The Ground-Set-Cost Budgeted Maximum Coverage Problem

We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph G = (V, E), where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges T subseteq E such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. Besides being a natural generalization of the well-studied maximum coverage problem, our motivation for investigating this problem originates from its application in the context of bid optimization in sponsored search auctions, such as Google AdWords. It is easily seen that this problem is strictly harder than budgeted max coverage, which means that the problem is (1-1/e)-inapproximable. The difference of our problem to the budgeted maximum coverage problem is that the costs are associated with the covered vertices instead of the selected hyperedges. As it turns out, this difference refutes the applicability of standard greedy approaches which are used to obtain constant factor approximation algorithms for several other variants of the maximum coverage problem. Our main results are as follows: - We obtain a (1 - 1/sqrt(e))/2-approximation algorithm for graphs. - We derive a fully polynomial-time approximation scheme (FPTAS) if the incidence graph of the hypergraph is a forest (i.e., the hypergraph is Berge-acyclic). We also extend this result to incidence graphs with a fixed-size feedback hyperedge node set. - We give a (1-epsilon)/(2d^2)-approximation algorithm for every epsilon > 0, where d is the maximum degree of a vertex in the hypergraph

A Nested Genetic Algorithm for Explaining Classification Data Sets with Decision Rules

Our goal in this paper is to automatically extract a set of decision rules (rule set) that best explains a classification data set. First, a large set of decision rules is extracted from a set of decision trees trained on the data set. The rule set should be concise, accurate, have a maximum coverage and minimum number of inconsistencies. This problem can be formalized as a modified version of the weighted budgeted maximum coverage problem, known to be NP-hard. To solve the combinatorial optimization problem efficiently, we introduce a nested genetic algorithm which we then use to derive explanations for ten public data sets

Concept-based Summarization using Integer Linear Programming: From Concept Pruning to Multiple Optimal Solutions

International audienceIn concept-based summarization, sentence selection is modelled as a budgeted maximum coverage problem. As this problem is NP-hard, pruning low-weight concepts is required for the solver to find optimal solutions efficiently. This work shows that reducing the number of concepts in the model leads to lower Rouge scores, and more importantly to the presence of multiple optimal solutions. We address these issues by extending the model to provide a single optimal solution, and eliminate the need for concept pruning using an approximation algorithm that achieves comparable performance to exact inference

Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design

We study a type of reverse (procurement) auction problems in the presence of budget constraints. The general algorithmic problem is to purchase a set of resources, which come at a cost, so as not to exceed a given budget and at the same time maximize a given valuation function. This framework captures the budgeted version of several well known optimization problems, and when the resources are owned by strategic agents the goal is to design truthful and budget feasible mechanisms, i.e. elicit the true cost of the resources and ensure the payments of the mechanism do not exceed the budget. Budget feasibility introduces more challenges in mechanism design, and we study instantiations of this problem for certain classes of submodular and XOS valuation functions. We first obtain mechanisms with an improved approximation ratio for weighted coverage valuations, a special class of submodular functions that has already attracted attention in previous works. We then provide a general scheme for designing randomized and deterministic polynomial time mechanisms for a class of XOS problems. This class contains problems whose feasible set forms an independence system (a more general structure than matroids), and some representative problems include, among others, finding maximum weighted matchings, maximum weighted matroid members, and maximum weighted 3D-matchings. For most of these problems, only randomized mechanisms with very high approximation ratios were known prior to our results

Sticky Seeding in Discrete-Time Reversible-Threshold Networks

When nodes can repeatedly update their behavior (as in agent-based models from computational social science or repeated-game play settings) the problem of optimal network seeding becomes very complex. For a popular spreading-phenomena model of binary-behavior updating based on thresholds of adoption among neighbors, we consider several planning problems in the design of \textit{Sticky Interventions}: when adoption decisions are reversible, the planner aims to find a Seed Set where temporary intervention leads to long-term behavior change. We prove that completely converting a network at minimum cost is $\Omega(\ln (OPT) )$-hard to approximate and that maximizing conversion subject to a budget is $(1-\frac{1}{e})$-hard to approximate. Optimization heuristics which rely on many objective function evaluations may still be practical, particularly in relatively-sparse networks: we prove that the long-term impact of a Seed Set can be evaluated in $O(|E|^2)$ operations. For a more descriptive model variant in which some neighbors may be more influential than others, we show that under integer edge weights from $\{0,1,2,...,k\}$ objective function evaluation requires only $O(k|E|^2)$ operations. These operation bounds are based on improvements we give for bounds on time-steps-to-convergence under discrete-time reversible-threshold updates in networks.Comment: 19 pages, 2 figure

Observer Placement for Source Localization: The Effect of Budgets and Transmission Variance

When an epidemic spreads in a network, a key question is where was its source, i.e., the node that started the epidemic. If we know the time at which various nodes were infected, we can attempt to use this information in order to identify the source. However, maintaining observer nodes that can provide their infection time may be costly, and we may have a budget $k$ on the number of observer nodes we can maintain. Moreover, some nodes are more informative than others due to their location in the network. Hence, a pertinent question arises: Which nodes should we select as observers in order to maximize the probability that we can accurately identify the source? Inspired by the simple setting in which the node-to-node delays in the transmission of the epidemic are deterministic, we develop a principled approach for addressing the problem even when transmission delays are random. We show that the optimal observer-placement differs depending on the variance of the transmission delays and propose approaches in both low- and high-variance settings. We validate our methods by comparing them against state-of-the-art observer-placements and show that, in both settings, our approach identifies the source with higher accuracy.Comment: Accepted for presentation at the 54th Annual Allerton Conference on Communication, Control, and Computin