Marek Adamczyk
Motivated by applications in online dating and kidney exchange, the stochastic matching problem was introduced by Chen, Immorlica, Karlin, Mahdian and Rudra (2009). They have proven a 4-approximation of a simple greedy strategy, but conjectured that it is in fact a 2-approximation. In this paper we confirm this hypothesis.
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We propose a new greedy algorithm for the maximum cardinality matching problem. We give experimental evidence that this algorithm is likely to find a maximum matching in random graphs with constant expected degree c>0, independent of the value of c. This is contrary to the behavior of commonly used greedy matching heuristics which are known to have some range of c where they probably fail to compute a maximum matching.
Matchings via Random Greedy Independent Set: A Simpler Algorithm and Analysis
本文提出一种简化的随机贪心独立集扩展算法，用于求解最大匹配问题，其分析无需涉及分数匹配，能更简洁直接地证明近似比。
We consider the problem of approximating a maximum weighted matching, when the edges of an underlying weighted graph $G(V,E)$ are revealed in a streaming fashion. We analyze a variant of the previously best-known $(4+ε)$-approximation algorithm due to Crouch and Stubbs (APPROX, 2014), and prove their conjecture that it achieves a tight approximation factor of $3.5+ε$. The algorithm splits the stream into substreams on which it runs a greedy maximum matching algorithm. At the end of the stream, the selected edges are given as input to an optimal maximum weighted matching algorithm. To analyze t…
Optimal Analysis of Greedy for Stochastic Online Euclidean Matching
本研究分析随机在线欧几里得匹配中贪心算法的竞争比，证明除二维外均为常数，二维为$\Theta(\sqrt{\log n})$，通过环面分析转移至立方体。
We study sublinear time algorithms for estimating the size of maximum matching in graphs. Our main result is a $(\frac{1}{2}+Ω(1))$-approximation algorithm which can be implemented in $O(n^{1+ε})$ time, where $n$ is the number of vertices and the constant $ε> 0$ can be made arbitrarily small. The best known lower bound for the problem is $Ω(n)$, which holds for any constant approximation. Existing algorithms either obtain the greedy bound of $\frac{1}{2}$-approximation [Behnezhad FOCS'21], or require some assumption on the maximum degree to run in $o(n^2)$-time [Yoshida, Yamamoto, and Ito STOC…
We study the average performance of online greedy matching algorithms on $G(n,n,p)$, the random bipartite graph with $n$ vertices on each side and edges occurring independently with probability $p=p(n)$. In the online model, vertices on one side of the graph are given up front while vertices on the other side arrive sequentially; when a vertex arrives its edges are revealed and it must be immediately matched or dropped. We begin by analyzing the \textsc{oblivious} algorithm, which tries to match each arriving vertex to a random neighbor, even if the neighbor has already been matched. The algor…
We perform an experimental study of algorithms for online bipartite matching under the known i.i.d. input model with integral types. In the last decade, there has been substantial effort in designing complex algorithms with the goal of improving worst-case approximation ratios. Our goal is to determine how these algorithms perform on more practical instances rather than worst-case instances. In particular, we are interested in whether the ranking of the algorithms by their worst-case performance is consistent with the ranking of the algorithms by their average-case/practical performance. We ar…
We consider the maximum bipartite matching problem in stochastic settings, namely the query-commit and price-of-information models. In the query-commit model, an edge e independently exists with probability $p_e$. We can query whether an edge exists or not, but if it does exist, then we have to take it into our solution. In the unweighted case, one can query edges in the order given by the classical online algorithm of Karp, Vazirani, and Vazirani to get a $(1-1/e)$-approximation. In contrast, the previously best known algorithm in the weighted case is the $(1/2)$-approximation achieved by the…
There has been a long history for studying randomized greedy matching algorithms since the work by Dyer and Frieze~(RSA 1991). We follow this trend and consider the problem formulated in the oblivious setting, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. We revisit the \textsf{Modified Randomized Greedy (MRG)} algorithm by Aronson et al.~(RSA 1995) that is proved to be $(0.5+ε)$-approximate. In particular, we study a weaker version of the algorithm named \textsf{Random Decision Order (RDO)} that in each step, randomly picks an unmatched ver…
Since Tinhofer proposed the MinGreedy algorithm for maximum cardinality matching in 1984, several experimental studies found the randomized algorithm to perform excellently for various classes of random graphs and benchmark instances. In contrast, only few analytical results are known. We show that MinGreedy cannot improve on the trivial approximation ratio 1/2 whp., even for bipartite graphs. Our hard inputs seem to require a small number of high-degree nodes. This motivates an investigation of greedy algorithms on graphs with maximum degree D: We show that MinGreedy achieves a (D-1)/(2D-3)-a…
We study adaptive greedy algorithms for the problems of stochastic set cover with perfect and imperfect coverages. In stochastic set cover with perfect coverage, we are given a set of items and a ground set B. Evaluating an item reveals its state which is a random subset of B drawn from the state distribution of the item. Every element in B is assumed to be present in the state of some item with probability 1. For this problem, we show that the adaptive greedy algorithm has an approximation ratio of H(|B|), the |B|th Harmonic number. In stochastic set cover with imperfect coverage, an element …
The Power of Greedy for Online Minimum Cost Matching on the Line
本文研究直线上在线最小费用匹配中贪心算法的性能，证明在随机请求和服务器下其常数竞争，并扩展到服务器过剩和半随机模型，给出紧下界。
Motivated by sequential budgeted allocation problems, we investigate online matching problems where connections between vertices are not i.i.d., but they have fixed degree distributions -- the so-called configuration model. We estimate the competitive ratio of the simplest algorithm, GREEDY, by approximating some relevant stochastic discrete processes by their continuous counterparts, that are solutions of an explicit system of partial differential equations. This technique gives precise bounds on the estimation errors, with arbitrarily high probability as the problem size increases. In partic…
We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs. We show that for $r$-regular graphs with $n$ nodes and girth at least $g$, the algorithm finds an independent set of expected cardinality $f(r)n - O\big(\frac{(r-1)^{\frac{g}{2}}}{\frac{g}{2}!} n\big)$, where $f(r)$ is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type f…
A Unified Framework for Analysis of Randomized Greedy Matching Algorithms
本文提出统一框架分析随机贪心匹配算法，改进Ranking和FRanking的近似比，同时为无短奇环图提供更优近似比。
Semi-Streaming Matching in a Single Pass II: Greedy is Optimal
研究最大匹配问题的单遍半流算法近似率，证明无算法能超二分之一近似，表明朴素贪心算法最优，还通过“蓝图框架”给出构造及结果，解决了带抢占在线匹配最优竞争率的问题。
Submodular maximization with a cardinality constraint can model various problems, and those problems are often very large in practice. For the case where objective functions are monotone, many fast approximation algorithms have been developed. The stochastic greedy algorithm (SG) is one such algorithm, which is widely used thanks to its simplicity, efficiency, and high empirical performance. However, its approximation guarantee has been proved only for monotone objective functions. When it comes to non-monotone objective functions, existing approximation algorithms are inefficient relative to …
We introduce a new random input model for bipartite matching which we call the Random Type Poisson Arrival Model. Just like in the known i.i.d. model (introduced by Feldman et al. 2009), online nodes have types in our model. In contrast to the adversarial types studied in the known i.i.d. model, following the random graphs studied in Mastin and Jaillet 2016, in our model each type graph is generated randomly by including each offline node in the neighborhood of an online node with probability $c/n$ independently. In our model, nodes of the same type appear consecutively in the input and the nu…
Potential-Based Greedy Matching for Dynamic Delivery Pooling
本文提出基于潜力的贪心算法，用于动态配送池化，通过优化即时节省和潜力计算，有效提升配送效率并减少计算成本。
Parallel Greedy Spanners
本文研究了并行贪心算法构造图的 $t$-spanner，证明即使并行添加匹配边，也能保证 spanner 的稀疏性，并利用长度受限扩展器分解的新进展进行分析。
A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $π$ over $V$, the other player imposes a permutation $σ$ over $U$. In the greedy matching algorithm, vertices of $U$ arrive in order $σ$ and each vertex is matched to the lowest (under $π$) yet unmatched neighbor in $V$ (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals $π$, and the second (min) player responds with the wo…
In this paper, we explicitly study the online vertex cover problem, which is a natural generalization of the well-studied ski-rental problem. In the online vertex cover problem, we are required to maintain a monotone vertex cover in a graph whose vertices arrive online. When a vertex arrives, all its incident edges to previously arrived vertices are revealed to the algorithm. For bipartite graphs with the left vertices offline (i.e. all of the left vertices arrive first before any right vertex), there are algorithms achieving the optimal competitive ratio of $\frac{1}{1-1/e}\approx 1.582$. Our…
Greedy Dynamic Matching
针对带放弃行为的动态匹配市场模型，提出基于新颖线性规划$LP^{ALG}$的贪心策略，证明其在两种设置下竞争比达最优1/2，改进现有下界并优化全知策略目标值上界。
Most practical scheduling applications involve some uncertainty about the arriving times and lengths of the jobs. Stochastic online scheduling is a well-established model capturing this. Here the arrivals occur online, while the processing times are random. For this model, Gupta, Moseley, Uetz, and Xie recently devised an efficient policy for non-preemptive scheduling on unrelated machines with the objective to minimize the expected total weighted completion time. We improve upon this policy by adroitly combining greedy job assignment with $α_j$-point scheduling on each machine. In this way we…
In the design of greedy algorithms for the maximum cardinality matching problem the utilization of degree information when selecting the next edge is a well established and successful approach. We define the class of "degree sensitive" greedy matching algorithms, which allows us to analyze many well-known heuristics, and provide tight approximation guarantees under worst case tie breaking. We exhibit algorithms in this class with optimal approximation guarantee for bipartite graphs. In particular the Karp-Sipser algorithm, which picks an edge incident with a degree-1 node if possible and other…
We consider the MINGREEDY strategy for Maximum Cardinality Matching. MINGREEDY repeatedly selects an edge incident with a node of minimum degree. For graphs of degree at most $Δ$ we show that MINGREEDY achieves approximation ratio at least $ \frac{Δ-1}{2Δ-3} $ in the worst case and that this performance is optimal among adaptive priority algorithms in the vertex model, which include many prominent greedy matching heuristics. Even when considering expected approximation ratios of randomized greedy strategies, no better worst case bounds are known for graphs of small degrees.
The greedy sequential algorithm for maximal independent set (MIS) loops over the vertices in arbitrary order adding a vertex to the resulting set if and only if no previous neighboring vertex has been added. In this loop, as in many sequential loops, each iterate will only depend directly on a subset of the previous iterates (i.e. knowing that any one of a vertices neighbors is in the MIS or knowing that it has no previous neighbors is sufficient to decide its fate). This leads to a dependence structure among the iterates. If this structure is shallow then running the iterates in parallel whil…
We study the online maximum matching problem in a model in which the edges are associated with a known recourse parameter $k$. An online algorithm for this problem has to maintain a valid matching while edges of the underlying graph are presented one after the other. At any moment the algorithm can decide to include an edge into the matching or to exclude it, under the restriction that at most $k$ such actions per edge take place, where $k$ is typically a small constant. This problem was introduced and studied in the context of general online packing problems with recourse by Avitabile et al. …
Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by [Gamlath et al. FOCS 2019], who showed that no online policy is better than the straightforward greedy algorithm, i.e., no online algorithm has a worst-case competitive ratio better than $0.5$. In this work, we consider the bipartite matching problem with edge arrivals in a natural stochastic framework, i.e., Bayesian setting where each edge of the graph is independently realized according to a known probability distribution. We focus on a natural class of prune & greed…
Setcover greedy algorithm is a natural approximation algorithm for test set problem. This paper gives a precise and tighter analysis of performance guarantee of this algorithm. The author improves the performance guarantee $2\ln n$ which derives from set cover problem to $1.1354\ln n$ by applying the potential function technique. In addition, the author gives a nontrivial lower bound $1.0004609\ln n$ of performance guarantee of this algorithm. This lower bound, together with the matching bound of information content heuristic, confirms the fact information content heuristic is slightly better …
In this paper we prove the efficacy of a simple greedy algorithm for a finite horizon online resource allocation/matching problem, when the corresponding static planning linear program (SPP) exhibits a non-degeneracy condition called the general position gap (GPG). The key intuition that we formalize is that the solution of the reward maximizing SPP is the same as a feasibility Linear Program restricted to the optimal basic activities, and under GPG this solution can be tracked with bounded regret by a greedy algorithm, i.e., without the commonly used technique of periodically resolving the SP…
Optimality of Non-Adaptive Algorithms in Online Submodular Welfare Maximization with Stochastic Outcomes
本文证明在随机结果下的在线子模福利最大化问题中，非自适应贪心算法在所有多项式时间算法（包括自适应算法）中达到最优竞争比，除非NP=RP，并引入一种将确定性结果推广到随机环境的技术。
We study the greedy-based online algorithm for edge-weighted matching with (one-sided) vertex arrivals in bipartite graphs, and edge arrivals in general graphs. This algorithm was first studied more than a decade ago by Korula and Pál for the bipartite case in the random-order model. While the weighted bipartite matching problem is solved in the random-order model, this is not the case in recent and exciting online models in which the online player is provided with a sample, and the arrival order is adversarial. The greedy-based algorithm is arguably the most natural and practical algorithm to…
Cost-efficient compressive sensing is challenging when facing large-scale data, {\em i.e.}, data with large sizes. Conventional compressive sensing methods for large-scale data will suffer from low computational efficiency and massive memory storage. In this paper, we revisit well-known solvers called greedy algorithms, including Orthogonal Matching Pursuit (OMP), Subspace Pursuit (SP), Orthogonal Matching Pursuit with Replacement (OMPR). Generally, these approaches are conducted by iteratively executing two main steps: 1) support detection and 2) solving least square problem. To reduce the co…
MapReduce has become the de facto standard model for designing distributed algorithms to process big data on a cluster. There has been considerable research on designing efficient MapReduce algorithms for clustering, graph optimization, and submodular optimization problems. We develop new techniques for designing greedy and local ratio algorithms in this setting. Our randomized local ratio technique gives $2$-approximations for weighted vertex cover and weighted matching, and an $f$-approximation for weighted set cover, all in a constant number of MapReduce rounds. Our randomized greedy techni…
When Stochastic Rewards Reduce to Deterministic Rewards in Online Bipartite Matching
研究带顶点权重的在线二分匹配随机奖励问题，通过将特例归约为确定性设定给出简洁证明，揭示了 Perturbed Greedy 算法在特定相关奖励下的 $(1-1/e)$ 竞争比保证。
Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including $2$-pass streaming algorithms that break the $1/2$-approximation barrier, and $(1-ε)$-approximation streaming algorithms that run in $O(\text{poly} \frac{1}ε)$ passes in bipartite graphs and in $O( (\frac{1}ε)^{\frac{1}ε})$ or $O(\text{poly} (\frac{1}ε) \cdot \log n)$ passes in general graphs, where $n$ is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for ex…
Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x is drawn from a given product distribution, and the goal is to minimize the expected cost. This problem has been studied in Operations Research, where it is known as "sequential testing" of Boolean functions. It has also been studied in learning theory in the context of learning with attribute costs. We consider the general problem of developing approximatio…
We analyze greedy algorithms for the Hierarchical Aggregation (HAG) problem, a strategy introduced in [Jia et al., KDD 2020] for speeding up learning on Graph Neural Networks (GNNs). The idea of HAG is to identify and remove redundancies in computations performed when training GNNs. The associated optimization problem is to identify and remove the most redundancies. Previous work introduced a greedy approach for the HAG problem and claimed a 1-1/e approximation factor. We show by example that this is not correct, and one cannot hope for better than a 1/2 approximation factor. We prove that thi…
Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time step a matching between the servers while maximizing the total satisfied demand over time. The crux of this scheduling problem is that once one shifts from one matching to a different one a fixed delay $δ$ is incurred during which no data can be transmitted. For the offline version of the problem we present a $(1-\frac{1}{e}-ε)$ approximation ratio (for any c…
A basic combinatorial online resource allocation problem is considered, where multiple servers have individual capacity constraints, and at each time slot, a set of jobs arrives, that have potentially different weights to different servers. At each time slot, a one-to-one matching has to be found between jobs and servers, subject to individual capacity constraints, in an online manner. The objective is to maximize the aggregate weight of jobs allotted to servers, summed across time slots and servers, subject to individual capacity constraints. This problem generalizes the well known adwords pr…
Within the context of stochastic probing with commitment, we consider the online stochastic matching problem; that is, the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching (if possible). We consider the competitiveness of online algorithms in both the adversarial order model (AOM) and the random order model (ROM). More specifically, we consider a bipartite stochastic graph $G = (U,V,E)$ …
Fast Stochastic Greedy Algorithm for $k$-Submodular Cover Problem
针对$k$-子模覆盖问题现有算法近似保证弱且查询复杂度高的问题，提出快速随机贪心算法，在实现强双标准近似的同时大幅降低查询复杂度与函数评估次数，适用于大规模AI应用。
We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2-competitive for welfare maximization with coverage valuations, unless $NP = RP$. Since the Greedy algorithm is known to be 1/2-competitive for monotone submodular valuations, of which coverage is a special case, this proves that Greedy provides the optimal competitive ratio. On the other hand, we prove that Greedy in a stochastic setting with i.i.d.items and valuations satisfying diminishing returns is $(1-1/e)$-competitive, which is optimal even for coverage valuations, unless $NP=RP$. For o…
We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is $1/2$, which is achieved by any greedy algorithm. Dürr et al. recently presented a $2$-pass algorithm called Category-Advice that achieves approximation ratio $3/5$. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the $k$-pass Category-Advice algorithm for all $k \ge 1$, and show that …
The maximization for the independence systems defined on graphs is a generalization of combinatorial optimization problems such as the maximum $b$-matching, the unweighted MAX-SAT, the matchoid, and the maximum timed matching problems. In this paper, we consider the problem under the local oracle model to investigate the global approximability of the problem by using the local approximability. We first analyze two simple algorithms FixedOrder and Greedy for the maximization under the model, which shows that they have no constant approximation ratio. Here algorithms FixedOrder and Greedy apply …
Is it possible to maximize a monotone submodular function faster than the widely used lazy greedy algorithm (also known as accelerated greedy), both in theory and practice? In this paper, we develop the first linear-time algorithm for maximizing a general monotone submodular function subject to a cardinality constraint. We show that our randomized algorithm, STOCHASTIC-GREEDY, can achieve a $(1-1/e-\varepsilon)$ approximation guarantee, in expectation, to the optimum solution in time linear in the size of the data and independent of the cardinality constraint. We empirically demonstrate the ef…
Sparse representation of a single measurement vector (SMV) has been explored in a variety of compressive sensing applications. Recently, SMV models have been extended to solve multiple measurement vectors (MMV) problems, where the underlying signal is assumed to have joint sparse structures. To circumvent the NP-hardness of the $\ell_0$ minimization problem, many deterministic MMV algorithms solve the convex relaxed models with limited efficiency. In this paper, we develop stochastic greedy algorithms for solving the joint sparse MMV reconstruction problem. In particular, we propose the MMV St…
We consider Online Minimum Bipartite Matching under the uniform metric. We show that Randomized Greedy achieves a competitive ratio equal to $(1+1/n) (H_{n+1}-1)$, which matches the lower bound. Comparing with the fact that RG achieves an optimal ratio of $Θ(\ln n)$ for the same problem but under the adversarial order, we find that the weaker arrival assumption of random order doesn't offer any extra algorithmic advantage for RG, or make the model strictly more tractable.
Motivated by an application in kidney exchange, we study the following query-commit problem: we are given the set of vertices of a non-bipartite graph G. The set of edges in this graph are not known ahead of time. We can query any pair of vertices to determine if they are adjacent. If the queried edge exists, we are committed to match the two endpoints. Our objective is to maximize the size of the matching. This restriction in the amount of information available to the algorithm constraints us to implement myopic, greedy-like algorithms. A simple deterministic greedy algorithm achieves a facto…
We present a unified framework for designing and analyzing algorithms for online budgeted allocation problems (including online matching) and their generalization, the Online Generalized Assignment Problem (OnGAP). These problems have been intensively studied as models of how to allocate impressions for online advertising. In contrast to previous analyses of online budgeted allocation algorithms (the so-called "balance" or "water-filling" family of algorithms) our analysis is based on the method of randomized dual fitting, analogous to the recent analysis of the RANKING algorithm for online ma…
We study the classic online bipartite matching problem with a twist: offline vertices, called resources, are $\textit{reusable}$. In particular, when a resource is matched to an online vertex it is unavailable for a deterministic time duration $d$ after which it becomes available again for a re-match. Thus, a resource can be matched to many different online vertices over a period of time. While recent work on the problem have resolved the asymptotic case where we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of $\textit{unit inventory}…
The Semi-Oblivious Cup Game: an Imperfect Information Setting
本文研究了贪心算法在杯游戏中的性能，证明其积压量下界为2.076，推翻了之前的猜想，并提出了一种混合算法在三种不同设置中均实现渐进最优性能。
Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of $2$ of the naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree $Δ= O(\log n)$, which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or…
Greedy-Based Online Fair Allocation with Adversarial Input: Enabling Best-of-Many-Worlds Guarantees
本文研究对抗性在线公平分配，分析 PACE 与整数贪心算法，证明在自然输入限制下二者可同时获得嫉妒与 Nash 福利的渐近保证，从而补全 PACE 的多世界最优性能。
Improved Approximation for Unsplittable CVRP via a Greedy Approach
针对不可分割容量约束车辆路径问题，提出平均贪心算法，结合行程划分与匹配，实现多项式时间3.159-近似。
Real-world problems such as ad allocation and matching have been extensively studied under the lens of combinatorial optimization. In several applications, uncertainty in the input appears naturally and this has led to the study of online stochastic optimization models for such problems. For the offline case, these constrained combinatorial optimization problems have been extensively studied, and Contention Resolution Schemes (CRSs), introduced by Chekuri, Vondrák, and Zenklusen, have emerged in recent years as a general framework to obtaining a solution. The idea behind a CRS is to first obta…
We design new serial and parallel approximation algorithms for computing a maximum weight $b$-matching in an edge-weighted graph with a submodular objective function. This problem is NP-hard; the new algorithms have approximation ratio $1/3$, and are relaxations of the Greedy algorithm that rely only on local information in the graph, making them parallelizable. We have designed and implemented Local Lazy Greedy algorithms for both serial and parallel computers. We have applied the approximate submodular $b$-matching algorithm to assign tasks to processors in the computation of Fock matrices i…
The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. We begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed is chosen uniformly at random from the collection of all remaining triangles. This stochastic process terminates once it arrives at a triangle-free graph. In this note we show that with high probability the number of edges in the final graph is at most $ O\big( n^{7/4}\log^{5/4}n \big) $.
It is known that greedy methods perform well for maximizing monotone submodular functions. At the same time, such methods perform poorly in the face of non-monotonicity. In this paper, we show - arguably, surprisingly - that invoking the classical greedy algorithm $O(\sqrt{k})$-times leads to the (currently) fastest deterministic algorithm, called Repeated Greedy, for maximizing a general submodular function subject to $k$-independent system constraints. Repeated Greedy achieves $(1 + O(1/\sqrt{k}))k$ approximation using $O(nr\sqrt{k})$ function evaluations (here, $n$ and $r$ denote the size o…
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