arXiv subjects

Viswanath Nagarajan

Publications and source records attributed to Viswanath Nagarajan.

At least 37 records · Page 2Linked to original sources

arXiv2020

Online Generalized Network Design Under (Dis)Economies of Scale

We consider a general online network design problem where a sequence of N requests arrive over time, each of which needs to use some subset of the available resources E. The cost incurred by any resource e is some function $f_e$ of the total load $L_e$ on that resource. The objective is to minimize the total cost $\sum_{e\in E} f_e(L_e)$. We focus on cost functions that exhibit (dis)economies of scale, that are of the form $f_e(x) = σ_e + ξ_e\cdot x^{α_e}$ if $x>0$ (and zero if $x=0$), where the exponent $α_e\ge 1$. Optimization problems under these functions have received significant recent attention due to applications in energy-efficient computing. Our main result is a deterministic online algorithm with tight competitive ratio $Θ\left(\max_{e\in E} \left(\frac{σ_e}{ξ_e}\right)^{1/α_e}\right)$ when $α_e$ is constant for all $e\in E$. This framework is applicable to a variety of network design problems in undirected and directed graphs, including multicommodity routing, Steiner tree/forest connectivity and set-connectivity. In fact, our online competitive ratio even matches the previous-best (offline) approximation ratio for generalized network design.

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arXiv2019

Stochastic Load Balancing on Unrelated Machines

We consider the problem of makespan minimization on unrelated machines when job sizes are stochastic. The goal is to find a fixed assignment of jobs to machines, to minimize the expected value of the maximum load over all the machines. For the identical machines special case when the size of a job is the same across all machines, a constant-factor approximation algorithm has long been known. Our main result is the first constant-factor approximation algorithm for the general case of unrelated machines. This is achieved by (i) formulating a lower bound using an exponential-size linear program that is efficiently computable, and (ii) rounding this linear program while satisfying only a specific subset of the constraints that still suffice to bound the expected makespan. We also consider two generalizations. The first is the budgeted makespan minimization problem, where the goal is to minimize the expected makespan subject to scheduling a target number (or reward) of jobs. We extend our main result to obtain a constant-factor approximation algorithm for this problem. The second problem involves $q$-norm objectives, where we want to minimize the expected q-norm of the machine loads. Here we give an $O(q/\log q)$-approximation algorithm, which is a constant-factor approximation for any fixed $q$.

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arXiv2019

Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Other Directed Network Design Problems

We consider the following general network design problem on directed graphs. The input is an asymmetric metric $(V,c)$, root $r^{*}\in V$, monotone submodular function $f:2^V\rightarrow \mathbb{R}_+$ and budget $B$. The goal is to find an $r^{*}$-rooted arborescence $T$ of cost at most $B$ that maximizes $f(T)$. Our main result is a simple quasi-polynomial time $O(\frac{\log k}{\log\log k})$-approximation algorithm for this problem, where $k\le |V|$ is the number of vertices in an optimal solution. To the best of our knowledge, this is the first non-trivial approximation ratio for this problem. As a consequence we obtain an $O(\frac{\log^2 k}{\log\log k})$-approximation algorithm for directed (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved $O(\frac{\log^2 k}{\log\log k})$-approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio [GLL19]. Our algorithm has the advantage of being deterministic and faster: the runtime is $\exp(O(\log n\, \log^{1+ε} k))$. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first non-trivial approximation ratio. All our approximation ratios are tight (up to constant factors) for quasi-polynomial algorithms.

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arXiv2019

Adaptive Submodular Ranking and Routing

We study a general stochastic ranking problem where an algorithm needs to adaptively select a sequence of elements so as to "cover" a random scenario (drawn from a known distribution) at minimum expected cost. The coverage of each scenario is captured by an individual submodular function, where the scenario is said to be covered when its function value goes above a given threshold. We obtain a logarithmic factor approximation algorithm for this adaptive ranking problem, which is the best possible (unless P=NP). This problem unifies and generalizes many previously studied problems with applications in search ranking and active learning. The approximation ratio of our algorithm either matches or improves the best result known in each of these special cases. Furthermore, we extend our results to an adaptive vehicle routing problem, where costs are determined by an underlying metric. This routing problem is a significant generalization of the previously-studied adaptive traveling salesman and traveling repairman problems. Our approximation ratio nearly matches the best bound known for these special cases. Finally, we present experimental results for some applications of adaptive ranking.

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arXiv2019

Approximation Algorithms for the A Priori TravelingRepairman

We consider the a priori traveling repairman problem, which is a stochastic version of the classic traveling repairman problem (also called the traveling deliveryman or minimum latency problem). Given a metric $(V,d)$ with a root $r\in V$, the traveling repairman problem (TRP) involves finding a tour originating from $r$ that minimizes the sum of arrival-times at all vertices. In its a priori version, we are also given independent probabilities of each vertex being active. We want to find a master tour $τ$ originating from $r$ and visiting all vertices. The objective is to minimize the expected sum of arrival-times at all active vertices, when $τ$ is shortcut over the inactive vertices. We obtain the first constant-factor approximation algorithm for a priori TRP under non-uniform probabilities. Previously, such a result was only known for uniform probabilities.

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arXiv2018

Approximating Max-Cut under Graph-MSO Constraints

We consider the max-cut and max-$k$-cut problems under graph-based constraints. Our approach can handle any constraint specified using monadic second-order (MSO) logic on graphs of constant treewidth. We give a $\frac{1}{2}$-approximation algorithm for this class of problems.

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arXiv2017

Online Covering with Sum of $\ell_q$-Norm Objectives

We consider fractional online covering problems with $\ell_q$-norm objectives. The problem of interest is of the form $\min\{ f(x) \,:\, Ax\ge 1, x\ge 0\}$ where $f(x)=\sum_{e} c_e \|x(S_e)\|_{q_e} $ is the weighted sum of $\ell_q$-norms and $A$ is a non-negative matrix. The rows of $A$ (i.e. covering constraints) arrive online over time. We provide an online $O(\log d+\log ρ)$-competitive algorithm where $ρ= \frac{\max a_{ij}}{\min a_{ij}}$ and $d$ is the maximum of the row sparsity of $A$ and $\max |S_e|$. This is based on the online primal-dual framework where we use the dual of the above convex program. Our result expands the class of convex objectives that admit good online algorithms: prior results required a monotonicity condition on the objective $f$ which is not satisfied here. This result is nearly tight even for the linear special case. As direct applications we obtain (i) improved online algorithms for non-uniform buy-at-bulk network design and (ii) the first online algorithm for throughput maximization under $\ell_p$-norm edge capacities.

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arXiv2017

Approximation Algorithms for Optimal Decision Trees and Adaptive TSP Problems

We consider the problem of constructing optimal decision trees: given a collection of tests which can disambiguate between a set of $m$ possible diseases, each test having a cost, and the a-priori likelihood of the patient having any particular disease, what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease? We settle the approximability of this problem by giving a tight $O(\log m)$-approximation algorithm. We also consider a more substantial generalization, the Adaptive TSP problem. Given an underlying metric space, a random subset $S$ of cities is drawn from a known distribution, but $S$ is initially unknown to us--we get information about whether any city is in $S$ only when we visit the city in question. What is a good adaptive way of visiting all the cities in the random subset $S$ while minimizing the expected distance traveled? For this problem, we give the first poly-logarithmic approximation, and show that this algorithm is best possible unless we can improve the approximation guarantees for the well-known group Steiner tree problem.

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arXiv2016

Approximation-Friendly Discrepancy Rounding

Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative rounding. We provide an extension of the discrepancy-based rounding algorithm due to Lovett-Meka that (i) combines the advantages of both randomized and iterated rounding, (ii) makes it applicable to settings with more general combinatorial structure such as matroids. As applications of this approach, we obtain new results for various classical problems such as linear system rounding, degree-bounded matroid basis and low congestion routing.

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arXiv2016

Approximation Algorithms for Stochastic k-TSP

We consider the stochastic $k$-TSP problem where rewards at vertices are random and the objective is to minimize the expected length of a tour that collects reward $k$. We present an adaptive $O(\log k)$-approximation algorithm, and a non-adaptive $O(\log^2k)$-approximation algorithm. We also show that the adaptivity gap of this problem is between $e$ and $O(\log^2k)$.

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arXiv2016

Adaptivity Gaps for Stochastic Probing: Submodular and XOS Functions

Suppose we are given a submodular function $f$ over a set of elements, and we want to maximize its value subject to certain constraints. Good approximation algorithms are known for such problems under both monotone and non-monotone submodular functions. We consider these problems in a stochastic setting, where elements are not all active and we can only get value from active elements. Each element $e$ is active independently with some known probability $p_e$, but we don't know the element's status \emph{a priori}. We find it out only when we \emph{probe} the element $e$---probing reveals whether it's active or not, whereafter we can use this information to decide which other elements to probe. Eventually, if we have a probed set $S$ and a subset $\text{active}(S)$ of active elements in $S$, we can pick any $T \subseteq \text{active}(S)$ and get value $f(T)$. Moreover, the sequence of elements we probe must satisfy a given \emph{prefix-closed constraint}---e.g., these may be given by a matroid, or an orienteering constraint, or deadline, or precedence constraint, or an arbitrary downward-closed constraint---if we can probe some sequence of elements we can probe any prefix of it. What is a good strategy to probe elements to maximize the expected value? In this paper we study the gap between adaptive and non-adaptive strategies for $f$ being a submodular or a fractionally subadditive (XOS) function. If this gap is small, we can focus on finding good non-adaptive strategies instead, which are easier to find as well as to represent. We show that the adaptivity gap is a constant for monotone and non-monotone submodular functions, and logarithmic for XOS functions of small \emph{width}. These bounds are nearly tight. Our techniques show new ways of arguing about the optimal adaptive decision tree for stochastic problems.

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arXiv2015

Max-Cut under Graph Constraints

An instance of the graph-constrained max-cut (GCMC) problem consists of (i) an undirected graph G and (ii) edge-weights on a complete undirected graph on the same vertex set. The objective is to find a subset of vertices satisfying some graph-based constraint in G that maximizes the total weight of edges in the cut. The types of graph constraints we can handle include independent set, vertex cover, dominating set and connectivity. Our main results are for the case when G is a graph with bounded treewidth, where we obtain a 0.5-approximation algorithm. Our algorithm uses an LP relaxation based on the Sherali-Adams hierarchy. It can handle any graph constraint for which there is a (certain type of) dynamic program that exactly optimizes linear objectives. Using known decomposition results, these imply essentially the same approximation ratio for GCMC under constraints such as independent set, dominating set and connectivity on a planar graph G (more generally for bounded-genus or excluded-minor graphs).

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arXiv2015

Approximation Algorithms for Inventory Problems with Submodular or Routing Costs

We consider the following two deterministic inventory optimization problems over a finite planning horizon $T$ with non-stationary demands. (a) Submodular Joint Replenishment Problem: This involves multiple item types and a single retailer who faces demands. In each time step, any subset of item-types can be ordered incurring a joint ordering cost which is submodular. Moreover, items can be held in inventory while incurring a holding cost. The objective is find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs. (b) Inventory Routing Problem: This involves a single depot that stocks items, and multiple retailer locations facing demands. In each time step, any subset of locations can be visited using a vehicle originating from the depot. There is also cost incurred for holding items at any retailer. The objective here is to satisfy all demands while minimizing the sum of routing and holding costs. We present a unified approach that yields $\mathcal{O}\left(\frac{\log T}{\log\log T}\right)$-factor approximation algorithms for both problems when the holding costs are polynomial functions. A special case is the classic linear holding cost model, wherein this is the first sub-logarithmic approximation ratio for either problem.

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arXiv2014

Online Packing and Covering Framework with Convex Objectives

We consider online fractional covering problems with a convex objective, where the covering constraints arrive over time. Formally, we want to solve $\min\,\{f(x) \mid Ax\ge \mathbf{1},\, x\ge 0\},$ where the objective function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex, and the constraint matrix $A_{m\times n}$ is non-negative. The rows of $A$ arrive online over time, and we wish to maintain a feasible solution $x$ at all times while only increasing coordinates of $x$. We also consider "dual" packing problems of the form $\max\,\{c^\intercal y - g(μ) \mid A^\intercal y \le μ,\, y\ge 0\}$, where $g$ is a convex function. In the online setting, variables $y$ and columns of $A^\intercal$ arrive over time, and we wish to maintain a non-decreasing solution $(y,μ)$. We provide an online primal-dual framework for both classes of problems with competitive ratio depending on certain "monotonicity" and "smoothness" parameters of $f$; our results match or improve on guarantees for some special classes of functions $f$ considered previously. Using this fractional solver with problem-dependent randomized rounding procedures, we obtain competitive algorithms for the following problems: online covering LPs minimizing $\ell_p$-norms of arbitrary packing constraints, set cover with multiple cost functions, capacity constrained facility location, capacitated multicast problem, set cover with set requests, and profit maximization with non-separable production costs. Some of these results are new and others provide a unified view of previous results, with matching or slightly worse competitive ratios.

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arXiv2014

On the Adaptivity Gap of Stochastic Orienteering

The input to the stochastic orienteering problem consists of a budget $B$ and metric $(V,d)$ where each vertex $v$ has a job with deterministic reward and random processing time (drawn from a known distribution). The processing times are independent across vertices. The goal is to obtain a non-anticipatory policy to run jobs at different vertices, that maximizes expected reward, subject to the total distance traveled plus processing times being at most $B$. An adaptive policy is one that can choose the next vertex to visit based on observed random instantiations. Whereas, a non-adaptive policy is just given by a fixed ordering of vertices. The adaptivity gap is the worst-case ratio of the expected rewards of the optimal adaptive and non-adaptive policies. We prove an $Ω(\log\log B)^{1/2}$ lower bound on the adaptivity gap of stochastic orienteering. This provides a negative answer to the $O(1)$ adaptivity gap conjectured earlier, and comes close to the $O(\log\log B)$ upper bound. This result holds even on a line metric. We also show an $O(\log\log B)$ upper bound on the adaptivity gap for the correlated stochastic orienteering problem, where the reward of each job is random and possibly correlated to its processing time. Using this, we obtain an improved quasi-polynomial time approximation algorithm for correlated stochastic orienteering.

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arXiv2013

Minimum Latency Submodular Cover

We study the Minimum Latency Submodular Cover problem (MLSC), which consists of a metric $(V,d)$ with source $r\in V$ and $m$ monotone submodular functions $f_1, f_2, ..., f_m: 2^V \rightarrow [0,1]$. The goal is to find a path originating at $r$ that minimizes the total cover time of all functions. This generalizes well-studied problems, such as Submodular Ranking [AzarG11] and Group Steiner Tree [GKR00]. We give a polynomial time $O(\log \frac{1}{\eps} \cdot \log^{2+δ} |V|)$-approximation algorithm for MLSC, where $ε>0$ is the smallest non-zero marginal increase of any $\{f_i\}_{i=1}^m$ and $δ>0$ is any constant. We also consider the Latency Covering Steiner Tree problem (LCST), which is the special case of \mlsc where the $f_i$s are multi-coverage functions. This is a common generalization of the Latency Group Steiner Tree [GuptaNR10a,ChakrabartyS11] and Generalized Min-sum Set Cover [AzarGY09, BansalGK10] problems. We obtain an $O(\log^2|V|)$-approximation algorithm for LCST. Finally we study a natural stochastic extension of the Submodular Ranking problem, and obtain an adaptive algorithm with an $O(\log 1/ \eps)$ approximation ratio, which is best possible. This result also generalizes some previously studied stochastic optimization problems, such as Stochastic Set Cover [GoemansV06] and Shared Filter Evaluation [MunagalaSW07, LiuPRY08].

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arXiv2013

A Stochastic Probing Problem with Applications

We study a general stochastic probing problem defined on a universe V, where each element e in V is "active" independently with probability p_e. Elements have weights {w_e} and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the p_e values-- to determine whether or not an element e is active, our algorithm must probe e. If element e is probed and happens to be active, then e must irrevocably be added to the chosen set S; if e is not active then it is not included in S. Moreover, the following conditions must hold in every random instantiation: (1) the set Q of probed elements satisfy an "outer" packing constraint, and (2) the set S of chosen elements satisfy an "inner" packing constraint. The kinds of packing constraints we consider are intersections of matroids and knapsacks. Our results provide a simple and unified view of results in stochastic matching and Bayesian mechanism design, and can also handle more general constraints. As an application, we obtain the first polynomial-time $Ω(1/k)$-approximate "Sequential Posted Price Mechanism" under k-matroid intersection feasibility constraints.

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arXiv2013

Thrifty Algorithms for Multistage Robust Optimization

We consider a class of multi-stage robust covering problems, where additional information is revealed about the problem instance in each stage, but the cost of taking actions increases. The dilemma for the decision-maker is whether to wait for additional information and risk the inflation, or to take early actions to hedge against rising costs. We study the "k-robust" uncertainty model: in each stage i = 0, 1,...,T, the algorithm is shown some subset of size k_i that completely contains the eventual demands to be covered; here k_1 > k_2 >...> k_T which ensures increasing information over time. The goal is to minimize the cost incurred in the worst-case possible sequence of revelations. For the multistage k-robust set cover problem, we give an O(log m + log n)-approximation algorithm, nearly matching the Ω(log n + log m/loglog m) hardness of approximation even for T=2 stages. Moreover, our algorithm has a useful "thrifty" property: it takes actions on just two stages. We show similar thrifty algorithms for multi-stage k-robust Steiner tree, Steiner forest, and minimum-cut. For these problems our approximation guarantees are O(min{T, log n, log L_{max}), where L_{max} is the maximum inflation over all the stages. We conjecture that these problems also admit O(1)-approximate thrifty algorithms.

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