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 $\sum_{\ell = 1}^m v_\ell(S_\ell)$. A simple greedy algorithm gives a $1/2$-approximation to SWM in the offline setting, and this was the best known until Vondrák's recent $(1-1/e)$-approximation algorithm [ Optimal approximation for the submodular welfare problem in the value oracle model, in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing (STOC '08), ACM, 2008, pp. 67--74]. 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. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary/worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm's performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: 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. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is $1/2 + 1/n$, which is barely better than the ratio for the adversarial order [S. Dobzinski, N. Nisan, and M. Schapira, Math. Oper. Res., 35 (2010), pp. 1--13; S. Dobzinski and M. Schapira, An improved approximation algorithm for combinatorial auctions with submodular bidders, in Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '06), ACM; SIAM, 2006, pp. 1064--1073]. Obtaining a competitive ratio of $1/2 + \Omega(1)$ 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 online SWM in the random order model. This is the first result showing a competitive ratio bounded above $1/2$ in the random order model, even for special cases such as the weighted matching or budgeted allocation problem (without the so-called large capacity assumptions). 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. We analyze the greedy algorithm using a factor-revealing linear program, bounding how the assignment of one item can decrease potential welfare from assigning future items. In addition to our new competitive ratios for online SWM, we make two further contributions: First, we define the classes of second-order modular, supermodular, and submodular functions, which are likely to be of independent interest in submodular optimization. Second, we obtain an improved competitive ratio via a technique we refer to as gain linearizing, which may be useful in other contexts (see [V. S. Mirrokni and M. Zadimoghaddam, Randomized composable core-sets for distributed submodular maximization, in Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing (STOC '15), ACM, 2015, pp. 153--162]): Essentially, we linearize the submodular function by dividing the gain of an optimal solution into gain from individual elements, compare the algorithm's gain when it assigns an element to the optimal solution's gain from the element, and, crucially, bound the extent to which assigning elements can affect the potential gain of other elements.

Online allocation problems have been widely studied due to their numerous practical applications (particularly to Internet advertising), as well as considerable theoretical interest. The main challenge in such problems is making assignment decisions in the face of uncertainty about future input; effective algorithms need to predict which constraints are most likely to bind, and learn the balance between short-term gain and the value of long-term resource availability. In many important applications, the algorithm designer is faced with multiple objectives to optimize. In particular, in online advertising it is fairly common to optimize multiple metrics, such as clicks, conversions, and impressions, as well as other metrics which may be largely uncorrelated such as ‘share of voice’, and ‘buyer surplus’. While there has been considerable work on multi-objective offline optimization (when the entire input is known in advance), very little is known about the online case, particularly in the case of adversarial input. In this paper, we give the first results for bi-objective online submodular optimization, providing almost matching upper and lower bounds for allocating items to agents with two submodular value functions. We also study practically relevant special cases of this problem related to Internet advertising, and obtain improved results. All our algorithms are nearly best possible, as well as being efficient and easy to implement in practice.

Linking accounts of the same user across datasets -- even when personally identifying information is removed or unavailable -- is an important open problem studied in many contexts. Beyond many practical applications, (such as cross domain analysis, recommendation, and link prediction), understanding this problem more generally informs us on the privacy implications of data disclosure. Previous work has typically addressed this question using either different portions of the same dataset or observing the same behavior across thematically similar domains. In contrast, the general cross-domain case where users have different profiles independently generated from a common but unknown pattern raises new challenges, including difficulties in validation, and remains under-explored.

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 an 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 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.

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 l has a valuation function vl, where vl(S) denotes the welfare obtained by this agent if she receives the set of items S. The functions vl are all submodular; as is standard, we assume that they are monotone and vl(∅) = 0. The goal is to partition the items into m disjoint subsets S1, S2, ... Sm in order to maximize the social welfare, defined as ∑l = 1m vl(Sl). A simple greedy algorithm gives a 1/2-approximation to SWM in the offline setting, and this was the best known until Vondrak's recent (1-1/e)-approximation algorithm [34]. 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. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary / worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm's performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: 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. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is 1/2 + 1/n [9,10], barely better than the ratio for the adversarial 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 online SWM in the random order model. This is the first result showing a competitive ratio bounded above 1/2 in the random order model, even for special cases such as the weighted matching or budgeted allocation problems (without the so-called 'large capacity' assumptions). 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. We analyze the greedy algorithm using a factor-revealing linear program, bounding how the assignment of one item can decrease potential welfare from assigning future items. We also formulate a natural conjecture which, if true, would improve the competitive ratio of the greedy algorithm to at least 0.567.

The impact of blogs and microblogging on the consumption of news is dramatic, as every day users rely more on these sources to decide what content to pay attention to. In this work, we empirically and theoretically analyze the dynamics of bloggers serving as intermediaries between the mass media and the general public.

Display ads on the Internet are often sold by publishers to advertisers in bundles of thousands or millions of impressions over a particular time period. The ad delivery systems assign ads to pages on behalf of publishers to satisfy these contracts, and at the same time, try to maximize the overall quality of assignment. This is usually modeled in the literature as an online allocation problem, where contracts are represented by overall delivery constraints.

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.

Given an undirected graph $G=(V,E)$ and a subset of vertices called terminals $T \subseteq V$, the element-connectivity $\elconn_G(u,v)$ of two terminals $u,v \in T$ is the maximum number of $u$-$v$ paths that are pairwise element-disjoint, that is, disjoint in both edges and nonterminals $V \setminus T$. (Element-connectivity was first (implicitly) defined by Frank, Ibaraki, and Nagamochi in [J. Graph Theory, 17 (1993), pp. 275--281].) (Element-disjoint paths need not be disjoint in terminals.) Hind and Oellermann [\em Congr. Numer.}, 113 (1996), pp. 179--204] gave a graph reduction step that preserves the global element-connectivity of the terminals. We show that one can also apply such a reduction step while preserving local connectivity, that is, all the pairwise element-connectivities of the terminals. We illustrate the usefulness of this more general reduction step by giving applications to packing element-disjoint Steiner trees and forests: Given a graph $G$ and disjoint terminal sets $T_1, T_2, \ldots, T_h$, 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 $\Omega(\frac{k}{\log |T| \log h})$ element-disjoint Steiner forests, where $T = \bigcup_i T_i$. If $G$ is planar (or has fixed genus), we show that there exist $\Omega(k)$ Steiner forests. Our proofs are constructive, giving poly-time algorithms to find these forests; these are the first nontrivial algorithms for packing element-disjoint Steiner forests.

In the context of online ad serving, display ads may appear on different types of web-pages, where each page includes several ad slots and therefore multiple ads can be shown on each page. The set of ads that can be assigned to ad slots of the same page needs to satisfy various pre-specified constraints including exclusion constraints, diversity constraints, and the like. Upon arrival of a user, the ad serving system needs to allocate a set of ads to the current web-page respecting these per-page allocation constraints. Previous slot-based settings ignore the important concept of a page, and may lead to highly suboptimal results in general. In this paper, motivated by these applications in display advertising and inspired by the submodular welfare maximization problem with online bidders, we study a general class of page-based ad allocation problems, present the first (tight) constant-factor approximation algorithms for these problems, and confirm the performance of our algorithms experimentally on real-world data sets.

In this article, we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to point-orienteering problem is the following: Given an edge-weighted graph G =( V, E ) (directed or undirected), two nodes s, t ∈ V and a time limit B , find an s - t walk in G of total length at most B that maximizes the number of distinct nodes visited by the walk. This problem is closely related to tour problems such as TSP as well as network design problems such as k -MST. Orienteering with time-windows is the more general problem in which 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. We design new and improved algorithms for the orienteering problem and orienteering with time-windows. Our main results are the following: — A (2+ϵ) approximation for orienteering in undirected graphs, improving upon the 3-approximation of Bansal et al. [2004]. — An O (log 2 OPT) approximation for orienteering in directed graphs, where OPT ≤ n is the number of vertices visited by an optimal solution. Previously, only a quasipolynomial-time algorithm due to Chekuri and Pál [2005] achieved a polylogarithmic approximation (a ratio of O (log OPT)). — Given an α approximation for orienteering, we show an O (α ċ max{log OPT, log l max / l min }) approximation for orienteering with time-windows, where l max and l min are the lengths of the longest and shortest time-windows respectively.

In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP), Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST), Prize-Collecting Steiner Forest (PCSF), and more generally Submodular Prize-Collecting Steiner Forest (SPCSF), on planar graphs (and also on bounded-genus graphs) to the corresponding problem on graphs of bounded treewidth. More precisely, for each of the mentioned problems, an α-approximation algorithm for the problem on graphs of bounded treewidth implies an (α + ε)-approximation algorithm for the problem on planar graphs (and also bounded-genus graphs), for any constant ε > 0. PCS, PCTSP, and PCST can be solved exactly on graphs of bounded treewidth and hence we obtain a PTAS for these problems on planar graphs and bounded-genus graphs. In contrast, we show that PCSF is APX-hard to approximate on series-parallel graphs, which are planar graphs of treewidth at most 2. Apart from ruling out a PTAS for PCSF on planar graphs and bounded treewidth graphs, this result is also interesting since it gives the first provable hardness separation between the approximability of a problem and its prize-collecting version. We also show that PCSF is APX-hard on Euclidean instances.

Given an edge-weighted undirected graph $G$ with a specified set of terminals, let the \emph{density} of any subgraph be the ratio of its weight/cost to the number of terminals it contains. If $G$ is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of $G$? We answer this question in the affirmative by giving an algorithm to \emph{prune} $G$ and find such subgraphs of any desired size, at the cost of only a logarithmic increase in density (plus a small additive factor). We apply the pruning techniques to give algorithms for two NP-Hard problems on finding large 2-vertex-connected subgraphs of low cost; no previous approximation algorithm was known for either problem. In the \kv 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. In the \bv\ problem, we are given the graph $G$ with edge costs, and a budget $B$; the goal is to find a 2-vertex-connected subgraph $H$ of $G$ with total edge cost at most $B$ that maximizes the number of vertices in $H$. We describe an $O(\log n \log k)$ approximation for the \kv problem, and a bicriteria approximation for the \bv\ problem that gives an $O(\frac{1}{\eps}\log^2 n)$ approximation, while violating the budget by a factor of at most $3+\eps$.

We study single-sink network design problems in undirected graphs with vertex connectivity requirements. The input to these problems is an edge-weighted undirected graph $G=(V,E)$, a sink/root vertex $r$, a set of terminals $T \subseteq V$, and integer $k$. The goal is to connect each terminal $t \in T$ to $r$ via $k$ \emph{vertex-disjoint} paths. In the {\em connectivity} problem, the objective is to find a min-cost subgraph of $G$ that contains the desired paths. There is a $2$-approximation for this problem when $k \le 2$ \cite{FleischerJW} but for $k \ge 3$, the first non-trivial approximation was obtained in the recent work of Chakraborty, Chuzhoy and Khanna \cite{ChakCK08}; they describe and analyze an algorithm with an approximation ratio of $O(k^{O(k^2)}\log^4 n)$ where $n=|V|$. In this paper, inspired by the results and ideas in \cite{ChakCK08}, we show an $O(k^{O(k)}\log |T|)$-approximation bound for a simple greedy algorithm. Our analysis is based on the dual of a natural linear program and is of independent technical interest. We use the insights from this analysis to obtain an $O(k^{O(k)}\log |T|)$-approximation for the more general single-sink {\em rent-or-buy} network design problem with vertex connectivity requirements. We further extend the ideas to obtain a poly-logarithmic approximation for the single-sink {\em buy-at-bulk} problem when $k=2$ and the number of cable-types is a fixed constant; we believe that this should extend to any fixed $k$. We also show that for the non-uniform buy-at-bulk problem, for each fixed $k$, a small variant of a simple algorithm suggested by Charikar and Kargiazova \cite{CharikarK05} for the case of $k=1$ gives an $2^{O(\sqrt{\log |T|})}$ approximation for larger $k$. These results show that for each of these problems, simple and natural algorithms that have been developed for $k=1$ have good performance for small $k > 1$.