Stochastic Budget Optimization in Internet Advertising

Internet advertising is a sophisticated game in which the many advertisers "play" to optimize their return on investment. There are many "targets" for the advertisements, and each "target" has a collection of games with a potentially different set of players involved. In this paper, we study the problem of how advertisers allocate their budget across these "targets". In particular, we focus on formulating their best response strategy as an optimization problem. Advertisers have a set of keywords ("targets") and some stochastic information about the future, namely a probability distribution over scenarios of cost vs click combinations. This summarizes the potential states of the world assuming that the strategies of other players are fixed. Then, the best response can be abstracted as stochastic budget optimization problems to figure out how to spread a given budget across these keywords to maximize the expected number of clicks. We present the first known non-trivial poly-logarithmic approximation for these problems as well as the first known hardness results of getting better than logarithmic approximation ratios in the various parameters involved. We also identify several special cases of these problems of practical interest, such as with fixed number of scenarios or with polynomial-sized parameters related to cost, which are solvable either in polynomial time or with improved approximation ratios. Stochastic budget optimization with scenarios has sophisticated technical structure. Our approximation and hardness results come from relating these problems to a special type of (0/1, bipartite) quadratic programs inherent in them. Our research answers some open problems raised by the authors in (Stochastic Models for Budget Optimization in Search-Based Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio

Uniqueness of radial solutions for the fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian $(-\Delta)^s$ with $s \in (0,1)$ for any space dimensions $N \geq 1$. By extending a monotonicity formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear equation $(-\Delta)^s u+ Vu = 0$ in $\mathbb{R}^N$ has at most one radial and bounded solution vanishing at infinity, provided that the potential $V$ is a radial and non-decreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schr\"odinger operator $H=(-\Delta)^s + V$ are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space $\mathbb{R}^{N+1}_+$, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation $(-\Delta)^s Q + Q - |Q|^{\alpha} Q = 0$ in $\mathbb{R}^N$ for arbitrary space dimensions $N \geq 1$ and all admissible exponents $\alpha >0$. This generalizes the nondegeneracy and uniqueness result for dimension N=1 recently obtained by the first two authors in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.Comment: 38 pages; revised version; various typos corrected; proof of Lemma 8.1 corrected; discussion of case \kappa_* =1 in the proof of Theorem 2 corrected with new Lemma A.2; accepted for publication in Comm. Pure. Appl. Mat

Stochastic 2-microlocal analysis

A lot is known about the H\"older regularity of stochastic processes, in particular in the case of Gaussian processes. Recently, a finer analysis of the local regularity of functions, termed 2-microlocal analysis, has been introduced in a deterministic frame: through the computation of the so-called 2-microlocal frontier, it allows in particular to predict the evolution of regularity under the action of (pseudo-) differential operators. In this work, we develop a 2-microlocal analysis for the study of certain stochastic processes. We show that moments of the increments allow, under fairly general conditions, to obtain almost sure lower bounds for the 2-microlocal frontier. In the case of Gaussian processes, more precise results may be obtained: the incremental covariance yields the almost sure value of the 2-microlocal frontier. As an application, we obtain new and refined regularity properties of fractional Brownian motion, multifractional Brownian motion, stochastic generalized Weierstrass functions, Wiener and stable integrals.Comment: 35 page

Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank $r$. Our main result is a deterministic algorithm to generate a matching which is an $O(r)$-approximation to the maximum weight matching, running in $\tilde O(r \log \Delta + \log^2 \Delta + \log^* n)$ rounds. (Here, the $\tilde O()$ notations hides $\text{polyloglog } \Delta$ and $\text{polylog } r$ factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). As a main application, we obtain nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a $(1+\epsilon)$ approximation algorithm using $\tilde O(\log \Delta / \epsilon^3 + \text{polylog}(1/\epsilon, \log \log n))$ randomized time and $\tilde O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon)$ deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for $(2 \Delta - 1)$-edge-list coloring in $\tilde O(\log^2 \Delta \log n)$ rounds deterministically or $\tilde O( (\log \log n)^3 )$ rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most $\lceil (1+\epsilon) \lambda \rceil$ for a graph of arboricity $\lambda$; for fixed $\epsilon$ this runs in $\tilde O(\log^6 n)$ rounds deterministically or $\tilde O(\log^3 n )$ rounds randomly

An Algorithmic Theory of Integer Programming

We study the general integer programming problem where the number of variables $n$ is a variable part of the input. We consider two natural parameters of the constraint matrix $A$: its numeric measure $a$ and its sparsity measure $d$. We show that integer programming can be solved in time $g(a,d)\textrm{poly}(n,L)$, where $g$ is some computable function of the parameters $a$ and $d$, and $L$ is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by $a$ and $d$, and is solvable in polynomial time for every fixed $a$ and $d$. Our results also extend to nonlinear separable convex objective functions. Moreover, for linear objectives, we derive a strongly-polynomial algorithm, that is, with running time $g(a,d)\textrm{poly}(n)$, independent of the rest of the input data. We obtain these results by developing an algorithmic framework based on the idea of iterative augmentation: starting from an initial feasible solution, we show how to quickly find augmenting steps which rapidly converge to an optimum. A central notion in this framework is the Graver basis of the matrix $A$, which constitutes a set of fundamental augmenting steps. The iterative augmentation idea is then enhanced via the use of other techniques such as new and improved bounds on the Graver basis, rapid solution of integer programs with bounded variables, proximity theorems and a new proximity-scaling algorithm, the notion of a reduced objective function, and others. As a consequence of our work, we advance the state of the art of solving block-structured integer programs. In particular, we develop near-linear time algorithms for $n$-fold, tree-fold, and $2$-stage stochastic integer programs. We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified proximity-scaling algorith

Better Approximation Bounds for the Joint Replenishment Problem

The Joint Replenishment Problem (JRP) deals with optimizing shipments of goods from a supplier to retailers through a shared warehouse. Each shipment involves transporting goods from the supplier to the warehouse, at a fixed cost C, followed by a redistribution of these goods from the warehouse to the retailers that ordered them, where transporting goods to a retailer $\rho$ has a fixed cost $c_\rho$. In addition, retailers incur waiting costs for each order. The objective is to minimize the overall cost of satisfying all orders, namely the sum of all shipping and waiting costs. JRP has been well studied in Operations Research and, more recently, in the area of approximation algorithms. For arbitrary waiting cost functions, the best known approximation ratio is 1.8. This ratio can be reduced to 1.574 for the JRP-D model, where there is no cost for waiting but orders have deadlines. As for hardness results, it is known that the problem is APX-hard and that the natural linear program for JRP has integrality gap at least 1.245. Both results hold even for JRP-D. In the online scenario, the best lower and upper bounds on the competitive ratio are 2.64 and 3, respectively. The lower bound of 2.64 applies even to the restricted version of JRP, denoted JRP-L, where the waiting cost function is linear. We provide several new approximation results for JRP. In the offline case, we give an algorithm with ratio 1.791, breaking the barrier of 1.8. In the online case, we show a lower bound of 2.754 on the competitive ratio for JRP-L (and thus JRP as well), improving the previous bound of 2.64. We also study the online version of JRP-D, for which we prove that the optimal competitive ratio is 2

Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics

In one and two spatial dimensions there is a logical possibility for identical quantum particles different from bosons and fermions, obeying intermediate or fractional (anyon) statistics. We consider applications of a recent Lieb-Thirring inequality for anyons in two dimensions, and derive new Lieb-Thirring inequalities for intermediate statistics in one dimension with implications for models of Lieb-Liniger and Calogero-Sutherland type. These inequalities follow from a local form of the exclusion principle valid for such generalized exchange statistics.Comment: Revised and accepted version. 49 pages, 2 figure