Approximation Guarantee 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.

Learning Discrete-Time Major-Minor Mean Field Games

Recent techniques based on Mean Field Games (MFGs) allow the scalable analysis of multi-player games with many similar, rational agents. However, standard MFGs remain limited to homogeneous players that weakly influence each other, and cannot model major players that strongly influence other players, severely limiting the class of problems that can be handled. We propose a novel discrete time version of major-minor MFGs (M3FGs), along with a learning algorithm based on fictitious play and partitioning the probability simplex. Importantly, M3FGs generalize MFGs with common noise and can handle not only random exogeneous environment states but also major players. A key challenge is that the mean field is stochastic and not deterministic as in standard MFGs. Our theoretical investigation verifies both the M3FG model and its algorithmic solution, showing firstly the well-posedness of the M3FG model starting from a finite game of interest, and secondly convergence and approximation guarantees of the fictitious play algorithm. Then, we empirically verify the obtained theoretical results, ablating some of the theoretical assumptions made, and show successful equilibrium learning in three example problems. Overall, we establish a learning framework for a novel and broad class of tractable games.

Solving Satisfiability Modulo Counting for Symbolic and Statistical AI Integration with Provable Guarantees

Satisfiability Modulo Counting (SMC) encompasses problems that require both symbolic decision-making and statistical reasoning. Its general formulation captures many real-world problems at the intersection of symbolic and statistical AI. SMC searches for policy interventions to control probabilistic outcomes. Solving SMC is challenging because of its highly intractable nature (NP^PP-complete), incorporating statistical inference and symbolic reasoning. Previous research on SMC solving lacks provable guarantees and/or suffers from suboptimal empirical performance, especially when combinatorial constraints are present. We propose XOR-SMC, a polynomial algorithm with access to NP-oracles, to solve highly intractable SMC problems with constant approximation guarantees. XOR-SMC transforms the highly intractable SMC into satisfiability problems by replacing the model counting in SMC with SAT formulae subject to randomized XOR constraints. Experiments on solving important SMC problems in AI for social good demonstrate that XOR-SMC outperforms several baselines both in solution quality and running time.

Nonparametric Additive Models for Billion Observations

The nonparametric additive model (NAM) is a widely used nonparametric regression method. Nevertheless, due to the high computational burden, classic statistical techniques for fitting NAMs are not well-equipped to handle massive data with billions of observations. To address this challenge, we develop a scalable element-wise subset selection method, referred to as Core-NAM, for fitting penalized regression spline based NAMs. Specifically, we first propose an approximation of the penalized least squares estimation, based on which we develop an efficient variant of generalized cross-validation (GCV) to select the smoothing parameter and approximate the Bayesian confidence intervals for statistical inference. Theoretically, we show that the proposed estimator approximately minimizes an upper bound of the estimation mean squared error. Moreover, we provide a non-asymptotic approximation guarantee for the proposed estimator and establish the asymptotic optimality of the proposed variant of GCV. Extensive simulations demonstrate the superior accuracy and efficiency of the Core-NAM method. We also apply the proposed method to a total column ozone dataset containing nearly one billion observations, and the results indicate a speed-up by almost a thousand times with comparable performance compared to the full data approach. Supplementary materials for this article are available online.

Minimum Strongly Connected Subgraph Collection in Dynamic Graphs

Real-world directed graphs are dynamically changing, and it is important to identify and maintain the strong connectivity information between nodes, which is useful in numerous applications. Given an input graph G , we study a new problem, minimum strongly connected subgraph collection (MSCSC), which asks for a complete collection of subgraphs, each of which contains a maximal set of nodes that are strongly connected to each other via minimum number of edges in G. MSCSC is NP-hard, and its computation and maintenance are challenging, especially on large-scale dynamic graphs. Thus, we resort to approximate MSCSC with theoretical guarantees. We develop a series of approximate MSCSC methods for both static and dynamic graphs. Specifically, we first develop a static MSCSC method MSC that only needs one scan of the graph G , runs in linear time w.r.t. , the number of edges, and provides rigorous approximation guarantees. Then, based on MSC, we leverage a reduced directed acyclic graph of G to design incremental MSCSC method MSC i with two variants to handle edge insertions efficiently. We further develop MSC d that updates MSCSC under edge deletions by efficiently scanning only locally affected subgraphs. Moreover, to demonstrate the high utility, we conduct two use case studies to apply our MSCSC methods to boost the efficiency of dynamic strongly connected component (SCC) maintenance and dynamic SCC-based reachability index maintenance. Extensive experiments on 8 large graphs, including 3 billion-edge graphs, validate the superior efficiency of our methods.

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.

Conformity-aware adoption maximization in competitive social networks

Influence maximization (IM) problem is an extensively studied problem in social networks. It aims to find a small set of users in the social network to initiate the diffusion process and maximize the expected influence spread. Existing works on conformity-aware IM focus on the interaction between influence and conformity in a single-influence setting and ignore the role of conformity in a competitive and multiple-influence setting. This paper proposes a conformity-aware independent cascade (C-IC) model that considers the competition among multiple influences as well as the role of conformity in a user’s decision-making. It is proved that the adoption of an influence under the C-IC model is monotone and submodular. Meanwhile, we formulate two adoption maximization (AM) problems, O-AM and S-AM, which are both NP-hard. Because estimating the adoption through diffusion simulations is very time-consuming, we propose a reverse adoption estimation (RAE) method based on a reverse multiple influence sampling (RMIS) technology for the C-IC model and integrate it into the D-SSA-fix (Nguyenet al., 2018) framework, DSSA for short, to compute a solution with approximation guarantee. To further boost the performance, we present a fast one-hop adoption estimation (OAE) method and develop a heuristic algorithm based on OAE, called GOAE. Extensive experiments on eight real-world social networks show that the C-IC model is superior to a non-conformity diffusion model and that RAE+DSSA and GOAE are efficient and effective. In most cases, GOAE finds comparable solutions to RAE+DSSA and CELF with less time and memory overhead. GOAE is five to six orders of magnitude faster than CELF and RAE+DSSA is up to three orders of magnitude faster than CELF on NetHEPT. GOAE runs up to four to five orders of magnitude faster than RAE+DSSA with at most two orders of magnitude less memory usage. GOAE is more scalable than RAE+DSSA in terms of the number of seeds and the size of the social network.

How vulnerable is an undirected planar graph with respect to max flow

AbstractWe study the problem of computing the vitality of edges and vertices with respect to the ‐max flow in undirected planar graphs, where the vitality of an edge/vertex is the ‐max flow decrease when the edge/vertex is removed from the graph. This allows us to establish the vulnerability of the graph with respect to the ‐max flow. We give efficient algorithms to compute an additive guaranteed approximation of the vitality of edges and vertices in planar undirected graphs. We show that in the general case high vitality values are well approximated in time close to the time currently required to compute ‐max flow . We also give improved, and sometimes optimal, results in the case of integer capacities. All our algorithms work in space.

Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

Efficient and Accurate SimRank-Based Similarity Joins: Experiments, Analysis, and Improvement

SimRank-based similarity joins, which mainly include threshold-based and top- k similarity joins, are important types of all-pair SimRank queries. Although a line of related algorithms have been proposed recently, they still fall short of providing approximation guarantee and suffer from scalability issues on medium and large graphs. Meanwhile, we also lack an extensive analysis of existing techniques in terms of accuracy and efficiency. Motivated by these challenges, we first conduct detailed analysis of state-of-the-art algorithms and provide additional theoretical results. Second, to address the limitations of existing techniques, we propose simple yet effective algorithm frameworks for both queries to theoretically guarantee the approximation bound, and present a more efficient all-pair algorithm inspired by randomized local push of Personalized PageRank. Next, we analyze the algorithmic complexity of threshold-based and top- k similarity joins by leveraging a reasonable assumption of SimRank distribution. Through extensive experiments, we find that our proposed methods far exceed existing ones with respect to query efficiency, approximation guarantee and practical accuracy, while our theoretical analysis nicely matches the empirical study.

Upper and lower bounds for complete linkage in general metric spaces

In a hierarchical clustering problem the task is to compute a series of mutually compatible clusterings of a finite metric space (P,dist)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(P,{{\\,\ extrm{dist}\\,}})$$\\end{document}. Starting with the clustering where every point forms its own cluster, one iteratively merges two clusters until only one cluster remains. Complete linkage is a well-known and popular algorithm to compute such clusterings: in every step it merges the two clusters whose union has the smallest radius (or diameter) among all currently possible merges. We prove that the radius (or diameter) of every k-clustering computed by complete linkage is at most by factor O(k) (or O(kln(3)/ln(2))=O(k1.59)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(k^{\\ln (3)/\\ln (2)})=O(k^{1{.}59})$$\\end{document}) worse than an optimal k-clustering minimizing the radius (or diameter). Furthermore we give a negative answer to the question proposed by Dasgupta and Long (J Comput Syst Sci 70(4):555–569, 2005. https://doi.org/10.1016/j.jcss.2004.10.006), who show a lower bound of Ω(log(k))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (\\log (k))$$\\end{document} and ask if the approximation guarantee is in fact Θ(log(k))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta (\\log (k))$$\\end{document}. We present instances where complete linkage performs poorly in the sense that the k-clustering computed by complete linkage is off by a factor of Ω(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (k)$$\\end{document} from an optimal solution for radius and diameter. We conclude that in general metric spaces complete linkage does not perform asymptotically better than single linkage, merging the two clusters with smallest inter-cluster distance, for which we prove an approximation guarantee of O(k).

Approximating sparse quadratic programs

Given a matrix A∈Rn×n, we consider the problem of maximizing xTAx subject to the constraint x∈{−1,1}n. This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an Ω(1/lg⁡n) approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an Ω(1) approximation when A corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is O˜(n1.5⋅min⁡{N,n1.5}), where N is the number of nonzero entries in A and O˜ ignores polylogarithmic factors.In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where A is sparse (i.e., has O(n) nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that:•MaxQP admits a (1/2Δ)-approximation in O(nlg⁡n) time, where Δ=O(1) is the maximum degree of the corresponding graph.•Unit MaxQP, where A∈{−1,0,1}n×n, admits a (1/2d)-approximation in O(n) time when the corresponding graph is d-degenerate, and a (1/3δ)-approximation in O(n1.5) time when the corresponding graph has δn edges for δ=O(1).•MaxQP admits a (1−ε)-approximation in O(n) time when the corresponding graph and each of its minors have bounded local treewidth.•Unit MaxQP admits a (1−ε)-approximation in O(n2) time for H-minor free graphs.

Maximizing the Diversity of Exposure in Online Social Networks by Identifying Users with Increased Susceptibility to Persuasion

Individuals may have a range of opinions on controversial topics. However, the ease of making friendships in online social networks tends to create groups of like-minded individuals, who propagate messages that reinforce existing opinions and ignore messages expressing opposite opinions. This creates a situation where there is a decrease in the diversity of messages to which users are exposed ( diversity of exposure ). This means that users do not easily get the chance to be exposed to messages containing alternative viewpoints; it is even more unlikely that they forward such messages to their friends. Increasing the chance that such messages are propagated implies that an individuals’ susceptibility to persuasion is increased, something that may ultimately increase the diversity of messages to which users are exposed. This article formulates a novel problem which aims to identify a small set of users for whom increasing susceptibility to persuasion maximizes the diversity of exposure of all users in the network. We study the properties of this problem and develop a method to find a solution with an approximation guarantee. For this, we first prove that the problem is neither submodular nor supermodular and then we develop submodular bounds for it. These bounds are used in the Sandwich framework to propose a method which approximates the solution using reverse sampling. The proposed method is validated using four real-world datasets. The obtained results demonstrate the superiority of the proposed method compared to baseline approaches.

Efficient Algorithm for Budgeted Adaptive Influence Maximization: An Incremental RR-set Update Approach

Given a graph G, a cost associated with each node, and a budget B, the budgeted influence maximization (BIM) aims to find the optimal set S of seed nodes that maximizes the influence among all possible sets such that the total cost of nodes in S is no larger than B. Existing solutions mainly follow the non-adaptive idea, i.e., determining all the seeds before observing any actual diffusion. Due to the absence of actual diffusion information, they may result in unsatisfactory influence spread. Motivated by the limitation of existing solutions, in this paper, we make the first attempt to solve the BIM problem under the adaptive setting, where seed nodes are iteratively selected after observing the diffusion result of the previous seeds. We design the first practical algorithm which achieves an expected approximation guarantee by probabilistically adopting a cost-aware greedy idea or a single influential node. Further, we develop an optimized version to improve its practical performance in terms of influence spread. Besides, the scalability issues of the adaptive IM-related problems still remain open. It is because they usually involve multiple rounds (e.g., equal to the number of seeds) and in each round, they have to construct sufficient new reverse-reachable set (RR-set) samples such that the claimed approximation guarantee can actually hold. However, this incurs prohibitive computation, imposing limitations on real applications. To solve this dilemma, we propose an incremental update approach. Specifically, it maintains extra construction information when building RR-sets, and then it can quickly correct a problematic RR-set from the very step where it is first affected. As a result, we recycle the RR-sets at a small computational cost, while still providing correctness guarantee. Finally, extensive experiments on large-scale real graphs demonstrate the superiority of our algorithms over baselines in terms of both influence spread and running time.

Communication-aware scheduling of precedence-constrained tasks on related machines

Scheduling precedence-constrained tasks is a classical problem that has been studied for more than fifty years. However, little progress has been made in the setting where there are non-uniform communication delays between tasks. In this work, we propose a new scheduler, Generalized Earliest Time First (GETF), and provide the first provable, worst-case approximation guarantees for the goals of minimizing both the makespan and total weighted completion time of tasks with precedence constraints on related machines with machine-dependent communication speeds.

Learning automata-accelerated greedy algorithms for stochastic submodular maximization

Submodular function maximization is a typical combinatorial optimization problem that arises in many areas of computer science, such as data summarization, batch mode active learning, and recommendations. We focus on the stochastic submodular maximization (SSM) problems, where an unknown monotonic submodular function is to be maximized under cardinality constraints. Previous research commonly relied on the Monte-Carlo method to assess the stochasticity in submodular functions, which is challenged by the accuracy–efficiency dilemma. Achieving high accuracy requires a large number of Monte-Carlo simulations, while efficiency necessitates the minimization of simulation costs. In this paper, we propose a new family of greedy algorithms based on learning automata (LA) for SSM problems. By exploring the solution space more efficiently, the proposed LA-based greedy family accelerates conventional greedy algorithms. Instead of massively evaluating the expected benefits of each element indiscriminately, an LA-based learning strategy is designed to quickly identify the optimal element among all candidates. Based on the strategy, we present a primitive greedy algorithm and two improved variants, along with proofs of the same approximation guarantee as conventional greedy algorithms. We conducted experiments on three representative SSM tasks, nonparametric learning for structured data, exemplar-based clustering for unstructured data, and influence maximization for graph data. The results verified the advantage of the proposed LA-based greedy family against previous greedy algorithms.

Revisiting maximum satisfiability and related problems in data streams

We revisit the maximum satisfiability problem (Max-SAT) in the data stream model. In this problem, the stream consists of m clauses that are disjunctions of literals drawn from n Boolean variables. The objective is to find an assignment to the variables that maximizes the number of satisfied clauses. Chou et al. (FOCS 2020) showed that Ω(n) space is necessary to yield a 2/2+ε approximation of the optimum value; they also presented an algorithm that yields a 2/2−ε approximation of the optimum value using O(ε−2log⁡n) space.In this paper, we not only focus on approximating the optimum value, but also on obtaining the corresponding Boolean assignment using sublinear o(mn) space. We present randomized single-pass algorithms that w.h.p.1 yield:•A 1−ε approximation using O˜(n/ε3) space and exponential post-processing time•A 3/4−ε approximation using O˜(n/ε) space and polynomial post-processing time. Our ideas also extend to dynamic streams. However, we show that the streaming k-SAT problem, which asks whether one can satisfy all size-k input clauses, must use Ω(nk) space.We also consider the related minimum satisfiability problem (Min-SAT), introduced by Kohli et al. (SIAM J. Discrete Math. 1994), that asks to find an assignment that minimizes the number of satisfied clauses. For this problem, we give a O˜(n2/ε2) space algorithm, which is sublinear when m=ω(n), that yields an α+ε approximation where α is the approximation guarantee of the offline algorithm. If each variable appears in at most f clauses, we show that a 2fn approximation using O˜(n) space is possible.Finally, for the Max-AND-SAT problem where clauses are conjunctions of literals, we show that any single-pass algorithm that approximates the optimal value up to a factor better than 1/2 with success probability at least 2/3 must use Ω(mn) space.

The limits of local search for weighted k-set packing

We consider the weighted k-set packing problem, where, given a collection S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {S}}$$\\end{document} of sets, each of cardinality at most k, and a positive weight function w:S→Q>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w:{\\mathcal {S}}\\rightarrow {\\mathbb {Q}}_{>0}$$\\end{document}, the task is to find a sub-collection of S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {S}}$$\\end{document} consisting of pairwise disjoint sets of maximum total weight. As this problem does not permit a polynomial-time o(klogk)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$o(\\frac{k}{\\log k})$$\\end{document}-approximation unless P=NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P=NP$$\\end{document} (Hazan et al. in Comput Complex 15:20–39, 2006. https://doi.org/10.1007/s00037-006-0205-6), most previous approaches rely on local search. For twenty years, Berman’s algorithm SquareImp (Berman, in: Scandinavian workshop on algorithm theory, Springer, 2000. https://doi.org/10.1007/3-540-44985-X_19), which yields a polynomial-time k+12+ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{k+1}{2}+\\epsilon $$\\end{document}-approximation for any fixed ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document}, has remained unchallenged. Only recently, it could be improved to k+12-163,700,993\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{k+1}{2}-\\frac{1}{63,700,993}$$\\end{document} by Neuwohner (38th International symposium on theoretical aspects of computer science (STACS 2021), Leibniz international proceedings in informatics (LIPIcs), 2021. https://doi.org/10.4230/LIPIcs.STACS.2021.53). In her paper, she showed that instances for which the analysis of SquareImp is almost tight are “close to unweighted” in a certain sense. But for the unit weight variant, the best known approximation guarantee is k+13+ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{k+1}{3}+\\epsilon $$\\end{document} (Fürer and Yu in International symposium on combinatorial optimization, Springer, 2014. https://doi.org/10.1007/978-3-319-09174-7_35). Using this observation as a starting point, we conduct a more in-depth analysis of close-to-tight instances of SquareImp. This finally allows us to generalize techniques used in the unweighted case to the weighted setting. In doing so, we obtain approximation guarantees of k+ϵk2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{k+\\epsilon _k}{2}$$\\end{document}, where limk→∞ϵk=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim _{k\\rightarrow \\infty } \\epsilon _k = 0$$\\end{document}. On the other hand, we prove that this is asymptotically best possible in that local improvements of logarithmically bounded size cannot produce an approximation ratio below k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{k}{2}$$\\end{document}.

A single factor approximation ratio algorithm for DR-submodular maximization on integer lattice beyond non-negativity and monotonicity

In this work, we investigate the problem of maximizing nonmonotone DR-submodular function (possibly negative) subject to cardinality constraints on an integer lattice space. We propose a M-Threshold Decrease Algorithm that uses a compensation constant M from a special decomposition h(x)=f(x)−g(x), which achieves a 1−e−δ−ϵ approximation guarantee. To ensure that the algorithm ends in finite time, the supremum of δ is δ(ϵ)=ϵmaxs∈Ω⁡h(χs|0)2kmaxs∈Ω⁡g(χs|0)+ϵmaxs∈Ω⁡h(χs|0). If maxs∈Ω⁡h(χs|0)>2kmaxs∈Ω⁡g(χs|0), there exists ϵ⁎=arg⁡maxϵ∈(0,1)⁡1−e−δ(ϵ)−ϵ such that 1−e−δ(ϵ⁎)−ϵ⁎>0. This implies that there is a valid δ such that 1−e−δ−ϵ>0. To accelerate the algorithm, we employ a random subset to replace the ground set and propose a Random M-Threshold Decrease Algorithm, where δ(ϵ)=ϵrmaxs∈Ω⁡h(χs|0)2nkmaxs∈Ω⁡g(χs|0)+[k(n−r)+ϵr]maxs∈Ω⁡h(χs|0) with r being the size of the random subset and n being the size of the ground set.Furthermore, we demonstrate that, based on the DR-ratio γd and the DS decomposition, the M-Threshold Decrease Algorithm is also suitable for solving the monotone non-DR-submodular maximization problem, achieving a 1−e−γdδ−O(ϵ) approximation guarantee. Because γd is a challenging metric to calculate, we refine the M-Threshold Decrease Algorithm, which can return an estimate of γd.To the best of our knowledge, this is the first time that a single factor approximation ratio has been proposed for the nonmonotone DR-submodular maximization problem (possibly negative) subject to cardinality constraints.

Optimization on the smallest eigenvalue of grounded Laplacian matrix via edge addition

The grounded Laplacian matrix L−S of a graph G=(V,E) with n=|V| nodes and m=|E| edges is a (n−s)×(n−s) submatrix of its Laplacian matrix L, obtained from L by deleting rows and columns corresponding to s=|S|≪n ground nodes forming set S⊂V. The smallest eigenvalue of L−S plays an important role in various practical scenarios, such as characterizing the convergence rate of leader-follower opinion dynamics, with a larger eigenvalue indicating faster convergence of opinion. In this paper, we study the problem of adding k≪n edges among all the nonexistent edges forming the candidate edge set Q=(V×V)﹨E, in order to maximize the smallest eigenvalue of the grounded Laplacian matrix. We show that the objective function of the combinatorial optimization problem is monotone but non-submodular. To solve the problem, we first simplify the problem by restricting the candidate edge set Q to be (S×(V﹨S))﹨E, and prove that it has the same optimal solution as the original problem, although the size of set Q is reduced from O(n2) to O(n). Then, we propose two greedy approximation algorithms. One is a simple greedy algorithm with an approximation ratio (1−e−αγ)/α and time complexity O(kn4), where γ and α are, respectively, submodularity ratio and curvature, whose bounds are provided for some particular cases. The other is a fast greedy algorithm without approximation guarantee, which has a running time O˜(km), where O˜(⋅) suppresses the poly(log⁡n) factors. Numerous experiments on various real networks are performed to validate the superiority of our algorithms, in terms of effectiveness and efficiency.