arXiv subjects

Nitish Korula

Publications and source records attributed to Nitish Korula.

13 recordsLinked to original sources

arXiv2017

Online Submodular Welfare Maximization: Greedy Beats 1/2 in Random Order

In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of $n$ items, each of which must be allocated to one of $m$ agents. Each agent $\ell$ has a valuation function $v_\ell$, where $v_\ell(S)$ denotes the welfare obtained by this agent if she receives the set of items $S$. The functions $v_\ell$ are all submodular; as is standard, we assume that they are monotone and $v_\ell(\emptyset) = 0$. The goal is to partition the items into $m$ disjoint subsets $S_1, S_2, \ldots S_m$ in order to maximize the social welfare, defined as $\sum_{\ell = 1}^m v_\ell(S_\ell)$. In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they receive. In the random order model, the adversary can construct a worst-case set of items and valuations, but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. Obtaining a competitive ratio of $1/2 + Ω(1)$ for the random order model has been an important open problem for several years. We solve this open problem by demonstrating that the greedy algorithm has a competitive ratio of at least $0.505$ for the Online Submodular Welfare Maximization problem in the random order model. For special cases of submodular functions including weighted matching, weighted coverage functions and a broader class of "second-order supermodular" functions, we provide a different analysis that gives a competitive ratio of $0.51$.

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arXiv2017

Online Allocation with Traffic Spikes: Mixing Adversarial and Stochastic Models

Motivated by Internet advertising applications, online allocation problems have been studied extensively in various adversarial and stochastic models. While the adversarial arrival models are too pessimistic, many of the stochastic (such as i.i.d or random-order) arrival models do not realistically capture uncertainty in predictions. A significant cause for such uncertainty is the presence of unpredictable traffic spikes, often due to breaking news or similar events. To address this issue, a simultaneous approximation framework has been proposed to develop algorithms that work well both in the adversarial and stochastic models; however, this framework does not enable algorithms that make good use of partially accurate forecasts when making online decisions. In this paper, we propose a robust online stochastic model that captures the nature of traffic spikes in online advertising. In our model, in addition to the stochastic input for which we have good forecasting, an unknown number of impressions arrive that are adversarially chosen. We design algorithms that combine a stochastic algorithm with an online algorithm that adaptively reacts to inaccurate predictions. We provide provable bounds for our new algorithms in this framework. We accompany our positive results with a set of hardness results showing that our algorithms are not far from optimal in this framework. As a byproduct of our results, we also present improved online algorithms for a slight variant of the simultaneous approximation framework.

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arXiv2013

An efficient reconciliation algorithm for social networks

People today typically use multiple online social networks (Facebook, Twitter, Google+, LinkedIn, etc.). Each online network represents a subset of their "real" ego-networks. An interesting and challenging problem is to reconcile these online networks, that is, to identify all the accounts belonging to the same individual. Besides providing a richer understanding of social dynamics, the problem has a number of practical applications. At first sight, this problem appears algorithmically challenging. Fortunately, a small fraction of individuals explicitly link their accounts across multiple networks; our work leverages these connections to identify a very large fraction of the network. Our main contributions are to mathematically formalize the problem for the first time, and to design a simple, local, and efficient parallel algorithm to solve it. We are able to prove strong theoretical guarantees on the algorithm's performance on well-established network models (Random Graphs, Preferential Attachment). We also experimentally confirm the effectiveness of the algorithm on synthetic and real social network data sets.

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arXiv2013

Connected Domatic Packings in Node-capacitated Graphs

A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. A dominating set is connected if the subgraph induced by its vertices is connected. The connected domatic partition problem asks for a partition of the nodes into connected dominating sets. The connected domatic number of a graph is the size of a largest connected domatic partition and it is a well-studied graph parameter with applications in the design of wireless networks. In this note, we consider the fractional counterpart of the connected domatic partition problem in \emph{node-capacitated} graphs. Let $n$ be the number of nodes in the graph and let $k$ be the minimum capacity of a node separator in $G$. Fractionally we can pack at most $k$ connected dominating sets subject to the capacities on the nodes, and our algorithms construct packings whose sizes are proportional to $k$. Some of our main contributions are the following: \begin{itemize} \item An algorithm for constructing a fractional connected domatic packing of size $Ω(k)$ for node-capacitated planar and minor-closed families of graphs. \item An algorithm for constructing a fractional connected domatic packing of size $Ω(k / \ln{n})$ for node-capacitated general graphs. \end{itemize}

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arXiv2010

Approximability of Capacitated Network Design

In the {\em capacitated} survivable network design problem (Cap-SNDP), we are given an undirected multi-graph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimum-cut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of Cap-SNDP is not well understood; even in very restricted settings no known algorithm achieves a $o(m)$ approximation, where $m$ is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of Cap-SNDP.

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arXiv2010

Overlap Number of Graphs

An {\it overlap representation} of a graph $G$ assigns sets to vertices so that vertices are adjacent if and only if their assigned sets intersect with neither containing the other. The {\it overlap number} $\ol(G)$ (introduced by Rosgen) is the minimum size of the union of the sets in such a representation. We prove the following: (1) An optimal overlap representation of a tree can be produced in linear time, and its size is the number of vertices in the largest subtree in which the neighbor of any leaf has degree 2. (2) If $δ(G)\ge 2$ and $G\ne K_3$, then $\ol(G)\le |E(G)|-1$, with equality when $G$ is connected and triangle-free and has no star-cutset. (3) If $G$ is an $n$-vertex plane graph with $n\ge5$, then $\ol(G)\le 2n-5$, with equality when every face has length 4 and there is no star-cutset. (4) If $G$ is an $n$-vertex graph with $n\ge 14$, then $\ol(G)\le \floor{n^2/4-n/2-1}$, and this is sharp (for even $n$, equality holds when $G$ arises from $K_{n/2,n/2}$ by deleting a perfect matching).

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arXiv2010

Prize-Collecting Steiner Tree and Forest in Planar Graphs

We obtain polynomial-time approximation-preserving reductions (up to a factor of 1 + ε) from the prize-collecting Steiner tree and prize-collecting Steiner forest problems in planar graphs to the corresponding problems in graphs of bounded treewidth. We also give an exact algorithm for the prize-collecting Steiner tree problem that runs in polynomial time for graphs of bounded treewidth. This, combined with our reductions, yields a PTAS for the prize-collecting Steiner tree problem in planar graphs and generalizes the PTAS of Borradaile, Klein and Mathieu for the Steiner tree problem in planar graphs. Our results build upon the ideas of Borradaile, Klein and Mathieu and the work of Bateni, Hajiaghayi and Marx on a PTAS for the Steiner forest problem in planar graphs. Our main technical result is on the properties of primal-dual algorithms for Steiner tree and forest problems in general graphs when they are run with scaled up penalties.

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arXiv2010

On k-Column Sparse Packing Programs

We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+o(k))-approximation algorithm for k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and $O(k^2)$. We also show that the integrality gap of our linear programming relaxation is at least 2k-1; it is known that k-column sparse PIPs are $Ω(k/ \log k)$-hard to approximate. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over k-column sparse packing constraints.

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arXiv2010

Online Stochastic Packing Applied to Display Ad Allocation

Inspired by online ad allocation, we study online stochastic packing linear programs from theoretical and practical standpoints. We first present a near-optimal online algorithm for a general class of packing linear programs which model various online resource allocation problems including online variants of routing, ad allocations, generalized assignment, and combinatorial auctions. As our main theoretical result, we prove that a simple primal-dual training-based algorithm achieves a (1 - o(1))-approximation guarantee in the random order stochastic model. This is a significant improvement over logarithmic or constant-factor approximations for the adversarial variants of the same problems (e.g. factor 1 - 1/e for online ad allocation, and \log m for online routing). We then focus on the online display ad allocation problem and study the efficiency and fairness of various training-based and online allocation algorithms on data sets collected from real-life display ad allocation system. Our experimental evaluation confirms the effectiveness of training-based primal-dual algorithms on real data sets, and also indicate an intrinsic trade-off between fairness and efficiency.

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arXiv2009

A Graph Reduction Step Preserving Element-Connectivity and Applications

Given an undirected graph G=(V,E) and subset of terminals T \subseteq V, the element-connectivity of two terminals u,v \in T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V \setminus T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellermann gave a graph reduction step that preserves the global element-connectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise element-connectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems: 1. Given a graph G and disjoint terminal sets T_1, T_2, ..., T_m, we seek a maximum number of element-disjoint Steiner forests where each forest connects each T_i. We prove that if each T_i is k-element-connected then there exist Ω(\frac{k}{\log h \log m}) element-disjoint Steiner forests, where h = |\bigcup_i T_i|. If G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving poly-time algorithms to find these forests; these are the first non-trivial algorithms for packing element-disjoint Steiner Forests. 2. We give a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna in the context of the single-sink k-vertex-connectivity problem; this yields a simple and alternative analysis of an O(k \log n) approximation. Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future.

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arXiv2008

Algorithms for Secretary Problems on Graphs and Hypergraphs

We examine several online matching problems, with applications to Internet advertising reservation systems. Consider an edge-weighted bipartite graph G, with partite sets L, R. We develop an 8-competitive algorithm for the following secretary problem: Initially given R, and the size of L, the algorithm receives the vertices of L sequentially, in a random order. When a vertex l \in L is seen, all edges incident to l are revealed, together with their weights. The algorithm must immediately either match l to an available vertex of R, or decide that l will remain unmatched. Dimitrov and Plaxton show a 16-competitive algorithm for the transversal matroid secretary problem, which is the special case with weights on vertices, not edges. (Equivalently, one may assume that for each l \in L, the weights on all edges incident to l are identical.) We use a similar algorithm, but simplify and improve the analysis to obtain a better competitive ratio for the more general problem. Perhaps of more interest is the fact that our analysis is easily extended to obtain competitive algorithms for similar problems, such as to find disjoint sets of edges in hypergraphs where edges arrive online. We also introduce secretary problems with adversarially chosen groups. Finally, we give a 2e-competitive algorithm for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph.

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arXiv2008

Min-Cost 2-Connected Subgraphs With k Terminals

In the k-2VC problem, we are given an undirected graph G with edge costs and an integer k; the goal is to find a minimum-cost 2-vertex-connected subgraph of G containing at least k vertices. A slightly more general version is obtained if the input also specifies a subset S \subseteq V of terminals and the goal is to find a subgraph containing at least k terminals. Closely related to the k-2VC problem, and in fact a special case of it, is the k-2EC problem, in which the goal is to find a minimum-cost 2-edge-connected subgraph containing k vertices. The k-2EC problem was introduced by Lau et al., who also gave a poly-logarithmic approximation for it. No previous approximation algorithm was known for the more general k-2VC problem. We describe an O(\log n \log k) approximation for the k-2VC problem.

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arXiv2007

Approximation Algorithms for Orienteering with Time Windows

Orienteering is the following optimization problem: given an edge-weighted graph (directed or undirected), two nodes s,t and a time limit T, find an s-t walk of total length at most T that maximizes the number of distinct nodes visited by the walk. One obtains a generalization, namely orienteering with time-windows (also referred to as TSP with time-windows), if each node v has a specified time-window [R(v), D(v)] and a node v is counted as visited by the walk only if v is visited during its time-window. For the time-window problem, an O(\log \opt) approximation can be achieved even for directed graphs if the algorithm is allowed quasi-polynomial time. However, the best known polynomial time approximation ratios are O(\log^2 \opt) for undirected graphs and O(\log^4 \opt) in directed graphs. In this paper we make some progress towards closing this discrepancy, and in the process obtain improved approximation ratios in several natural settings. Let L(v) = D(v) - R(v) denote the length of the time-window for v and let \lmax = \max_v L(v) and \lmin = \min_v L(v). Our results are given below with αdenoting the known approximation ratio for orienteering (without time-windows). Currently α= (2+\eps) for undirected graphs and α= O(\log^2 \opt) in directed graphs. 1. An O(α\log \lmax) approximation when R(v) and D(v) are integer valued for each v. 2. An O(α\max{\log \opt, \log \frac{\lmax}{\lmin}}) approximation. 3. An O(α\log \frac{\lmax}{\lmin}) approximation when no start and end points are specified. In particular, if \frac{\lmax}{\lmin} is poly-bounded, we obtain an O(\log n) approximation for the time-window problem in undirected graphs.

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