arXiv subjects

Debmalya Panigrahi

Publications and source records attributed to Debmalya Panigrahi.

65 records · Page 4Linked to original sources

arXiv2016

On the Price of Stability of Undirected Multicast Games

In multicast network design games, a set of agents choose paths from their source locations to a common sink with the goal of minimizing their individual costs, where the cost of an edge is divided equally among the agents using it. Since the work of Anshelevich et al. (FOCS 2004) that introduced network design games, the main open problem in this field has been the price of stability (PoS) of multicast games. For the special case of broadcast games (every vertex is a terminal, i.e., has an agent), a series of works has culminated in a constant upper bound on the PoS (Bilo` et al., FOCS 2013). However, no significantly sub-logarithmic bound is known for multicast games. In this paper, we make progress toward resolving this question by showing a constant upper bound on the PoS of multicast games for quasi-bipartite graphs. These are graphs where all edges are between two terminals (as in broadcast games) or between a terminal and a nonterminal, but there is no edge between nonterminals. This represents a natural class of intermediate generality between broadcast and multicast games. In addition to the result itself, our techniques overcome some of the fundamental difficulties of analyzing the PoS of general multicast games, and are a promising step toward resolving this major open problem.

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arXiv2016

Online Budgeted Allocation with General Budgets

We study the online budgeted allocation (also called ADWORDS) problem, where a set of impressions arriving online are allocated to a set of budget-constrained advertisers to maximize revenue. Motivated by connections to Internet advertising, several variants of this problem have been studied since the seminal work of Mehta, Saberi, Vazirani, and Vazirani (FOCS 2005). However, this entire body of work focuses on a single budget for every advertising campaign, whereas in order to fully represent the actual agenda of an advertiser, an advertising budget should be expressible over multiple tiers of user-attribute granularity. A simple example is an advertising campaign that is constrained by an overall budget but is also accompanied by a set of sub-budgets for each target demographic. In such a contract scheme, an advertiser can specify their true user-targeting goals, allowing the publisher to fulfill them through relevant allocations. In this paper, we give a complete characterization of the ADWORDS problem for general advertising budgets. In the most general setting, we show that, unlike in the single-budget ADWORDS problem, obtaining a constant competitive ratio is impossible and give asymptotically tight upper and lower bounds. However for our main result, we observe that in many real-world scenarios (as in the above example), multi-tier budgets have a laminar structure, since most relevant consumer or product classifications are hierarchical. For laminar budgets, we obtain a competitive ratio of e/(e-1) in the small bids case, which matches the best known ADWORDS result for single budgets. Our algorithm has a primal-dual structure and generalizes the primal-dual analysis for single- budget ADWORDS first given by Buchbinder, Jain, and Naor (ESA 2007).

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arXiv2015

Online Buy-at-Bulk Network Design

We present the first non-trivial online algorithms for the non-uniform, multicommodity buy-at-bulk (MC-BB) network design problem in undirected and directed graphs. Our competitive ratios qualitatively match the best known approximation factors for the corresponding offline problems. The main engine for our results is an online reduction theorem of MC-BB problems to their single-sink (SS-BB) counterparts. We use the concept of junction-tree solutions (Chekuri et al., FOCS 2006) that play an important role in solving the offline versions of the problem via a greedy subroutine -- an inherently offline procedure. Our main technical contribution is in designing an online algorithm using only the existence of good junction-trees to reduce an MC-BB instance to multiple SS-BB sub-instances. Along the way, we also give the first non-trivial online node-weighted/directed single-sink buy-at-bulk algorithms. In addition to the new results, our generic reduction also yields new proofs of recent results for the online node-weighted Steiner forest and online group Steiner forest problems.

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arXiv2015

Tight Bounds for Online Vector Scheduling

Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multi-dimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and $L_r$ norms of loads are among the most popular objectives considered. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna (SIAM J of Comp. 2006), vector scheduling is a generalization of classical load balancing, where every job has a vector load instead of a scalar load. In this paper, we resolve the online complexity of the vector scheduling problem and its important generalizations. Our main results are: -For identical machines, we show that the optimal competitive ratio is $Θ(\log d / \log \log d)$ by giving an online lower bound and an algorithm with an asymptotically matching competitive ratio. The lower bound is technically challenging, and is obtained via an online lower bound for the minimum mono-chromatic clique problem using a novel online coloring game and randomized coding scheme. -For unrelated machines, we show that the optimal competitive ratio is $Θ(\log m + \log d)$ by giving an online lower bound that matches a previously known upper bound. Unlike identical machines, however, extending these results, particularly the upper bound, to general $L_r$ norms requires new ideas. In particular, we use a carefully constructed potential function that balances the individual $L_r$ objectives with the overall (convexified) min-max objective to guide the online algorithm and track the changes in potential to bound the competitive ratio.

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arXiv2014

Online Covering with Convex Objectives and Applications

We give an algorithmic framework for minimizing general convex objectives (that are differentiable and monotone non-decreasing) over a set of covering constraints that arrive online. This substantially extends previous work on online covering for linear objectives (Alon {\em et al.}, STOC 2003) and online covering with offline packing constraints (Azar {\em et al.}, SODA 2013). To the best of our knowledge, this is the first result in online optimization for generic non-linear objectives; special cases of such objectives have previously been considered, particularly for energy minimization. As a specific problem in this genre, we consider the unrelated machine scheduling problem with startup costs and arbitrary $\ell_p$ norms on machine loads (including the surprisingly non-trivial $\ell_1$ norm representing total machine load). This problem was studied earlier for the makespan norm in both the offline (Khuller~{\em et al.}, SODA 2010; Li and Khuller, SODA 2011) and online settings (Azar {\em et al.}, SODA 2013). We adapt the two-phase approach of obtaining a fractional solution and then rounding it online (used successfully to many linear objectives) to the non-linear objective. The fractional algorithm uses ideas from our general framework that we described above (but does not fit the framework exactly because of non-positive entries in the constraint matrix). The rounding algorithm uses ideas from offline rounding of LPs with non-linear objectives (Azar and Epstein, STOC 2005; Kumar {\em et al.}, FOCS 2005). Our competitive ratio is tight up to a logarithmic factor. Finally, for the important special case of total load ($\ell_1$ norm), we give a different rounding algorithm that obtains a better competitive ratio than the generic rounding algorithm for $\ell_p$ norms. We show that this competitive ratio is asymptotically tight.

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arXiv2014

Precedence-constrained Scheduling of Malleable Jobs with Preemption

Scheduling jobs with precedence constraints on a set of identical machines to minimize the total processing time (makespan) is a fundamental problem in combinatorial optimization. In practical settings such as cloud computing, jobs are often malleable, i.e., can be processed on multiple machines simultaneously. The instantaneous processing rate of a job is a non-decreasing function of the number of machines assigned to it (we call it the processing function). Previous research has focused on practically relevant concave processing functions, which obey the law of diminishing utility and generalize the classical (non-malleable) problem. Our main result is a $(2+ε)$-approximation algorithm for concave processing functions (for any $ε> 0$), which is the best possible under complexity theoretic assumptions. The approximation ratio improves to $(1 + ε)$ for the interesting and practically relevant special case of power functions, i.e., $p_j(z) = c_j \cdot z^γ$.

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arXiv2014

Online Algorithms for Machine Minimization

In this paper, we consider the online version of the machine minimization problem (introduced by Chuzhoy et al., FOCS 2004), where the goal is to schedule a set of jobs with release times, deadlines, and processing lengths on a minimum number of identical machines. Since the online problem has strong lower bounds if all the job parameters are arbitrary, we focus on jobs with uniform length. Our main result is a complete resolution of the deterministic complexity of this problem by showing that a competitive ratio of $e$ is achievable and optimal, thereby improving upon existing lower and upper bounds of 2.09 and 5.2 respectively. We also give a constant-competitive online algorithm for the case of uniform deadlines (but arbitrary job lengths); to the best of our knowledge, no such algorithm was known previously. Finally, we consider the complimentary problem of throughput maximization where the goal is to maximize the sum of weights of scheduled jobs on a fixed set of identical machines (introduced by Bar-Noy et al. STOC 1999). We give a randomized online algorithm for this problem with a competitive ratio of e/e-1; previous results achieved this bound only for the case of a single machine or in the limit of an infinite number of machines.

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arXiv2012

Online Load Balancing on Unrelated Machines with Startup Costs

Motivated by applications in energy-efficient scheduling in data centers, Khuller, Li, and Saha introduced the {\em machine activation} problem as a generalization of the classical optimization problems of set cover and load balancing on unrelated machines. In this problem, a set of $n$ jobs have to be distributed among a set of $m$ (unrelated) machines, given the processing time of each job on each machine, where each machine has a startup cost. The goal is to produce a schedule of minimum total startup cost subject to a constraint $\bf L$ on its makespan. While Khuller {\em et al} considered the offline version of this problem, a typical scenario in scheduling is one where jobs arrive online and have to be assigned to a machine immediately on arrival. We give an $(O(\log (mn)\log m), O(\log m))$-competitive randomized online algorithm for this problem, i.e. the schedule produced by our algorithm has a makespan of $O({\bf L} \log m)$ with high probability, and a total expected startup cost of $O(\log (mn)\log m)$ times that of an optimal offline schedule with makespan $\bf L$. The competitive ratios of our algorithm are (almost) optimal. Our algorithms use the online primal dual framework introduced by Alon {\em et al} for the online set cover problem, and subsequently developed further by Buchbinder, Naor, and co-authors. To the best of our knowledge, all previous applications of this framework have been to linear programs (LPs) with either packing or covering constraints. One novelty of our application is that we use this framework for a mixed LP that has both covering and packing constraints. We hope that the algorithmic techniques developed in this paper to simultaneously handle packing and covering constraints will be useful for solving other online optimization problems as well.

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arXiv2011

Provenance Views for Module Privacy

Scientific workflow systems increasingly store provenance information about the module executions used to produce a data item, as well as the parameter settings and intermediate data items passed between module executions. However, authors/owners of workflows may wish to keep some of this information confidential. In particular, a module may be proprietary, and users should not be able to infer its behavior by seeing mappings between all data inputs and outputs. The problem we address in this paper is the following: Given a workflow, abstractly modeled by a relation R, a privacy requirement Γand costs associated with data. The owner of the workflow decides which data (attributes) to hide, and provides the user with a view R' which is the projection of R over attributes which have not been hidden. The goal is to minimize the cost of hidden data while guaranteeing that individual modules are Γ-private. We call this the "secureview" problem. We formally define the problem, study its complexity, and offer algorithmic solutions.

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arXiv2010

A Linear-time Algorithm for Sparsification of Unweighted Graphs

Given an undirected graph $G$ and an error parameter $ε> 0$, the {\em graph sparsification} problem requires sampling edges in $G$ and giving the sampled edges appropriate weights to obtain a sparse graph $G_ε$ with the following property: the weight of every cut in $G_ε$ is within a factor of $(1\pm ε)$ of the weight of the corresponding cut in $G$. If $G$ is unweighted, an $O(m\log n)$-time algorithm for constructing $G_ε$ with $O(n\log n/ε^2)$ edges in expectation, and an $O(m)$-time algorithm for constructing $G_ε$ with $O(n\log^2 n/ε^2)$ edges in expectation have recently been developed (Hariharan-Panigrahi, 2010). In this paper, we improve these results by giving an $O(m)$-time algorithm for constructing $G_ε$ with $O(n\log n/ε^2)$ edges in expectation, for unweighted graphs. Our algorithm is optimal in terms of its time complexity; further, no efficient algorithm is known for constructing a sparser $G_ε$. Our algorithm is Monte-Carlo, i.e. it produces the correct output with high probability, as are all efficient graph sparsification algorithms.

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arXiv2010

A General Framework for Graph Sparsification

Given a weighted graph $G$ and an error parameter $ε> 0$, the {\em graph sparsification} problem requires sampling edges in $G$ and giving the sampled edges appropriate weights to obtain a sparse graph $G_ε$ (containing O(n\log n) edges in expectation) with the following property: the weight of every cut in $G_ε$ is within a factor of $(1\pm ε)$ of the weight of the corresponding cut in $G$. We provide a generic framework that sets out sufficient conditions for any particular sampling scheme to result in good sparsifiers, and obtain a set of results by simple instantiations of this framework. The results we obtain include the following: (1) We improve the time complexity of graph sparsification from O(m\log^3 n) to O(m + n\log^4 n) for graphs with polynomial edge weights. (2) We improve the time complexity of graph sparsification from O(m\log^3 n) to O(m\log^2 n) for graphs with arbitrary edge weights. (3) If the size of the sparsifier is allowed to be O(n\log^2 n/ε^2) instead of O(n\log n/ε^2), we improve the time complexity of sparsification to O(m) for graphs with polynomial edge weights. (4) We show that sampling using standard connectivities results in good sparsifiers, thus resolving an open question of Benczur and Karger. As a corollary, we give a simple proof of (a slightly weaker version of) a result due to Spielman and Srivastava showing that sampling using effective resistances produces good sparsifiers. (5) We give a simple proof showing that sampling using strong connectivities results in good sparsifiers, a result obtained previously using a more involved proof by Benczur and Karger. A key ingredient of our proofs is a generalization of bounds on the number of small cuts in an undirected graph due to Karger; this generalization might be of independent interest.

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