The Maximum Traveling Salesman Problem with Submodular Rewards

In this paper, we look at the problem of finding the tour of maximum reward on an undirected graph where the reward is a submodular function, that has a curvature of $\kappa$, of the edges in the tour. This problem is known to be NP-hard. We analyze two simple algorithms for finding an approximate solution. Both algorithms require $O(|V|^3)$ oracle calls to the submodular function. The approximation factors are shown to be $\frac{1}{2+\kappa}$ and $\max\set{\frac{2}{3(2+\kappa)},2/3(1-\kappa)}$, respectively; so the second method has better bounds for low values of $\kappa$. We also look at how these algorithms perform for a directed graph and investigate a method to consider edge costs in addition to rewards. The problem has direct applications in monitoring an environment using autonomous mobile sensors where the sensing reward depends on the path taken. We provide simulation results to empirically evaluate the performance of the algorithms.Comment: Extended version of ACC 2013 submission (including p-system greedy bound with curvature

The Price of Information in Combinatorial Optimization

Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay. Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility/disutility? A classical example of the above setting is Weitzman's "Pandora's box" problem where we are given probability distributions on values of $n$ independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our utility/disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.Comment: SODA 201

How to Influence People with Partial Incentives

We study the power of fractional allocations of resources to maximize influence in a network. This work extends in a natural way the well-studied model by Kempe, Kleinberg, and Tardos (2003), where a designer selects a (small) seed set of nodes in a social network to influence directly, this influence cascades when other nodes reach certain thresholds of neighbor influence, and the goal is to maximize the final number of influenced nodes. Despite extensive study from both practical and theoretical viewpoints, this model limits the designer to a binary choice for each node, with no way to apply intermediate levels of influence. This model captures some settings precisely, e.g. exposure to an idea or pathogen, but it fails to capture very relevant concerns in others, for example, a manufacturer promoting a new product by distributing five "20% off" coupons instead of giving away one free product. While fractional versions of problems tend to be easier to solve than integral versions, for influence maximization, we show that the two versions have essentially the same computational complexity. On the other hand, the two versions can have vastly different solutions: the added flexibility of fractional allocation can lead to significantly improved influence. Our main theoretical contribution is to show how to adapt the major positive results from the integral case to the fractional case. Specifically, Mossel and Roch (2006) used the submodularity of influence to obtain their integral results; we introduce a new notion of continuous submodularity, and use this to obtain matching fractional results. We conclude that we can achieve the same greedy $(1-1/e-\epsilon)$-approximation for the fractional case as the integral case. In practice, we find that the fractional model performs substantially better than the integral model, according to simulations on real-world social network data

Validating Network Value of Influencers by means of Explanations

Recently, there has been significant interest in social influence analysis. One of the central problems in this area is the problem of identifying influencers, such that by convincing these users to perform a certain action (like buying a new product), a large number of other users get influenced to follow the action. The client of such an application is a marketer who would target these influencers for marketing a given new product, say by providing free samples or discounts. It is natural that before committing resources for targeting an influencer the marketer would be interested in validating the influence (or network value) of influencers returned. This requires digging deeper into such analytical questions as: who are their followers, on what actions (or products) they are influential, etc. However, the current approaches to identifying influencers largely work as a black box in this respect. The goal of this paper is to open up the black box, address these questions and provide informative and crisp explanations for validating the network value of influencers. We formulate the problem of providing explanations (called PROXI) as a discrete optimization problem of feature selection. We show that PROXI is not only NP-hard to solve exactly, it is NP-hard to approximate within any reasonable factor. Nevertheless, we show interesting properties of the objective function and develop an intuitive greedy heuristic. We perform detailed experimental analysis on two real world datasets - Twitter and Flixster, and show that our approach is useful in generating concise and insightful explanations of the influence distribution of users and that our greedy algorithm is effective and efficient with respect to several baselines

Minimizing Seed Set Selection with Probabilistic Coverage Guarantee in a Social Network

A topic propagating in a social network reaches its tipping point if the number of users discussing it in the network exceeds a critical threshold such that a wide cascade on the topic is likely to occur. In this paper, we consider the task of selecting initial seed users of a topic with minimum size so that with a guaranteed probability the number of users discussing the topic would reach a given threshold. We formulate the task as an optimization problem called seed minimization with probabilistic coverage guarantee (SM-PCG). This problem departs from the previous studies on social influence maximization or seed minimization because it considers influence coverage with probabilistic guarantees instead of guarantees on expected influence coverage. We show that the problem is not submodular, and thus is harder than previously studied problems based on submodular function optimization. We provide an approximation algorithm and show that it approximates the optimal solution with both a multiplicative ratio and an additive error. The multiplicative ratio is tight while the additive error would be small if influence coverage distributions of certain seed sets are well concentrated. For one-way bipartite graphs we analytically prove the concentration condition and obtain an approximation algorithm with an $O(\log n)$ multiplicative ratio and an $O(\sqrt{n})$ additive error, where $n$ is the total number of nodes in the social graph. Moreover, we empirically verify the concentration condition in real-world networks and experimentally demonstrate the effectiveness of our proposed algorithm comparing to commonly adopted benchmark algorithms.Comment: Conference version will appear in KDD 201