Provable Approximation Guarantees Research Articles

Towards a Theoretical Understanding of Why Local Search Works for Clustering with Fair-Center Representation

The representative k-median problem generalizes the classical clustering formulations in that it partitions the data points into several disjoint demographic groups and poses a lower-bound constraint on the number of opened facilities from each group, such that all the groups are fairly represented by the opened facilities. Due to its simplicity, the local-search heuristic that optimizes an initial solution by iteratively swapping at most a constant number of closed facilities for the same number of opened ones (denoted by the O(1)-swap heuristic) has been frequently used in the representative k-median problem. Unfortunately, despite its good performance exhibited in experiments, whether the O(1)-swap heuristic has provable approximation guarantees for the case where the number of groups is more than 2 remains an open question for a long time. As an answer to this question, we show that the O(1)-swap heuristic (1) is guaranteed to yield a constant-factor approximation solution if the number of groups is a constant, and (2) has an unbounded approximation ratio otherwise. Our main technical contribution is a new approach for theoretically analyzing local-search heuristics, which derives the approximation ratio of the O(1)-swap heuristic via linearly combining the increased clustering costs induced by a set of hierarchically organized swaps.

Co-Activity Maximization in Online Social Networks

Social media with online social networks has risen to be a prevalent force in information diffusion and public discourse. Despite its popularity and convenience, social media has been criticized for contributing to societal and ideological polarization as the result of trapping users in an echo chamber and filter bubbles. An emerging line of research focuses on ways to redesign content or link recommendation algorithms to mitigate the polarization phenomenon. However, existing works mainly concentrate on node-level balancing, while omitting the balancing effect that can be incurred by edge interaction in social networks. In this article, we take the first step to study the problem (CoAM) that assuming two campaigns are present in a network, how we should select seeds for each so as to maximize the interaction/activity between the followers of two campaigns (co-activity) after the diffusion process is finished. We begin our analysis by showing the hardness of CoAM under two diffusion models that are generalized from wildly used diffusion models and its objective function is neither submodular nor supermodular. This encourages us to design a submodular function that acts as a lower bound to the objective, by exploiting which we are able to devise a greedy algorithm with a provable approximation guarantee. To overcome the #P-hardness of diffusion calculation, we further extend the notion of random reverse-reachable (RR) set to devise a scalable instantiation of our approximation algorithm. We experimentally demonstrate the quality of our approximation algorithm on datasets collected from real-world social networks.

Maximization of nonsubmodular functions under multiple constraints with applications

We consider the problem of maximizing a monotone nondecreasing set function under multiple constraints, where the constraints are also characterized by monotone nondecreasing set functions. We propose two greedy algorithms to solve the problem with provable approximation guarantees. The first algorithm exploits the structure of a special class of the general problem instance to obtain a better time complexity. The second algorithm is suitable for the general problem. We characterize the approximation guarantees of the two algorithms, leveraging the notions of submodularity ratio and curvature introduced for set functions. We then discuss particular applications of the general problem formulation to problems that have been considered in the literature. We validate our theoretical results using numerical examples.

Regularized nonmonotone submodular maximization

In this paper, we present a thorough study of the regularized submodular maximization problem, in which the objective f := g − ℓ can be expressed as the difference between a submodular function and a modular function. This problem has drawn much attention in recent years. While existing works focuses on the case of g being monotone, we investigate the problem with a nonmonotone g. The main technique we use is to introduce a distorted objective function, which varies weights of the submodular component g and the modular component ℓ during the iterations of the algorithm. By combining the weighting technique and measured continuous greedy algorithm, we present an algorithm for the matroid-constrained problem, which has a provable approximation guarantee. In the cardinality-constrained case, we utilize random greedy algorithm and sampling technique together with the weighting technique to design two efficient algorithms. Moreover, we consider the unconstrained problem and propose a much simpler and faster algorithm compared with the algorithms for solving the problem with a cardinality constraint.

Classify nodes based on their degree distribution: A more scalable method for influence maximization

AbstractOne of the main problems in viral marketing is influence maximization (IM). With a social network and a predefined propagation model, the aim is to seek a subset of nodes that spread the influence widely into the network. Most scalable methods with provable approximation guarantees are presented for this problem based on the reverse influence sampling (RIS) framework. The RIS framework has two phases: sampling and node selection. The sampling phase encountered two challenges in the sampling phase: the number of required samples and the sampling method. Most methods have focused on the first challenge, that is, sample size, and have tried to provide a rigid sample size. In this paper, we focus on the second challenge: how to improve the precision of sampling. We propose to use stratified sampling rather than simple random sampling. Since the degree of each node is one of the affecting factors in the diffusion process. This issue leads us to use stratified sampling based on a degree distribution. The results show that with the application of the proposed method, the solution can estimate with fewer samples, which is faster than the state‐of‐the‐art methods.

Distributionally Robust Fair Transit Resource Allocation During a Pandemic

This paper studies the distributionally robust fair transit resource allocation model (DrFRAM) under the Wasserstein ambiguity set to optimize the public transit resource allocation during a pandemic. We show that the proposed DrFRAM is highly nonconvex and nonlinear, and it is NP-hard in general. Fortunately, we show that DrFRAM can be reformulated as a mixed integer linear programming (MILP) by leveraging the equivalent representation of distributionally robust optimization and monotonicity properties, binarizing integer variables, and linearizing nonconvex terms. To improve the proposed MILP formulation, we derive stronger ones and develop valid inequalities by exploiting the model structures. Additionally, we develop scenario decomposition methods using different MILP formulations to solve the scenario subproblems and introduce a simple yet effective no one left-based approximation algorithm with a provable approximation guarantee to solve the model to near optimality. Finally, we numerically demonstrate the effectiveness of the proposed approaches and apply them to real-world data provided by the Blacksburg Transit. History: This paper has been accepted for the Transportation Science Special Issue on Emerging Topics in Transportation Science and Logistics. Funding: This work was supported by the Division of Computing and Communication Foundations [Grant 2153607] and the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046426]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.1159 .

An $O(\log \log m)$ Prophet Inequality for Subadditive Combinatorial Auctions

Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.

Efficient Combinatorial Optimization for Word-Level Adversarial Textual Attack

Over the past few years, various word-level textual attack approaches have been proposed to reveal the vulnerability of deep neural networks used in natural language processing. Typically, these approaches involve an important optimization step to determine which substitute to be used for each word in the original input. However, current research on this step is still rather limited, from the perspectives of both problem-understanding and problem-solving. In this paper, we address these issues by uncovering the theoretical properties of the problem and proposing an efficient local search algorithm (LS) to solve it. We establish the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">first</i> provable approximation guarantee on solving the problem in general cases. Extensive experiments involving 5 NLP tasks, 8 datasets and 26 NLP models show that LS can largely reduce the number of queries usually by an order of magnitude to achieve high attack success rates. Further experiments show that the adversarial examples crafted by LS usually have higher quality, exhibit better transferability, and can bring more robustness improvement to victim models by adversarial training.

Multi-Type Sensor Placements in Gaussian Spatial Fields for Environmental Monitoring.

As citizens are increasingly concerned about the surrounding environment, it is important for modern cities to provide sufficient and accurate environmental information to the public for decision making in the era of smart cities. Due to the limited budget, we often need to optimize the sensor placement in order to maximize the overall information gain according to certain criteria. Existing work is primarily concerned with single-type sensor placement; however, the environment usually requires accurate measurements of multiple types of environmental characteristics. In this paper, we focus on the optimal multi-type sensor placement in Gaussian spatial field for environmental monitoring. We study two representative cases: the one-with-all case when each station is equipped with all types of sensors and the general case when each station is equipped with at least one type of sensor. We propose two greedy algorithms accordingly, each with a provable approximation guarantee. We evaluated the proposed approach via an application in air quality monitoring scenario in Hong Kong and experimental results demonstrate the effectiveness of the proposed approach.

Input matrix construction and approximation using a graphic approach

ABSTRACT Given a state transition matrix (STM), we reinvestigate the problem of constructing the sparsest input matrix with a fixed number of inputs to guarantee controllability of the associated system. A new and simple graph-theoretic characterisation is obtained for the sparsity pattern of input matrices to guarantee controllability for a general STM admitting multiple eigenvalues, and a deterministic procedure with polynomial time complexity is suggested for constructing real valued input matrices satisfying the controllability requirement with an arbitrarily prescribed sparsity pattern. Based on this criterion, some novel results on sparsely controlling a system are obtained. It is proven that the minimal number of inputs to guarantee controllability equals the maximum geometric multiplicity of the STM under the constraint that some states can not be directly actuated by inputs (provided such problem is feasible), extending the existing results. Moreover, the minimal sparsity of an input matrix to ensure system controllability can be affected by the number of independent inputs. Furthermore, a graphic submodular function is built, leading to a greedy algorithm which efficiently approximates the minimal states needed to be directly actuated by inputs to ensure controllability for general STMs. To approximate the sparsest input matrices with a fixed number of inputs, we propose a simple greedy algorithm (non-submodular) and a two-stage algorithm, and demonstrate that the latter algorithm, inspired from graph colouring, has a provable approximation guarantee. Finally, we present some numerical results to show the efficiency and effectiveness of our approaches.

The Densest Subgraph Problem with a Convex/Concave Size Function

In the densest subgraph problem, given an edge-weighted undirected graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes the density, i.e., $w(S)/|S|$, where $w(S)$ is the sum of weights of the edges in the subgraph induced by $S$. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem by generalizing the density of $S\subseteq V$. Specifically, we introduce the $f$-density of $S\subseteq V$, which is defined as $w(S)/f(|S|)$, where $f:\mathbb{Z}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}$ is a monotonically non-decreasing function. In the $f$-densest subgraph problem ($f$-DS), we aim to find $S\subseteq V$ that maximizes the $f$-density $w(S)/f(|S|)$. Although $f$-DS does not explicitly specify the size of the output subset of vertices, we can handle the above size issue using a convex/concave size function $f$ appropriately. For $f$-DS with convex function $f$, we propose a nearly-linear-time algorithm with a provable approximation guarantee. On the other hand, for $f$-DS with concave function $f$, we propose an LP-based exact algorithm, a flow-based $O(|V|^3)$-time exact algorithm for unweighted graphs, and a nearly-linear-time approximation algorithm.

Clustering uncertain graphs

An uncertain graph 𝒢 = (V, E, p : E → (0, 1]) can be viewed as a probability space whose outcomes (referred to as possible worlds ) are subgraphs of 𝒢 where any edge e ε E occurs with probability p ( e ), independently of the other edges. These graphs naturally arise in many application domains where data management systems are required to cope with uncertainty in interrelated data, such as computational biology, social network analysis, network reliability, and privacy enforcement, among the others. For this reason, it is important to devise fundamental querying and mining primitives for uncertain graphs. This paper contributes to this endeavor with the development of novel strategies for clustering uncertain graphs. Specifically, given an uncertain graph 𝒢 and an integer k , we aim at partitioning its nodes into k clusters, each featuring a distinguished center node, so to maximize the minimum/average connection probability of any node to its cluster's center, in a random possible world. We assess the NP-hardness of maximizing the minimum connection probability, even in the presence of an oracle for the connection probabilities, and develop efficient approximation algorithms for both problems and some useful variants. Unlike previous works in the literature, our algorithms feature provable approximation guarantees and are capable to keep the granularity of the returned clustering under control. Our theoretical findings are complemented with several experiments that compare our algorithms against some relevant competitors, with respect to both running-time and quality of the returned clusterings.

Revenue maximization in incentivized social advertising

Incentivized social advertising, an emerging marketing model, provides monetization opportunities not only to the owners of the social networking platforms but also to their influential users by offering a "cut" on the advertising revenue. We consider a social network (the host) that sells ad-engagements to advertisers by inserting their ads, in the form of promoted posts, into the feeds of carefully selected "initial endorsers" or seed users: these users receive monetary incentives in exchange for their endorsements. The endorsements help propagate the ads to the feeds of their followers. Whenever any user engages with an ad, the host is paid some fixed amount by the advertiser, and the ad further propagates to the feed of her followers, potentially recursively. In this context, the problem for the host is is to allocate ads to influential users, taking into account the propensity of ads for viral propagation, and carefully apportioning the monetary budget of each of the advertisers between incentives to influential users and ad-engagement costs, with the rational goal of maximizing its own revenue. We show that, taking all important factors into account, the problem of revenue maximization in incentivized social advertising corresponds to the problem of monotone submodular function maximization, subject to a partition matroid constraint on the ads-to-seeds allocation, and submodular knapsack constraints on the advertisers' budgets. We show that this problem is NP-hard and devise two greedy algorithms with provable approximation guarantees, which differ in their sensitivity to seed user incentive costs. Our approximation algorithms require repeatedly estimating the expected marginal gain in revenue as well as in advertiser payment. By exploiting a connection to the recent advances made in scalable estimation of expected influence spread, we devise efficient and scalable versions of our two greedy algorithms. An extensive experimental assessment confirms the high quality of our proposal.

Traffic Engineering With Equal-Cost-MultiPath: An Algorithmic Perspective

To efficiently exploit the network resources operators, do traffic engineering (TE), i.e., adapt the routing of traffic to the prevailing demands. TE in large IP networks typically relies on configuring static link weights and splitting traffic between the resulting shortest paths via the Equal-Cost-MultiPath (ECMP) mechanism. Yet, despite its vast popularity, crucial operational aspects of TE via ECMP are still little-understood from an algorithmic viewpoint. We embark upon a systematic algorithmic study of TE with ECMP. We consider the standard model of TE with ECMP and prove that, in general, even approximating the optimal link-weight configuration for ECMP within any constant ratio is an intractable feat, settling a long-standing open question. We establish, in contrast, that ECMP can provably achieve optimal traffic flow for the important category of Clos datacenter networks. We last consider a well-documented shortcoming of ECMP: suboptimal routing of large (“elephant”) flows. We present algorithms for scheduling “elephant” flows on top of ECMP (as in, e.g., Hedera) with provable approximation guarantees. Our results complement and shed new light on past experimental and empirical studies of the performance of TE with ECMP.

Jamming-Resistant Learning in Wireless Networks

We consider capacity maximization in wireless networks under adversarial interference conditions. There are $n$ links, i.e., sender-receiver pairs, which repeatedly try to perform a successful transmission. In each time step, the success of attempted transmissions depends on interference conditions, which are captured by an interference model (e.g., the SINR model). Additionally, an adversarial jammer can render a $(1-\delta)$ -fraction of time steps in a time window unsuccessful. For this scenario, we analyze a framework for distributed no-regret learning algorithms to get provable approximation guarantees. We obtain an $O(1/\delta)$ -approximation for the problem of maximizing the number of successful transmissions. Our approach provides even a constant-factor approximation when the jammer exactly blocks a $(1-\delta)$ -fraction of time steps. In addition, we consider the parameters of the jammer being partially unknown to the algorithm, and we also consider a stochastic jammer, for which we obtain a constant-factor approximation after a polynomial number of time steps. We extend our results to more general settings, in which links arrive and depart dynamically, and where each sender tries to reach multiple receivers. Our algorithms perform favorably in simulations.

Constrained Resource Assignments: Fast Algorithms and Applications in Wireless Networks

Resource assignment problems occur in a vast variety of applications, from scheduling problems over image recognition to communication networks. Often these problems can be modeled by a maximum weight matching problem in (bipartite) graphs or generalizations thereof, and efficient and practical algorithms are known for these problems. Although in some of the applications an assignment of the resources may be needed only once, in many of these applications, the assignment has to be computed more often for different scenarios. In that case it is often essential that the assignments can be computed very fast. Moreover, implementing different assignments in different scenarios may come with a certain cost for the reconfiguration of the system. In this paper, we consider the problem of determining optimal assignments sequentially over a given time horizon, where consecutive assignments are coupled by constraints that control the cost of reconfiguration. We develop fast approximation and online algorithms for this problem with provable approximation guarantees and competitive ratios. Moreover, we present an extensive computational study about the applicability of our model and our algorithms in the context of orthogonal frequency division multiple access (OFDMA) wireless networks, finding a significant performance improvement for the total bandwidth of the system using our algorithms. For this application (the downlink of an OFDMA wireless cell) , the run time of matching algorithms is extremely important, having an acceptable range of a few milliseconds only. For the considered realistic instances, our algorithms perform extremely well: the solution quality is, on average, within a factor of 0.8–0.9 of optimal off-line solutions, and the running times are at most 5 ms per phase even in the worst case. Thus, our algorithms are well suited to be applied in the context of OFDMA systems. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2221 . This paper was accepted by Teck-Hua Ho, optimization.

Optimizing Video Request Routing in Mobile Networks with Built-in Content Caching

Built-in content caching in mobile core networks can help improve quality of service, reduce operation expenses, simplify inter-network cooperation, and thus is a promising approach for more efficient networking architectures. In addition to the complexity of content placement as revealed in the literature, routing video requests remains a challenging issue. Two problems must be addressed: (i) how to distribute video requests among multiple internal servers (i.e., server selection ); and (ii) how to route so-generated video flows (i.e., flow routing ). In this work, we jointly formulate these two problems with two traffic-engineering objectives considered, namely, minimizing maximum link utilization and minimizing total link cost. We develop fast algorithms to solve the problems with provable approximation guarantees. We then propose a hop-by-hop routing protocol, which implements the optimization solutions by generating a set of flow-splitting and routing decisions for each router/caching node. Simulation results show that our algorithms significantly outperform existing routing schemes under various system settings, reducing up to 68 percent of maximum link utilization and more than 50 percent of link cost, and supporting over 60 percent more of traffic load.

Influence maximization by probing partial communities in dynamic online social networks

AbstractWith the rapid development of online social networks, exploring influence maximization for product publicity and advertisement marketing has attracted strong interests from both academia and industry. However, because of the continuous change of network topology, updating the variation of an entire network moment by moment is resource intensive and often insurmountable. On the other hand, the classical influence maximization models Independent Cascade and Linear Threshold together with their derived varieties are all computationally intensive. Thus, developing a solution for dynamic networks with lower cost and higher accuracy is in an urgent necessity. In this paper, a practical framework is proposed by only probing partial communities to explore the real changes of a network. Our framework minimizes the possible difference between the observed topology and the real network through several representative communities. Based on the framework, an algorithm that takes full advantage of our divide‐and‐conquer strategy, which reduces the computational overhead, is proposed. The systemically theoretical analysis shows that the proposed effective algorithm could achieve provable approximation guarantees. Empirical studies on synthetic and real large‐scale social networks demonstrate that our framework has better practicality compared with most existing works and provides a regulatory mechanism for enhancing influence maximization. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Approximation Algorithms for Stochastic Combinatorial Optimization Problems

Stochastic optimization has established itself as a major method to handle uncertainty in various optimization problems by modeling the uncertainty by a probability distribution over possible realizations. Traditionally, the main focus in stochastic optimization has been various stochastic mathematical programming (such as linear programming, convex programming). In recent years, there has been a surge of interest in stochastic combinatorial optimization problems from the theoretical computer science community. In this article, we survey some of the recent results on various stochastic versions of classical combinatorial optimization problems. Since most problems in this domain are NP-hard (or #P-hard, or even PSPACE-hard), we focus on the results which provide polynomial time approximation algorithms with provable approximation guarantees. Our discussions are centered around a few representative problems, such as stochastic knapsack, stochastic matching, multi-armed bandit etc. We use these examples to introduce several popular stochastic models, such as the fixed-set model, 2-stage stochastic optimization model, stochastic adaptive probing model etc, as well as some useful techniques for designing approximation algorithms for stochastic combinatorial optimization problems, including the linear programming relaxation approach, boosted sampling, content resolution schemes, Poisson approximation etc. We also provide some open research questions along the way. Our purpose is to provide readers a quick glimpse to the models, problems, and techniques in this area, and hopefully inspire new contributions .

Approximation Algorithms for Dynamic Assortment Optimization Models

We consider a single-period joint assortment and inventory planning problem with stochastic demand and dynamic substitution across products, motivated by inventory management applications in highly differentiated markets, such as retailing, airlines, and consumer electronics. In its utmost generality, this problem is extremely hard to approximate with respect to various parameters. In fact, prior to the present paper, only a handful of modeling approaches were known to admit efficient algorithms with provably-good performance guarantees.Our main contribution is to provide the first polynomial-time algorithms with provable approximation guarantees for a broad class of dynamic assortment optimization models that capture real-life purchasing behaviors. From a technical perspective, we introduce a number of novel algorithmic ideas of independent interest, possibly applicable in broader settings, and unravel hidden relations to submodular maximization. In addition, our algorithms employ a mixture of greedy procedures, efficient enumerations methods, and low-dimensional dynamic programs, that are suitable for solving instances of practical nature and scale.