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.

We show that the multiplicative weight update method provides a simple recipe for designing and analyzing optimal Bayesian Incentive Compatible (BIC) auctions, and reduces the time complexity of the problem to pseudo-polynomial in parameters that depend on single agent instead of depending on the size of the joint type space. We use this framework to design computationally efficient optimal auctions that satisfy ex-post Individual Rationality in the presence of constraints such as (hard, private) budgets and envy-freeness. We also design optimal auctions when buyers and a seller's utility functions are non-linear. Scenarios with such functions include (a) auctions with "quitting rights", (b) cost to borrow money beyond budget, (c) a seller's and buyers' risk aversion. Finally, we show how our framework also yields optimal auctions for variety of auction settings considered in Alaei et al and Cai et al, albeit with pseudo-polynomial running times.

Display ads on the Internet are often sold in bundles of thousands or millions of impressions over a particular time period, typically weeks or months. Ad serving systems that assign ads to pages on behalf of publishers must satisfy these contracts, but at the same time try to maximize overall quality of placement. This is usually modeled in the literature as an online allocation problem, where contracts are represented by overall delivery constraints over a finite time horizon. However this model misses an important aspect of ad delivery: time homogeneity. Advertisers who buy these packages expect their ad to be shown smoothly throughout the purchased time period, in order to reach a wider audience, to have a sustained impact, and to support the ads they are running on other media (e.g., television). In this paper we formalize this problem using several nested packing constraints, and develop a tight (1-1/e)-competitive online algorithm for this problem. Our algorithms and analysis require novel techniques as they involve online computation of multiple dual variables per ad. We then show the effectiveness of our algorithms through exhaustive simulation studies on real data sets.

With the advent of social networks such as Facebook and LinkedIn, and online offers/deals web sites, network externalties raise the possibility of marketing and advertising to users based on influence they derive from their neighbors in such networks. Indeed, a user's knowledge of which of his neighbors "liked" the product, changes his valuation for the product. Much of the work on the mechanism design under network externalities has addressed the setting when there is only one product. We consider a more natural setting when there are multiple competing products, and each node in the network is a unit-demand agent.

In the stochastic knapsack problem, we are given a set of items each associated with a probability distribution on sizes and a profit, and a knapsack of unit capacity. The size of an item is revealed as soon as it is inserted into the knapsack, and the goal is to design a policy that maximizes the expected profit of items that are successfully inserted into the knapsack. The stochastic knapsack problem is a natural generalization of the classical knapsack problem, and arises in many applications, including bandwidth allocation, budgeted learning, and scheduling. An adaptive policy for stochastic knapsack specifies the next item to be inserted based on observed sizes of the items inserted thus far. The adaptive policy can have an exponentially large explicit description and is known to be PSPACE-hard to compute. The best known approximation for this problem is a (3 + ∊)-approximation for any ∊ > 0. Our first main result is a relaxed PTAS (Polynomial Time Approximation Scheme) for the adaptive policy, that is, for any ∊ > 0, we present a poly-time computable (1 + ∊)-approximate adaptive policy when knapsack capacity is relaxed to 1+e. At a high-level, the proof is based on transforming an arbitrary collection of item size distributions to canonical item size distributions that admit a compact description. We then establish a coupling that shows a (1 + ∊)-approximation can be achieved for the original problem by a canonical policy that makes decisions at each step by observing events drawn from the sample space of canonical size distributions. Finally, we give a mechanism for approximating the optimal canonical policy. Our second main result is an (8/3 + ∊)-approximate adaptive policy for any ∊ > 0 without relaxing the knapsack capacity, improving the earlier (3 + ∊)-approximation result. Interestingly, we obtain this result by using the PTAS described above. We establish an existential result that the optimal policy for the knapsack with capacity 1 can be folded to get a policy with expected profit 3OPT/8 for a knapsack with capacity (1 − ∊), with capacity relaxed to 1 only for the first item inserted. We then use our PTAS result to compute the (1 + ∊)-approximation to such policy. Our techniques also yield a relaxed PTAS for non-adaptive policies. Finally, we also show that our ideas can be extended to yield improved approximation guarantees for multi-dimensional and fixed set variants of the stochastic knapsack problem.

Cut games and party affiliation games are well-known classes of potential games. Schaffer and Yannakakis showed that computing pure Nash equilibrium in these games is PLS-complete. In general potential games, even the problem of computing any finite approximation to a pure equilibrium is also PLS-complete. We show that for any ∈ > 0, we design an algorithm to compute in polynomial time a (3+∈)-approximate pure Nash equilibrium for cut and party affiliation games. Prior to our work, only a trivial polynomial factor approximation was known for these games. Our approach extends beyond cut and party affiliation games to a more general class of satisfiability games.

We consider an unweighted undirected graph with n vertices, m edges, and edge-connectivity 2k. The weak edge orientation problem requires that the edges of this graph be oriented so the resulting directed graph is at least k edge-connected. Nash-Williams proved the existence of such orientations and subsequently Frank [6], Gabow [7], and Nagamochi-Ibaraki [12] gave algorithmic constructions. All of these algorithms took time at least quadratic in n. We provide the first sub-quadratic (in n) algorithm for this problem. Our algorithm takes Õ(nk4 + m) time. This improves the previous best bounds of Õ(n2k2 +m) by Gabow [7] and Õ(n2m) by Nagamochi-Ibaraki [12] when k ≤ √n. Indeed, many real networks have k ≪ n. Our algorithm uses the fast edge splitting paradigm introduced by Bhalgat et al. [2]. We seek to split out a large fraction of the vertices, recurse on the resulting graph, and then put back the split-off vertices. The main challenge we face is that only vertices with even degree may be split-off in an undirected graph and there may not be any such vertex in the current graph. The edge orientation algorithms of Gabow and Nagamochi-Ibaraki as well as Frank's proof are based on showing the existence of at least two even degree vertices (in fact, vertices with degree 2k) in a 2k minimally connected graph. We generalize this to show that in any edge minimal 2k edge-connected graph, there are at least n/3 even degree vertices. These vertices are then split-off. Our next challenge is to drop edges from the given graph so it remains 2k connected and yet has Ω(n) even degree vertices. We provide an algorithm that discards edges specifically to produce Ω(n) even degree vertices while maintaining connectivity 2k and takes time O(nk 4 +m). Note that this algorithm does not necessarily make the graph edge-minimally 2k edge-connected. We also briefly outline an Õ(nk5 + m) time algorithm that achieves edge-minimality which improves the previous best bound of O(m + n2k2) by Gabow [7].

We present a fast algorithm for computing a Gomory-Hu tree or cut tree for an unweighted undirected graph G = (V,E). The expected running time of our algorithm is Õ(mc) where |E| = m and c is the maximum u-vedge connectivity, where u,v ∈ V. When the input graph is also simple (i.e., it has no parallel edges), then the u-v edge connectivity for each pair of vertices u and v is at most n-1; so the expected running time of our algorithm for simple unweighted graphs is Õ(mn).