Matroid Constraint Research Articles

Submodular Optimization for Coupled Task Allocation and Intermittent Deployment Problems

In this paper, we demonstrate a formulation for optimizing coupled submodular maximization problems with provable sub-optimality bounds. In robotics applications, it is quite common that optimization problems are coupled with one another and therefore cannot be solved independently. Specifically, we consider two problems coupled if the outcome of the first problem affects the solution of a second problem that operates over a longer time scale. For example, in our motivating problem of environmental monitoring, we posit that multi-robot task allocation will potentially impact environmental dynamics and thus influence the quality of future monitoring, here modeled as a multi-robot intermittent deployment problem. The general theoretical approach for solving this type of coupled problem is demonstrated through this motivating example. Specifically, we propose a method for solving coupled problems modeled by submodular set functions with matroid constraints. A greedy algorithm for solving this class of problem is presented, along with sub-optimality guarantees. Finally, practical optimality ratios are shown through Monte Carlo simulations to demonstrate that the proposed algorithm can generate near-optimal solutions with high efficiency.

On packing spanning arborescences with matroid constraint

AbstractLet be a digraph with a designated root vertex . Edmonds’ seminal result (see J. Edmonds [4]) implies that has a packing of spanning ‐arborescences if and only if has a packing of ‐paths for all , where a packing means arc‐disjoint subgraphs. Let be a matroid on the set of arcs leaving . A packing of ‐paths is called ‐based if their arcs leaving form a base of while a packing of ‐arborescences is called ‐based if, for all , the packing of ‐paths provided by the arborescences is ‐based. Durand de Gevigney, Nguyen, and Szigeti proved in [3] that has an ‐based packing of ‐arborescences if and only if has an ‐based packing of ‐paths for all . Bérczi and Frank conjectured that this statement can be strengthened in the sense of Edmonds’ theorem such that each ‐arborescence is required to be spanning. Specifically, they conjectured that has an ‐based packing of spanning ‐arborescences if and only if has an ‐based packing of ‐paths for all . In this paper we disprove this conjecture in its general form and we prove that the corresponding decision problem is NP‐complete. We also prove that the conjecture holds for several fundamental classes of matroids, such as graphic matroids and transversal matroids. For all the results presented in this paper, the undirected counterpart also holds.

Matroid Constrained Fair Allocation Problem

We consider the problem of allocating a set of indivisible goods among a group of homogeneous agents under matroid constraints and additive valuations, in a fair manner. We propose a novel algorithm that computes a fair allocation for instances with additive and identical valuations, even under matroid constraints. Our result provides a computational anchor to the existential result of the fairness notion, called EF1 (envy-free up to one good) by Biswas and Barman in this setting. We further provide examples to show that the fairness notions stronger than EF1 does not always exist in this setting.

Migration as Submodular Optimization

Migration presents sweeping societal challenges that have recently attracted significant attention from the scientific community. One of the prominent approaches that have been suggested employs optimization and machine learning to match migrants to localities in a way that maximizes the expected number of migrants who find employment. However, it relies on a strong additivity assumption that, we argue, does not hold in practice, due to competition effects; we propose to enhance the data-driven approach by explicitly optimizing for these effects. Specifically, we cast our problem as the maximization of an approximately submodular function subject to matroid constraints, and prove that the worst-case guarantees given by the classic greedy algorithm extend to this setting. We then present three different models for competition effects, and show that they all give rise to submodular objectives. Finally, we demonstrate via simulations that our approach leads to significant gains across the board.

Greedy Maximization of Functions with Bounded Curvature under Partition Matroid Constraints

We investigate the performance of a deterministic GREEDY algorithm for the problem of maximizing functions under a partition matroid constraint. We consider non-monotone submodular functions and monotone subadditive functions. Even though constrained maximization problems of monotone submodular functions have been extensively studied, little is known about greedy maximization of non-monotone submodular functions or monotone subadditive functions.
 We give approximation guarantees for GREEDY on these problems, in terms of the curvature. We find that this simple heuristic yields a strong approximation guarantee on a broad class of functions.
 We discuss the applicability of our results to three real-world problems: Maximizing the determinant function of a positive semidefinite matrix, and related problems such as the maximum entropy sampling problem, the constrained maximum cut problem on directed graphs, and combinatorial auction games.
 We conclude that GREEDY is well-suited to approach these problems. Overall, we present evidence to support the idea that, when dealing with constrained maximization problems with bounded curvature, one needs not search for (approximate) monotonicity to get good approximate solutions.

Parametric monotone function maximization with matroid constraints

We study the problem of maximizing an increasing function $$f:2^N\rightarrow \mathcal {R}_{+}$$ subject to matroid constraints. Gruia Calinescu, Chandra Chekuri, Martin Pal and Jan Vondrak have shown that, if f is nondecreasing and submodular, the continuous greedy algorithm and pipage rounding technique can be combined to find a solution with value at least $$1-1/e$$ of the optimal value. But pipage rounding technique have strong requirement for submodularity. Chandra Chekuri, Jan Vondrak and Rico Zenklusen proposed a rounding technique called contention resolution schemes. They showed that if f is submodular, the objective value of the integral solution rounding by the contention resolution schemes is at least $$1-1/e$$ times of the value of the fractional solution. Let $$f:2^N\rightarrow \mathcal {R}_{+}$$ be an increasing function with generic submodularity ratio $$\gamma \in (0,1]$$ , and let $$(N,\mathcal {I})$$ be a matroid. In this paper, we consider the problem $$\max _{S\in \mathcal {I}}f(S)$$ and provide a $$\gamma (1-e^{-1})(1-e^{-\gamma }-o(1))$$ -approximation algorithm. Our main tools are the continuous greedy algorithm and contention resolution schemes which are the first time applied to nonsubmodular functions.

Mobility-Aware Content Placement for Device-to-Device Caching Systems

User mobility has a large effect on optimal content placement in device-to-device (D2D) caching networks. Since a typical user can communicate neighboring users who stay in the D2D communication area of the typical user, the optimal content placement should be changed according to the user mobility. Under consideration of randomness of incoming and outgoing users, we formulate an optimization problem to minimize the average data load of a BS. It is proved that minimization of the average data load of a BS can be transformed to maximization of a monotonic submodular function with a matroid constraint, for which a greedy algorithm can find near-optimal solutions. Moreover, when motions of neighboring users are rapid, the optimal content placement is derived in closed-form, aided by reasonable approximation and relaxation. In the high mobility regime, the optimal content placement is shown to cache partial amounts of the most popular contents.

Maximizing submodular or monotone approximately submodular functions by multi-objective evolutionary algorithms

Evolutionary algorithms (EAs) are a kind of nature-inspired general-purpose optimization algorithm, and have shown empirically good performance in solving various real-word optimization problems. During the past two decades, promising results on the running time analysis (one essential theoretical aspect) of EAs have been obtained, while most of them focused on isolated combinatorial optimization problems, which do not reflect the general-purpose nature of EAs. To provide a general theoretical explanation of the behavior of EAs, it is desirable to study their performance on general classes of combinatorial optimization problems. To the best of our knowledge, the only result towards this direction is the provably good approximation guarantees of EAs for the problem class of maximizing monotone submodular functions with matroid constraints. The aim of this work is to contribute to this line of research. Considering that many combinatorial optimization problems involve non-monotone or non-submodular objective functions, we study the general problem classes, maximizing submodular functions with/without a size constraint and maximizing monotone approximately submodular functions with a size constraint. We prove that a simple multi-objective EA called GSEMO-C can generally achieve good approximation guarantees in polynomial expected running time.

Controlled Islanding via Weak Submodularity

Cascading failures typically occur following a large disturbance in power systems, such as tripping of a generating unit or a transmission line. Such failures can propagate and destabilize the entire power system, potentially leading to widespread outages. One approach to mitigate impending cascading failures is through controlled islanding, in which a set of transmission lines is deliberately tripped to partition the unstable system into several disjoint, internally stable islands. Selecting such a set of transmission lines is inherently a combinatorial optimization problem. Current approaches address this problem in two steps: first classify coherent generators into groups and then separate generator groups into different islands with minimal load-generation imbalance. These methods, however, are based on computationally expensive heuristics that do not provide optimality guarantees. In this paper, we propose a novel approach to controlled islanding based on weak submodularity. Our formulation jointly captures the minimal generator non-coherency and minimal load-generation imbalance in one objective function. We relax the problem to a formulation with bounded submodularity ratio and a matroid constraint, and propose an approximation algorithm which achieves a provable optimality bound on non-coherency and load-generation imbalance. The proposed framework is tested on IEEE 39-bus and 118-bus power systems.

Packing of arborescences with matroid constraints via matroid intersection

Edmonds’ arborescence packing theorem characterizes directed graphs that have arc-disjoint spanning arborescences in terms of connectivity. Later he also observed a characterization in terms of matroid intersection. Since these fundamental results, intensive research has been done for understanding and extending these results. In this paper we shall extend the second characterization to the setting of reachability-based packing of arborescences. The reachability-based packing problem was introduced by Cs. Király as a common generalization of two different extensions of the spanning arborescence packing problem, one is due to Kamiyama, Katoh, and Takizawa, and the other is due to Durand de Gevigney, Nguyen, and Szigeti. Our new characterization of the arc sets of reachability-based packing in terms of matroid intersection gives an efficient algorithm for the minimum weight reachability-based packing problem, and it also enables us to unify further arborescence packing theorems and Edmonds’ matroid intersection theorem. For the proof, we also show how a new class of matroids can be defined by extending an earlier construction of matroids from intersecting submodular functions to bi-set functions based on an idea of Frank.

Matroidal Choice Functions

In some game-theoretic models, an agent is supposed to choose a subset of available items under a matroid constraint. If an agent always chooses a subset according to the standard greedy algorithm for matroids, then this choice rule fulfills the “substitutability," an essential property for the existence of equilibria in matching market models. In this paper, we introduce a notion of “matroidal choice functions" to capture the entire class of substitutable choice rules under a matroid constraint. For such functions, we provide two characterizations: one is by the behavior of an online greedy algorithm, and the other is by a local condition. We also show that matroidal choice functions extend choice rules defined by the maximization algorithm for valuated matroids and discrete concave functions.

Resilient Active Target Tracking With Multiple Robots

The problem of target tracking with multiple robots consists of actively planning the motion of the robots to track the targets. A major challenge for practical deployments is to make the robots resilient to failures. In particular, robots may be attacked in adversarial scenarios, or their sensors may fail or get occluded. In this letter, we introduce planning algorithms for multi-target tracking that are resilient to such failures. In general, resilient target tracking is computationally hard. Contrary to the case where there are no failures, no scalable approximation algorithms are known for resilient target tracking when the targets are indistinguishable, or unknown in number, or with unknown motion model. In this letter, we provide the first such algorithm, which also has the following properties: First, it achieves maximal resiliency, since the algorithm is valid for any number of failures. Second, it is scalable, as our algorithm terminates with the same running time as state-of-the-art algorithms for (non-resilient) target tracking. Third, it provides provable approximation bounds on the tracking performance, since our algorithm guarantees a solution that is guaranteed to be close to the optimal. We quantify our algorithm's approximation performance using a novel notion of curvature for monotone set functions subject to matroid constraints. Finally, we demonstrate the efficacy of our algorithm through MATLAB and Gazebo simulations and a sensitivity analysis; we focus on scenarios that involve a known number of distinguishable targets.

Pareto Stable Matchings under One-Sided Matroid Constraints

The Pareto stability is one of the solution concepts in two-sided matching markets with ties. It is known that there always exists a Pareto stable matching in the many-to-many setting. In this pape...

Unit Cost Buyback Problem

In this paper, we study the unit cost buyback problem, i.e., the buyback problem with a fixed cancellation cost for each canceled element. The input of the problem is a sequence of elements e1, e2, . . . , en where each element ei has a weight w(ei). We assume that the weights are in a known range [l, u], i.e., l ≤ w(ei) ≤ u for any i. Given the i th element ei, we either accept ei or reject it with no cost, where we can keep a set of elements that satisfies a certain constraint. In order to accept a new element ei, we can cancel some previously selected elements at a cost which is proportional to the number of elements canceled. Our goal is to maximize the profit, i.e., the total weights of elements accepted (and not canceled) minus the total cancellation cost occurred. We construct optimal online algorithms and prove that they are the best possible when the constraint is a matroid constraint or the unweighted knapsack constraint.

Constrained submodular maximization via greedy local search

We present a simple combinatorial 1−e−22-approximation algorithm for maximizing a monotone submodular function subject to a knapsack and a matroid constraint. This classic problem is known to be hard to approximate within factor better than 1−1∕e. We extend the algorithm to yield 1−e−(k+1)k+1 approximation for submodular maximization subject to a single knapsack and k matroid constraints, for any fixed k>1.Our algorithms, which combine the greedy algorithm of Khuller et al. (1999) and Sviridenko (2004) with local search, show the power of this natural framework in submodular maximization with combined constraints.

QoE-driven social aware caching placement for terrestrial-satellite networks

In remote terrestrial-satellite networks, caching is a very promising technique to alleviate the burden of space cloudlet (e.g., cache-enabled satellite user terminal) and to improve subscribers' quality of experience (QoE) in terms of buffering delay and achievable video streaming rate. In this paper, we studied a QoE-driven caching placement optimization problem for video streaming that takes into account the required video streaming rate and the social relationship among users. Social ties between users are used to designate a set of helpers with caching capability, which can cache popular files pro-actively when the cloudlet is idle. We model the utility function of QoE as a logarithmic function. Then, the caching placement problem is formulated as an optimization problem to maximize the user's average QoE subject to the storage capacity constraints of the helpers and the cloudlets. Furthermore, we reformulate the problem into a monotone submodular optimization problem with a partition matroid constraint, and an efficient greedy algorithm with 1 − 1/6 approximation ratio is proposed to solve it. Simulation results show that the proposed caching placement approach significantly outperforms the traditional approaches in terms of QoE, while yields about the same delay and hit ratio performance compare to the delay-minimized scheme.

Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

We study the revenue maximization problem of a seller withnheterogeneous items for sale to a single buyer whose valuation function for sets of items is unknown and drawn from some distributionD. We show that ifDis a distribution over subadditive valuations with independent items, then the better of pricing each item separately or pricing only the grand bundle achieves a constant-factor approximation to the revenue of the optimal mechanism. This includes buyers who arek-demand, additive up to a matroid constraint, or additive up to constraints of any downward-closed set system (and whose values for the individual items are sampled independently), as well as buyers who are fractionally subadditive with item multipliers drawn independently. Our proof makes use of the core-tail decomposition framework developed in prior work showing similar results for the significantly simpler class of additive buyers.In the second part of the article, we develop a connection between approximately optimal simple mechanisms and approximate revenue monotonicity with respect to buyers’ valuations. Revenue non-monotonicity is the phenomenon that sometimes strictlyincreasingbuyers’ values for every set can strictlydecreasethe revenue of the optimal mechanism. Using our main result, we derive a bound on how bad this degradation can be (and dub such a bound a proof ofapproximaterevenue monotonicity); we further show that better bounds on approximate monotonicity imply a better analysis of our simple mechanisms.

Online Submodular Maximization with Free Disposal

We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k -uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches 1/α ∞ ≈ 0.3178 as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015), and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k -uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1/α ∞ ≈ 0.3178.

Inter-Cluster Cooperation for Wireless D2D Caching Networks

Proactive wireless caching and device to device (D2D) communication have emerged as promising techniques for enhancing users’ quality of service and network performance. In this paper, we propose a new architecture for D2D caching with inter-cluster cooperation. We study a cellular network in which users cache popular files and share them with other users either in their proximity via D2D communication or with remote users using cellular transmission. We characterize the network average delay per request from a queuing perspective. Specifically, we formulate the delay minimization problem and show that it is NP-hard. Furthermore, we prove that the delay minimization problem is equivalent to the minimization of a non-increasing monotone supermodular function subject to a uniform partition matroid constraint. A computationally efficient greedy algorithm is proposed which is proven to be locally optimal within a factor $(1 - e^{-1})\approx 0.63$ of the optimum. We analyze the average per request throughput for different caching schemes and conduct the scaling analysis for the average sum throughput. We show how throughput scaling depends on video content popularity when the number of files grows asymptotically large. Simulation results show a delay reduction of 45% to 80% compared to a D2D caching system without inter-cluster cooperation.

Improved bounds for the greedy strategy in optimization problems with curvature

Consider the problem of choosing a set of actions to optimize a real-valued polymatroid function subject to matroid constraints. The greedy strategy, an approximate solution, is known to satisfy some bounds in terms of the total curvature. The total curvature depends on function values on sets outside the constraint matroid. If the function is defined only on the matroid, the problem still makes sense, but the existing bounds involving the total curvature do not apply, which is puzzling. This motivates an alternative formulation of such bounds. The first question we address is whether it is possible to extend a polymatroid function defined on a matroid to one on the entire power set. This was recently shown to be negative in general. Here, we provide necessary and sufficient conditions for the existence of an incremental extension of a polymatroid function defined on the uniform matroid of rank k to one with rank $$k+1$$ , together with an algorithm for constructing the extension. Whenever a polymatroid function defined on a matroid can be extended to the entire power set, the bounds involving the total curvature of the extension apply. However, these bounds still depend on sets outside the constraint matroid. Motivated by this, we define a new notion of curvature called partial curvature, involving only sets in the matroid. We derive necessary and sufficient conditions for an extension to have a total curvature equal to the partial curvature. Moreover, we prove that the bounds in terms of the partial curvature are in general improved over the previous ones. We illustrate our results with two contrasting examples motivated by practical problems.