/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Given a water distribution network, where should we place sensors toquickly detect contaminants? Or, which blogs should we read to avoid missing important stories?.These seemingly different problems share common structure: Outbreak detection can be modeled as selecting nodes (sensor locations, blogs) in a network, in order to detect the spreading of a virus or information asquickly as possible. We present a general methodology for near optimal sensor placement in these and related problems. We demonstrate that many realistic outbreak detection objectives (e.g., detection likelihood, population affected) exhibit the property of "submodularity". We exploit submodularity to develop an efficient algorithm that scales to large problems, achieving near optimal placements, while being 700 times faster than a simple greedy algorithm. We also derive online bounds on the quality of the placements obtained by any algorithm. Our algorithms and bounds also handle cases where nodes (sensor locations, blogs) have different costs.We evaluate our approach on several large real-world problems,including a model of a water distribution network from the EPA, andreal blog data. The obtained sensor placements are provably near optimal, providing a constant fraction of the optimal solution. We show that the approach scales, achieving speedups and savings in storage of several orders of magnitude. We also show how the approach leads to deeper insights in both applications, answering multicriteria trade-off, cost-sensitivity and generalization questions.

/pdf/cost-effective-outbreak-detection-in-networks-23m9hhyi9o.pdf

Cost-effective outbreak detection in networks

We study fairness in classification, where individuals are classified, e.g., admitted to a university, and the goal is to prevent discrimination against individuals based on their membership in some group, while maintaining utility for the classifier (the university). The main conceptual contribution of this paper is a framework for fair classification comprising (1) a (hypothetical) task-specific metric for determining the degree to which individuals are similar with respect to the classification task at hand; (2) an algorithm for maximizing utility subject to the fairness constraint, that similar individuals are treated similarly. We also present an adaptation of our approach to achieve the complementary goal of "fair affirmative action," which guarantees statistical parity (i.e., the demographics of the set of individuals receiving any classification are the same as the demographics of the underlying population), while treating similar individuals as similarly as possible. Finally, we discuss the relationship of fairness to privacy: when fairness implies privacy, and how tools developed in the context of differential privacy may be applied to fairness.

/pdf/fairness-through-awareness-2pxxfbtoi2.pdf

Fairness through awareness

Fairness Through Awareness

When monitoring spatial phenomena, which can often be modeled as Gaussian processes (GPs), choosing sensor locations is a fundamental task. There are several common strategies to address this task, for example, geometry or disk models, placing sensors at the points of highest entropy (variance) in the GP model, and A-, D-, or E-optimal design. In this paper, we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected. We prove that the problem of finding the configuration that maximizes mutual information is NP-complete. To address this issue, we describe a polynomial-time approximation that is within (1-1/e) of the optimum by exploiting the submodularity of mutual information. We also show how submodularity can be used to obtain online bounds, and design branch and bound search procedures. We then extend our algorithm to exploit lazy evaluations and local structure in the GP, yielding significant speedups. We also extend our approach to find placements which are robust against node failures and uncertainties in the model. These extensions are again associated with rigorous theoretical approximation guarantees, exploiting the submodularity of the objective function. We demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real-world data sets.

/pdf/near-optimal-sensor-placements-in-gaussian-processes-theory-b4fe67xmxz.pdf

Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies

http://www.cs.toronto.edu/~eidan/papers/submod-knapsack-constraint.pdf

A note on maximizing a submodular set function subject to a knapsack constraint

The paper presents a general method of designing constant-factor approximation algorithms for some discrete optimization problems with assignment-type constraints. The core of the method is a simple deterministic procedure of rounding of linear relaxations (referred to as pipage rounding). With the help of the method we design approximation algorithms with better performance guarantees for some well-known problems including MAXIMUM COVERAGE, MAX CUT with given sizes of parts and some of their generalizations.

Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee

We consider the following problem: The Santa Claus has n presents that he wants to distribute among m kids. Each kid has an arbitrary value for each present. Let pij be the value that kid i has for present j. The Santa's goal is to distribute presents in such a way that the least lucky kid is as happy as possible, i.e he tries to maximize mini=1,...,m sumj ∈ Si pij where Si is a set of presents received by the i-th kid.Our main result is an O(log log m/log log log m) approximation algorithm for the restricted assignment case of the problem when pij ∈ pj,0 (i.e. when present j has either value pj or 0 for each kid). Our algorithm is based on rounding a certain natural exponentially large linear programming relaxation usually referred to as the configuration LP. We also show that the configuration LP has an integrality gap of Ω(m1/2) in the general case, when pij can be arbitrary.

The Santa Claus problem

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin - ie for bin i, a family Li of subsets of items that fit in bin i The goal is to pack items into bins to maximize the aggregate value This class of problems includes the maximum generalized assignment problem (GAP)1) and a distributed caching problem (DCP) described in this paperGiven a β-approximation algorithm for finding the highest value packing of a single bin, we give1 A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm2 A simple polynomial-time local search (β/β+1 - e) - approximation algorithm, for any e > 0Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 - 1/e - e)-approximation and a local search algorithm with (1/2-e)-approximation guarantee Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 - 1/e The best previously known approximation algorithm for GAP is a 1/2-approximation by Shmoys and Tardos; and Chekuri and Khanna Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gapTo complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 -1/e unless NP⊆ DTIME(nO(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problemWe extend the (1 - 1/e)-approximation algorithm to a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem We generalize the local search algorithm to yield a 1/2-e approximation algorithm for the k-median problem with hard capacities Finally, we study naturally defined game-theoretic versions of these problems, and show that they have price of anarchy of 2 We also prove the existence of cycles of best response moves, and exponentially long best-response paths to (pure or sink) equilibria

Tight approximation algorithms for maximum general assignment problems

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for non-monotone submodular functions. In particular, for any constant k, we present a (1/k+2+1/k+e)-approximation for the submodular maximization problem under k matroid constraints, and a (1/5-e)-approximation algorithm for this problem subject to k knapsack constraints (e>0 is any constant). We improve the approximation guarantee of our algorithm to 1/k+1+{1/k-1}+e for k≥2 partition matroid constraints. This idea also gives a ({1/k+e)-approximation for maximizing a monotone submodular function subject to k≥2 partition matroids, which improves over the previously best known guarantee of 1/k+1.

/pdf/non-monotone-submodular-maximization-under-matroid-and-403igjvfyp.pdf

Maxim Sviridenko

Papers

A note on maximizing a submodular set function subject to a knapsack constraint

Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee

The Santa Claus problem

Tight approximation algorithms for maximum general assignment problems

Non-monotone submodular maximization under matroid and knapsack constraints