (loss compared to the optimal strategy with the knowledge of distributions) is presented, and comparisons with the optimal schemes are provided for exponential ON and OFF times.

cs.IT↗