Matroid Constraint Research Articles

Fully Dynamic Submodular Maximization over Matroids

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this significant problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an \({\tilde{O}({k^{2}}{\varepsilon})}\) amortized update time (in the number of insertions and deletions) and yields a \({(4+O(\varepsilon))}\) -approximate solution with respect to the dynamic optimum, where \(k\) is the rank of the matroid.

On submodular prophet inequalities and correlation gap

Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. Rubinstein and Singla [31] developed a notion of combinatorial prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and subadditive functions. For non-negative submodular functions they demonstrated a constant factor prophet inequality for matroid constraints. Along the way they showed a variant of the correlation gap for non-negative submodular functions.In this paper we revisit their notion of correlation gap as well as the standard notion of correlation gap and prove much tighter and cleaner bounds. Via these bounds and other insights we obtain substantially improved constant factor combinatorial prophet inequalities for both monotone and non-monotone submodular functions over any constraint that admits an Online Contention Resolution Scheme. In addition to improved bounds we describe efficient polynomial-time algorithms that achieve these bounds.

Maximizing stochastic set function under a matroid constraint from decomposition

Maximizing stochastic set function under a matroid constraint from decomposition

Solving the Maximum Popular Matching Problem with Matroid Constraints

Solving the Maximum Popular Matching Problem with Matroid Constraints

A Semi-streaming Algorithm for Monotone Regularized Submodular Maximization with a Matroid Constraint

A Semi-streaming Algorithm for Monotone Regularized Submodular Maximization with a Matroid Constraint

Online Submodular Maximization via Online Convex Optimization

We study monotone submodular maximization under general matroid constraints in the online setting. We prove that online optimization of a large class of submodular functions, namely, threshold potential functions, reduces to online convex optimization (OCO). This is precisely because functions in this class admit a concave relaxation; as a result, OCO policies, coupled with an appropriate rounding scheme, can be used to achieve sublinear regret in the combinatorial setting. We also show that our reduction extends to many different versions of the online learning problem, including the dynamic regret, bandit, and optimistic-learning settings.

Improved deterministic algorithms for non-monotone submodular maximization

Submodular maximization is one of the central topics in combinatorial optimization. It has found numerous applications in the real world. In the past decades, a series of algorithms have been proposed for this problem. However, most of the state-of-the-art algorithms are randomized. There remain non-negligible gaps with respect to approximation ratios between deterministic and randomized algorithms in submodular maximization.In this paper, we propose deterministic algorithms with improved approximation ratios for non-monotone submodular maximization. Specifically, for the matroid constraint, we provide a deterministic 0.283−o(1) approximation algorithm, while the previous best deterministic algorithm only achieves a 1/4 approximation ratio. For the knapsack constraint, we provide a deterministic 1/4 approximation algorithm, while the previous best deterministic algorithm only achieves a 1/6 approximation ratio. For the linear packing constraints with large widths, we provide a deterministic 1/6−ϵ approximation algorithm. To the best of our knowledge, there is currently no deterministic approximation algorithm for the constraints.

Exchange of indivisible goods under matroid constraints

We study the problem of reallocating indivisible goods among a set of strategic agents by generalizing the original Sharpley-Scarf market to the setting where an object can be allocated to multiple agents but subject to an associated matroid constraint. We refer to this modified market as the matroid Sharpley-Scarf market. For this general market, we present a Pareto-optimal, individually rational and group strategy-proof mechanism. Our mechanism is simple and natural generalization of the Top Trading Cycle mechanism and the Serial Dictator mechanism. In our analysis, we demonstrate that the weak core may be absent in the matroid Sharpley-Scarf market, while a relaxed version called the constrained core does exist. Furthermore, we extend the concept called competitive equilibrium to the matroid Sharpley-Scarf market and show that an allocation is a competitive equilibrium if and only if it can be produced by our mechanism.

User-Assisted Base Station Caching and Cooperative Prefetching for High-Speed Railway Systems

With the aim of reducing the transmission latency, this paper proposes a scheme of user-assisted base station (BS) caching and cooperative prefetching for high-speed railway (HSR) communications. In the model under consideration, neighboring BSs periodically exchange information, including the coverage area and communication rate, to facilitate content caching and prefetching. As an additional means, users can cache contents that are different from BSs. Specifically, we construct an optimization problem for content caching and prefetching that minimizes the overall transmission latency. Moreover, this problem is decomposed into two subproblems, namely the one aiming to maximize the available throughput of BSs and the other attempting to maximize the user cache utilization subject to the constraint of the BS cache. We demonstrate that the objective function of the first subproblem is equivalent to a monotone submodular function subject to matroid constraints. Therefore, it can be efficiently solved with a greedy algorithm. Meanwhile, the user cache issue can be viewed as a weighted sum maximization problem. Simulation results are presented to verify that the proposed scheme can not only reduce the average transmission latency but also improve the hit probability and system throughput.

Clustering what Matters: Optimal Approximation for Clustering with Outliers

Clustering with outliers is one of the most fundamental problems in Computer Science. Given a set X of n points and two numbers k, m, the clustering with outliers aims to exclude m points from X and partition the remaining points into k clusters that minimizes a certain cost function. In this paper, we give a general approach for solving clustering with outliers, which results in a fixed-parameter tractable (FPT) algorithm in k and m—i.e., an algorithm with running time of the form f(k, m) · nO(1) for some function f—that almost matches the approximation ratio for its outlier-free counterpart. As a corollary, we obtain FPT approximation algorithms with optimal approximation ratios for k-Median and k-Means with outliers in general and Euclidean metrics. We also exhibit more applications of our approach to other variants of the problem that impose additional constraints on the clustering, such as fairness or matroid constraints.

Clustering What Matters: Optimal Approximation for Clustering with Outliers

Clustering with outliers is one of the most fundamental problems in Computer Science. Given a set X of n points and two numbers k and m, the clustering with outliers aims to exclude m points from X, and partition the remaining points into k clusters that minimizes a certain cost function. In this paper, we give a general approach for solving clustering with outliers, which results in a fixed-parameter tractable (FPT) algorithm in k and m (i.e., an algorithm with running time of the form f(k, m) * poly(n) for some function f), that almost matches the approximation ratio for its outlier-free counterpart. As a corollary, we obtain FPT approximation algorithms with optimal approximation ratios for k-Median and k-Means with outliers in general and Euclidean metrics. We also exhibit more applications of our approach to other variants of the problem that impose additional constraints on the clustering, such as fairness or matroid constraints.

Two-stage non-submodular maximization

The sheer size of modern datasets has led to an urgent need for summarization techniques that can identify representative elements of the data set. Fortunately, the vast majority of data summarization tasks satisfy an intuitive diminishing returns condition known as submodularity, which allows us to find nearly-optimal solutions in linear time. However, for many applications in practice, including experimental design and sparse Gaussian processes, the objective is in general not submodular. To solve these optimization problems, an important research method is to describe the characteristics of the non-submodular functions. The non-submodular function is a hot research topic in the study of nonlinear combinatorial optimizations. In this paper, we combine and generalize the curvature and the generic submodularity ratio to design an approximation algorithm for two-stage non-submodular maximization under a matroid constraint.

DCEC: D2D-Enabled Cost-Aware Cooperative Caching in MEC Networks

Various kinds of powerful intelligent mobile devices (MDs) need to access multimedia content anytime and anywhere, which places enormous pressure on mobile wireless networks. Fetching content from remote sources may introduce overly long accessing delays, which will result in a poor quality of experience (QoE). In this article, we considered the advantages of combining mobile/multi-access edge computing (MEC) with device-to-device (D2D) technologies. We propose a D2D-enabled cooperative edge caching (DCEC) architecture to reduce the delay of accessing content. We designed the DCEC caching management scheme through the maximization of a monotone submodular function under matroid constraints. The DCEC scheme includes a proactive cache placement algorithm and a reactive cache replacement algorithm. Thus, we obtained an optimal content caching and content update, which minimized the average delay cost of fetching content files. Finally, simulations compared the DCEC network architecture with the MEC and D2D networks and the DCEC caching management scheme with the least-frequently used and least-recently used scheme. The numerical results verified that the proposed DCEC scheme was effective at improving the cache hit ratio and the average delay cost. Therefore, the users’ QoE was improved.

Distributed strategy selection: A submodular set function maximization approach

Joint utility-maximization problems for multi-agent systems often should be addressed by distributed strategy-selection formulation. Constrained by discrete feasible strategy sets, these problems are broadly formulated as NP-hard combinatorial optimization problems. In many cases, these problems can be cast as constrained submodular set function maximization problems, which also belong to the NP-hard domain of problems. A prominent example is the problem of multi-agent mobile sensor dispatching over a discrete domain. This paper considers a class of submodular optimization problems that consist of maximization of a monotone and submodular set function subject to a partition matroid constraint over a group of networked agents that communicate over a connected undirected graph. We work with the value oracle model. Consequently, the only access of the agents to the utility function is through a black box that returns the utility function value given a specific strategy set. We propose a distributed suboptimal polynomial-time algorithm that enables each agent to obtain its respective strategy via local interactions with its neighboring agents. Our solution is a fully distributed gradient-based algorithm using the submodular set functions’ multilinear extension followed by a distributed stochastic Pipage rounding procedure. This algorithm results in a strategy set that when the team utility function is evaluated at the worst case, the utility function value is in 1c(1−e−c−O(1/T)) of the optimal solution with c being the curvature of the submodular function. An example demonstrates our results.

On maximizing monotone or non-monotone k-submodular functions with the intersection of knapsack and matroid constraints

On maximizing monotone or non-monotone k-submodular functions with the intersection of knapsack and matroid constraints

On Fair Division under Heterogeneous Matroid Constraints

We study fair allocation of indivisible goods among additive agents with feasibility constraints. In these settings, every agent is restricted to get a bundle among a specified set of feasible bundles. Such scenarios have been of great interest to the AI community due to their applicability to real-world problems. Following some impossibility results, we restrict attention to matroid feasibility constraints that capture natural scenarios, such as the allocation of shifts to medical doctors and the allocation of conference papers to referees. We focus on the common fairness notion of envy-freeness up to one good (EF1). Previous algorithms for finding EF1 allocations are either restricted to agents with identical feasibility constraints or allow free disposal of items. An open problem is the existence of EF1 complete allocations among agents who differ both in their valuations and in their feasibility constraints. In this work, we make progress on this problem by providing positive and negative results for several matroid and valuation types. Among other results, we devise polynomial-time algorithms for finding EF1 allocations in the following settings: (i) n agents with heterogeneous (non-identical) binary valuations and partition matroids with heterogeneous capacities; (ii) two agents with heterogeneous additive valuations and partition matroids with heterogeneous capacities; and (iii) three agents with heterogeneous binary valuations and identical base-orderable matroid constraints.

Improved Approximation Algorithms for Matroid and Knapsack Means Problems

Both matroid means and knapsack means are variations of the classic [Formula: see text]-means problem in which we replace the cardinality constraint by matroid constraint or knapsack constraint respectively. In this paper, we give a 64-approximation algorithm for the matroid means problem and a [Formula: see text]-approximation algorithm for the knapsack means problem by using a simpler and more efficient rounding method. We improve previous 304 approximate ratio for the former and 20016 approximate ratio for the latter. In the rounding process, the application of integrality of the intersection of submodular (or matroid) polyhedra provides strong theoretical support. Moreover, we extend this method to matroid means problem with penalties, and give 64 and 880-approximate algorithms for uniform penalties and nonuniform penalties problem.

Resilient Service Provisioning for Edge Computing

We study the problem of resilient service provisioning for edge computing (RSPE), i.e., how to determine a service placement strategy to maximize the expected overall utility by service provisioning, in the presence of uncertain service failures. RSPE is extremely challenging to tackle, because the explicit expression of its objective function is difficult to obtain, and it is a resilient max–min problem subject to knapsack constraints, which is unexplored so far and cannot be addressed by existing resilient optimization techniques. We first explore the potential properties of the implicit objective function, and reveal that it is monotone submodular under certain conditions. We further prove that the knapsack constraints form a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -independence system constraint, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q&gt;0$ </tex-math></inline-formula> is a constant related to the constraints. We propose two novel solutions for the general RSPE and homogeneous case, respectively. First, for the general problem, we propose a “two-step greedy” algorithm achieving a constant approximation ratio within polynomial time. Second, for the homogeneous case where one of the knapsack constraints reduces to a matroid constraint, we propose an improved “first-greedy-then-local search” polynomial time algorithm achieving better approximation ratio than the previous one. Both extensive simulations and field experiments validate the effectiveness of our proposed algorithms.

An adaptive algorithm for maximization of non-submodular function with a matroid constraint

&lt;p style='text-indent:20px;'&gt;In the paper, we design an adaptive algorithm for non-submodular set function maximization over a single matroid constraint. We measure the deviation of the set function from fully submodular with the help of the generic submodularity ratio &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \gamma $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. The adaptivity quantifies the information theoretic complexity of oracle optimization for parallel computations. We propose a new approximation algorithm based on the continuous greedy approach and prove that the algorithm could obtain a fractional solution with approximation ratio &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ (1-e^{-{\gamma}^2}-\mathcal{O}(\epsilon)) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathcal{O}\left(\frac{\log n\log(k/\gamma\epsilon)} {\epsilon^{-2} \log n\log(1/\gamma)-\epsilon^{-1}\log(1-\epsilon)}\right) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; adaptive rounds. We then use the contention resolution schemes to convert the fractional solution to a discrete one with &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \gamma(1-e^{-1})(1-e^{-{\gamma}^2}-\mathcal{O}(\epsilon)) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-approximation.&lt;/p&gt;

Greedy cooperative cache placement for mobile edge networks with user preferences prediction and adaptive clustering

Caching the most likely to be requested content at the base stations in a cooperative manner can facilitate direct content delivery without fetching content from the remote content server in a mobile edge network. This alleviates the user-perceived latency, reduces the burden on backhaul and minimizes the duplicated content transmissions. In addition to static content popularity, it is also important to consider the users’ dynamic behaviour for real-time applications. Unevenly distributed users, heterogeneity of the user preferences, and activity level in large-scale mobile edge networks are important issues that have been addressed in this work. The user preference has been predicted with the long short-term memory (LSTM) to capture the dynamic user behaviour and then users are clustered based on the predicted user preferences. The cache placement problem is formulated as Integer linear programming to maximize the saved delay with deadline and capacity constraints. Further, we devise a greedy algorithm by proving the submodular property of the objective function with matroid constraints, which can achieve (1−1e) optimality with linear computational complexity. Extensive simulation results demonstrate that the proposed cooperative caching mechanism significantly improves the performance in terms of acceleration ratio, hit ratio, and cache utilization compared with existing mechanisms.