We consider the following class of problems. The value of an outcome to a society is measured via a submodular utility function (submodularity has a natural economic interpretation: decreasing marginal utility). Decisions, however are controlled by noncooperative agents who seek to maximise their own private utility. We present, under some basic assumptions, guarantees on the social performance of Nash equilibria. For submodular utility functions, any Nash equilibrium gives an expected social utility within a factor 2 of optimal, subject to a function-dependent additive term. For non-decreasing, submodular utility functions, any Nash equilibrium gives an expected social utility within a factor 1 + of optimal, where 0 1 is a number based upon the discrete curvature of the function. A condition under which all sets of social and private utility functions induce pure strategy Nash equilibria is presented. The case in which agents, themselves, make use of approximation algorithms in decision making is discussed and performance guarantees given. Finally we present some speci c problems that fall into our framework. These include the competitive versions of the facility location problem and k-median problem, a maximisation version of the trac routing problem of Roughgarden and Tardos [16], and multiple-item auctions.

Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard. In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative submodular functions. In particular, we give a deterministic local search 1 2-approximation and a randomized-approximation algo-

Submodular-function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximum-entropy sampling, and maximum facility-location problems. Our main result is that for any k ≥ 2 and any ε> 0, there is a natural local-search algorithm which has approximation guarantee of 1/(k + ε) for the problem of maximizing a monotone submodular function subject to k matroid constraints. This improves a 1/(k + 1)-approximation of Nemhauser, Wolsey and Fisher, obtained more than 30 years ago. Also, our analysis can be applied to the problem of maximizing a linear objective function and even a general non-monotone submodular function subject to k matroid constraints. We show that in these cases the approximation guarantees of our algorithms are 1/(k − 1 + ε) and 1/(k + 1 + 1/k + ε), respectively.

A number of recent results on optimization problems involving submodular functions have made use of the ”multilinear relaxation” of the problem [3], [8], [24], [14], [13]. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of ”symmetry gap”. Our main result is that for any fixed instance that exhibits a certain ”symmetry gap ” in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1 − 1/e)-approximation for monotone submodular maximization under a cardinality constraint [20], [7], and the impossibility of ( 1 +ɛ)-approximation for uncon-2 strained (non-monotone) submodular maximization [8]. It follows from our result that ( 1 + ɛ)-approximation is also impossible for 2 non-monotone submodular maximization subject to a (non-trivial) matroid constraint. On the algorithmic side, we present a 0.309approximation for this problem, improving the previously known factor of 1 − o(1) [14]. 4 As another application, we consider the problem of maximizing a non-monotone submodular function over the bases of a matroid. A ( 1 − o(1))-approximation has been developed for this problem, 6 assuming that the matroid contains two disjoint bases [14]. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k ≥ 2, there is a class of matroids of fractional base packing number k k−1 ν = , such that any algorithm achieving a better than (1 − 1)-approximation for this class would require exponentially many

Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max |S|≤k f(S), the greedy algorithm yields a (1 − 1/e)-approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f(S) for a given set S) [28], and also for explicitly posed instances assuming P � = NP [10]. In this paper, we provide a randomized (1 − 1/e)-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires |X | to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.

To detect multiple objects of interest, the methods based on Hough transform use non-maxima supression or mode seeking in order to locate and to distinguish peaks in Hough images. Such postprocessing requires tuning of extra parameters and is often fragile, especially when objects of interest tend to be closely located. In the paper, we develop a new probabilistic framework that is in many ways related to Hough transform, sharing its simplicity and wide applicability. At the same time, the framework bypasses the problem of multiple peaks identification in Hough images, and permits detection of multiple objects without invoking nonmaximum suppression heuristics. As a result, the experiments demonstrate a significant improvement in detection accuracy both for the classical task of straight line detection and for a more modern category-level (pedestrian) detection problem. 1. Hough Transform in Object Detection The Hough transform [10] is one of the classical computer vision techniques which dates 50 years back. It was initially suggested as a method for line detection in edge maps of images but was then extended to detect general low-parametric objects such as circles [2]. In recent years, Hough-based methods were successful adapted to the problem of part-based category-level object detection where they have obtained state-of-the-art results for some popular datasets [12, 13, 7, 8, 15, 3]. Both the classical Hough transform and its more modern variants proceed by converting the input image into a new representation called the Hough image which lives in a domain called the Hough space (Figure 1). Each point in the Hough space corresponds to a hypothesis about the object of interest being present in the original image at a particular location and configuration. Any Hough transform based method essentially works by splitting the input image into a set of voting elements. Each such element votes for the hypotheses that might have generated this element. For instance, a feature that fires ∗ The first two authors were with Microsoft Research through the initial stages of the work and are currently supported by Microsoft Research programs in Russia. Victor Lempitsky is also supported by EU under ERC

Quicklinks for a website are navigational shortcuts displayed below the website homepage on a search results page, and that let the users directly jump to selected points inside the website. Since the real-estate on a search results page is constrained and valuable, picking the best set of quicklinks to maximize the benefits for a majority of the users becomes an important problem for search engines. Using user browsing trails obtained from browser toolbars, and a simple probabilistic model, we formulate the quicklink selection problem as a combinatorial optimizaton problem. We first demonstrate the hardness of the objective, and then propose an algorithm that is provably within a factor of (1 − 1/e) of the optimal. We also propose a different algorithm that works on trees and that can find the optimal solution; unlike the previous algorithm, this algorithm can incorporate natural constraints on the set of chosen quicklinks. The efficacy of our methods is demonstrated via empirical results on both a manually labeled set of websites and a set for which quicklink click-through rates for several webpages were obtained from a real-world search engine.

We present concentration bounds for linear functions of random variables arising from the pipage rounding procedure on matroid polytopes. As an application, we give a (1 − 1/e − ɛ)-approximation algorithm for the problem of maximizing a monotone submodular function subject to 1 matroid and k linear constraints, for any constant k ≥ 1 and ɛ> 0. This generalizes the result for k linear constraints by Kulik et al. [11]. We also give the same result for a super-constant number k of ”loose ” linear constraints, where the right-hand side dominates the matrix entries by an Ω(ɛ −2 log k) factor. As another application, we present a general result on minimax packing problems that involve a matroid base constraint. An example is the multi-path routing problem with integer demands for pairs of vertices; the goal is to minimize congestion. We give an O(log m / log log m)approximation for the general problem min{λ: ∃x ∈ {0, 1} N, x ∈ B(M), Ax ≤ λb} where m is the number of packing constraints.

Can airline Yield Management strategies be used to generate additional revenue from spare capacity in telecom networks? Pundits believe “yes”, based on several analogies between the industries such as, for instance, perishable inventory and negligible marginal cost of usage. However, no one has yet described how, one of the chief difficulties being the vastly different nature of airlines products and telecom services. Motivated to show how Operations Research can play a role in structuring this area, we: (i) argue that telecom Yield Management should be based on ’innovative ’ services explicitly designed to use only spare capacity, (ii) propose, borrowing from airlines, a framework to simplify related decision modeling, and (iii) demonstrate both our argument and the framework by articulating several ’innovative ’ telecom services and modeling them to varying degrees of depth. This thesis focuses only on the decision-making that will be required within a large infrastructure for operating new ’Yield Management ’ services. For each service, several decision variables can be considered to maximize revenue from available capacity, e.g. pricing, capacity limits and admission control, among others. Incorporating all such decisions in a single model usually leads to complicated

Let (X, I) be a matroid and let f: 2 X → R+ be a monotone submodular function. The curvature of f is a parameter c ∈ [0, 1] such that for any S ⊂ X and j ∈ X \ S, f(S ∪ {j}) − f(S) ≥ (1 − c)f({j}). We consider the optimization problem max{f(S) : S ∈ I}. It is known that the greedy algorithm yields a 1/2-approximation for this problem [11], and 1-approximation when f has curvature c [4]. For the uniform matroid, it was known 1+c that the greedy algorithm yields an improved 1 c (1 − e−c)-approximation [4]. In this paper, we analyze the continuous greedy algorithm [23] and prove that it gives a 1 c (1 − e−c)-approximation for any matroid. Moreover, we show that this holds for a relaxed notion of curvature, curvature with respect to the optimum, and we prove that any better approximation under these conditions would require an exponential number of value queries.