We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the above-mentioned connections, have multiple implications in the data stream and robust setting.
Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above $1/2$ can be achieved in our model, if only queries to feasible sets are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight $2/3$-approximation taking exponential time, and an efficient $0.514$-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the above-mentioned link to the robust setting, both of these algorithms improve on the current state-of-the-art for robust submodular maximization, showing that approximation factors beyond $1/2$ are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight $1/2+\varepsilon$ hardness result, based on the construction of a new family of coverage functions. This improves on a prior $1-1/e+\varepsilon$ hardness and matches, up to an arbitrarily small margin, the best known approximation algorithm.
Submodularity is one of the most important properties in combinatorial optimization, and $k$-submodularity is a generalization of submodularity. Maximization of a $k$-submodular function requires an exponential number of value oracle queries, and approximation algorithms have been studied. For unconstrained $k$-submodular maximization, Iwata et al. gave randomized $k/(2k-1)$-approximation algorithm for monotone functions, and randomized $1/2$-approximation algorithm for nonmonotone functions. In this paper, we present improved randomized algorithms for nonmonotone functions. Our algorithm give…
While greedy algorithms have long been observed to perform well on a wide variety of problems, up to now approximation ratios have only been known for their application to problems having submodular objective functions $f$. Since many practical problems have non-submodular $f$, there is a critical need to devise new techniques to bound the performance of greedy algorithms in the case of non-submodularity. Our primary contribution is the introduction of a novel technique for estimating the approximation ratio of the greedy algorithm for maximization of monotone non-decreasing functions based on…
The simplex algorithm for linear programming is based on the fact that any local optimum with respect to the polyhedral neighborhood is also a global optimum. We show that a similar result carries over to submodular maximization. In particular, every local optimum of a constrained monotone submodular maximization problem yields a $1/2$-approximation, and we also present an appropriate extension to the non-monotone setting. However, reaching a local optimum quickly is a non-trivial task. Moreover, we describe a fast and very general local search procedure that applies to a wide range of constra…
We present a simple combinatorial $\frac{1 -e^{-2}}{2}$-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 show that the algorithm can be extended to yield a ratio of $\frac{1 - e^{-(k+1)}}{k+1}$ for the problem with a single knapsack and the intersection of $k$ matroid constraints, for any fixed $k > 1$. Our algorithms, which combine the greedy algorithm of [Khuller, Moss and Naor, 1999] and [Sviridenko, 2004] with local search, sho…
We present a polynomial time algorithm for online maximization of $k$-submodular maximization. For online (nonmonotone) $k$-submodular maximization, our algorithm achieves a tight approximate factor in an approximate regret. For online monotone $k$-submodular maximization, our approximate-regret matches to the best-known approximation ratio, which is tight asymptotically as $k$ tends to infinity. Our approach is based on the Blackwell approachability theorem and online linear optimization.
In this paper we consider the problem of maximizing a non-negative submodular function subject to a cardinality constraint in the data stream model. Previously, the best known algorithm for this problem was a $5.828$-approximation semi-streaming algorithm based on a local search technique (Feldman et al., 2018). For the special case of this problem in which the objective function is also monotone, the state-of-the-art semi-streaming algorithm is an algorithm known as Sieve-Streaming, which is based on a different technique (Badanidiyuru, 2014). Adapting the technique of Sieve-Streaming to non-…
Submodularity is one of the most important property of combinatorial optimization, and $k$-submodularity is a generalization of submodularity. Maximization of $k$-submodular function is NP-hard, and approximation algorithms are studied. For monotone $k$-submodular function, [Iwata, Tanigawa, and Yoshida 2016] gave $k/(2k-1)$-approximation algorithm. In this paper, we give a deterministic algorithm by derandomizing that algorithm. Derandomization scheme is from [Buchbinder and Feldman 2016]. Our algorithm is $k/(2k-1)$-approximation and polynomial-time algorithm.
Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject to $d$ knapsack constraints, where $d$ is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through {\em extension by expectation} of the submodular function. Formally, we show that, for any non-negative submodular function, an $α$…
Weak submodularity is a natural relaxation of the diminishing return property, which is equivalent to submodularity. Weak submodularity has been used to show that many (monotone) functions that arise in practice can be efficiently maximized with provable guarantees. In this work we introduce two natural generalizations of weak submodularity for non-monotone functions. We show that an efficient randomized greedy algorithm has provable approximation guarantees for maximizing these functions subject to a cardinality constraint. We then provide a more refined analysis that takes into account that …
Seminal work by Edmonds and Lovasz shows the strong connection between submodularity and convexity. Submodular functions have tight modular lower bounds, and subdifferentials in a manner akin to convex functions. They also admit poly-time algorithms for minimization and satisfy the Fenchel duality theorem and the Discrete Seperation Theorem, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular maximization, though NP hard, admits constant factor approximation guarantees. Concave functions composed with modular …
Despite a surge of interest in submodular maximization in the data stream model, there remain significant gaps in our knowledge about what can be achieved in this setting, especially when dealing with multiple constraints. In this work, we nearly close several basic gaps in submodular maximization subject to $k$ matroid constraints in the data stream model. We present a new hardness result showing that super polynomial memory in $k$ is needed to obtain an $o(k / \log k)$-approximation. This implies near optimality of prior algorithms. For the same setting, we show that one can nevertheless obt…
A $k$-submodular function is an extension of a submodular function in that its input is given by $k$ disjoint subsets instead of a single subset. For unconstrained nonnegative $k$-submodular maximization, Ward and Živný proposed a constant-factor approximation algorithm, which was improved by the recent work of Iwata, Tanigawa and Yoshida presenting a $1/2$-approximation algorithm. Iwata et al. also provided a $k/(2k-1)$-approximation algorithm for monotone $k$-submodular maximization and proved that its approximation ratio is asymptotically tight. More recently, Ohsaka and Yoshida proposed co…
In this work, we consider robust submodular maximization with matroid constraints. We give an efficient bi-criteria approximation algorithm that outputs a small family of feasible sets whose union has (nearly) optimal objective value. This algorithm theoretically performs less function calls than previous works at cost of adding more elements to the final solution. We also provide significant implementation improvements showing that our algorithm outperforms the algorithms in the existing literature. We finally assess the performance of our contributions in three real-world applications.
Data-dependent Evaluations for Budgeted Submodular Maximization
College of Computing and Data Science, Nanyang Technological University, Singapore（新加坡南洋理工大学计算与数据科学学院）；Data Science and Analytics Thrust, The Hong Kong University of Science and Technology (Guangzhou), China（香港理工大学（广州）数据科学与分析方向）
We consider the problem of maximizing a nonnegative (possibly non-monotone) submodular set function with or without constraints. Feige et al. [FOCS'07] showed a 2/5-approximation for the unconstrained problem and also proved that no approximation better than 1/2 is possible in the value oracle model. Constant-factor approximation was also given for submodular maximization subject to a matroid independence constraint (a factor of 0.309 Vondrak [FOCS'09]) and for submodular maximization subject to a matroid base constraint, provided that the fractional base packing number is at least 2 (a 1/4-ap…
We initiate the study of the classical Submodular Cover (SC) problem in the data streaming model which we refer to as the Streaming Submodular Cover (SSC). We show that any single pass streaming algorithm using sublinear memory in the size of the stream will fail to provide any non-trivial approximation guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only seek to find a partial cover. We design the first Efficient bicriteria Submodular Cover Streaming (ESC-Streaming) algorithm for this problem, and provide theoretical guarantees for its performance supported by numeri…
In this paper we study the extraction of representative elements in the data stream model in the form of submodular maximization. Different from the previous work on streaming submodular maximization, we are interested only in the recent data, and study the maximization problem over sliding windows. We provide a general reduction from the sliding window model to the standard streaming model, and thus our approach works for general constraints as long as there is a corresponding streaming algorithm in the standard streaming model. As a consequence, we obtain the first algorithms in the sliding …
The optimization of submodular functions on the integer lattice has received much attention recently, but the objective functions of many applications are non-submodular. We provide two approximation algorithms for maximizing a non-submodular function on the integer lattice subject to a cardinality constraint; these are the first algorithms for this purpose that have polynomial query complexity. We propose a general framework for influence maximization on the integer lattice that generalizes prior works on this topic, and we demonstrate the efficiency of our algorithms in this context.
We consider the problem of maximizing a nonnegative submodular set function $f:2^{\mathcal{N}} \rightarrow \mathbb{R}^+$ subject to a $p$-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are {\em non-monotone}. We describe deterministic and randomized algorithms that obtain a $Ω(\frac{1}{p})$-approximation using $O(k \log k)$-space, where $k$ is an upper bound on the cardinality of the desired set. The model assumes…
Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this di…
We study the problem of maximizing a monotone submodular function subject to a matroid constraint and present a deterministic algorithm that achieves (1/2 + ε)-approximation for the problem. This algorithm is the first deterministic algorithm known to improve over the 1/2-approximation ratio of the classical greedy algorithm proved by Nemhauser, Wolsely and Fisher in 1978.
In this paper, we study the non-monotone DR-submodular function maximization over integer lattice. Functions over integer lattice have been defined submodular property that is similar to submodularity of set functions. DR-submodular is a further extended submodular concept for functions over the integer lattice, which captures the diminishing return property. Such functions find many applications in machine learning, social networks, wireless networks, etc. The techniques for submodular set function maximization can be applied to DR-submodular function maximization, e.g., the double greedy alg…
Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algorithms. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. This makes it hard, for example, to apply standar…
In this work, we present a new algorithm for maximizing a non-monotone submodular function subject to a general constraint. Our algorithm finds an approximate fractional solution for maximizing the multilinear extension of the function over a down-closed polytope. The approximation guarantee is 0.372 and it is the first improvement over the 1/e approximation achieved by the unified Continuous Greedy algorithm [Feldman et al., FOCS 2011].
Simon Fraser University（西蒙菲莎大学）；National Institute of Informatics（信息科学研究所）
The problem of maximizing nonnegative monotone submodular functions under a certain constraint has been intensively studied in the last decade, and a wide range of efficient approximation algorithms have been developed for this problem. Many machine learning problems, including data summarization and influence maximization, can be naturally modeled as the problem of maximizing monotone submodular functions. However, when such applications involve sensitive data about individuals, their privacy concerns should be addressed. In this paper, we study the problem of maximizing monotone submodular f…
We study the recently introduced idea of worst-case sensitivity for monotone submodular maximization with cardinality constraint $k$, which captures the degree to which the output argument changes on deletion of an element in the input. We find that for large classes of algorithms that non-trivial sensitivity of $o(k)$ is not possible, even with bounded curvature, and that these results also hold in the distributed framework. However, we also show that in the regime $k = Ω(n)$ that we can obtain $O(1)$ sensitivity for sufficiently low curvature.
A deterministic approximation algorithm is presented for the maximization of non-monotone submodular functions over a ground set of size $n$ subject to cardinality constraint $k$; the algorithm is based upon the idea of interlacing two greedy procedures. The algorithm uses interlaced, thresholded greedy procedures to obtain tight ratio $1/4 - ε$ in $O \left( \frac{n}ε \log \left( \frac{k}ε \right) \right)$ queries of the objective function, which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem. The algorithm is v…
We consider non-monotone DR-submodular function maximization, where DR-submodularity (diminishing return submodularity) is an extension of submodularity for functions over the integer lattice based on the concept of the diminishing return property. Maximizing non-monotone DR-submodular functions has many applications in machine learning that cannot be captured by submodular set functions. In this paper, we present a $\frac{1}{2+ε}$-approximation algorithm with a running time of roughly $O(\frac{n}ε\log^2 B)$, where $n$ is the size of the ground set, $B$ is the maximum value of a coordinate, an…
We consider the problem of maximizing a monotone submodular function in a $k$-exchange system. These systems, introduced by Feldman et al., generalize the matroid k-parity problem in a wide class of matroids and capture many other combinatorial optimization problems. Feldman et al. show that a simple non-oblivious local search algorithm attains a $(k + 1)/2$ approximation ratio for the problem of linear maximization in a $k$-exchange system. Here, we extend this approach to the case of monotone submodular objective functions. We give a deterministic, non-oblivious local search algorithm that a…
The Open University of Israel（以色列开放大学）；Yale Institute for Network Science, Yale University（耶鲁大学网络科学研究所）；University of Haifa（海法大学）
In this paper, we propose a novel framework that converts streaming algorithms for monotone submodular maximization into streaming algorithms for non-monotone submodular maximization. This reduction readily leads to the currently tightest deterministic approximation ratio for submodular maximization subject to a $k$-matchoid constraint. Moreover, we propose the first streaming algorithm for monotone submodular maximization subject to $k$-extendible and $k$-set system constraints. Together with our proposed reduction, we obtain $O(k\log k)$ and $O(k^2\log k)$ approximation ratio for submodular …
In this work, we study the problem of monotone non-submodular maximization with partition matroid constraint. Although a generalization of this problem has been studied in literature, our work focuses on leveraging properties of partition matroid constraint to (1) propose algorithms with theoretical bound and efficient query complexity; and (2) provide better analysis on theoretical performance guarantee of some existing techniques. We further investigate those algorithms' performance in two applications: Boosting Influence Spread and Video Summarization. Experiments show our algorithms return…
In this paper we consider parallelization for applications whose objective can be expressed as maximizing a non-monotone submodular function under a cardinality constraint. Our main result is an algorithm whose approximation is arbitrarily close to $1/2e$ in $O(\log^2 n)$ adaptive rounds, where $n$ is the size of the ground set. This is an exponential speedup in parallel running time over any previously studied algorithm for constrained non-monotone submodular maximization. Beyond its provable guarantees, the algorithm performs well in practice. Specifically, experiments on traffic monitoring …
We consider a robust formulation, introduced by Krause et al. (2008), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of up to $τ$ elements from the chosen set. For the fundamental case of $τ=1$, we give a deterministic $(1-1/e)-1/Θ(m)$ approximation algorithm, where $m$ is an input parameter and number of queries scale as $O(n^{m+1})$. In the process, we develop a deterministic $(1-1/e)-1/Θ(m)$ approximate greedy algorithm for bi-objecti…
The subspace selection problem seeks a subspace that maximizes an objective function under some constraint. This problem includes several important machine learning problems such as the principal component analysis and sparse dictionary selection problem. Often, these problems can be solved by greedy algorithms. Here, we are interested in why these problems can be solved by greedy algorithms, and what classes of objective functions and constraints admit this property. To answer this question, we formulate the problems as optimization problems on lattices. Then, we introduce a new class of func…
In this work, we consider the maximization of submodular functions constrained by independence systems. Because of the wide applicability of submodular functions, this problem has been extensively studied in the literature, on specialized independence systems. For general independence systems, even when all of the bases of the independence system have the same size, we show that for any $ε> 0$, the problem is hard to approximate within $(2/n)^{1-ε}$, where $n$ is the size of the ground set. In the same context, we show the greedy algorithm does obtain a ratio of $2/n$ under an additional mild …
In this paper, we study the non-monotone adaptive submodular maximization problem subject to a knapsack and a $k$-system constraints. The input of our problem is a set of items, where each item has a particular state drawn from a known prior distribution. However, the state of an item is initially unknown, one must select an item in order to reveal the state of that item. There is a utility function which is defined over items and states. Our objective is to sequentially select a group of items to maximize the expected utility. Although the cardinality-constrained non-monotone adaptive submodu…
Khoury College of Computer Sciences, Northeastern University（东北大学胡里计算机科学学院）
We study the problem of differentially private constrained maximization of decomposable submodular functions. A submodular function is decomposable if it takes the form of a sum of submodular functions. The special case of maximizing a monotone, decomposable submodular function under cardinality constraints is known as the Combinatorial Public Projects (CPP) problem [Papadimitriou et al., 2008]. Previous work by Gupta et al. [2010] gave a differentially private algorithm for the CPP problem. We extend this work by designing differentially private algorithms for both monotone and non-monotone d…
We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Since a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a $1 - 1/e$ approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of a distributive lattice and define the multilinear e…
We consider the problem of maximizing a non-negative submodular function under the $b$-matching constraint, in the semi-streaming model. When the function is linear, monotone, and non-monotone, we obtain the approximation ratios of $2+\varepsilon$, $3 + 2 \sqrt{2} \approx 5.828$, and $4 + 2 \sqrt{3} \approx 7.464$, respectively. We also consider a generalized problem, where a $k$-uniform hypergraph is given, along with an extra matroid or a $k'$-matchoid constraint imposed on the edges, with the same goal of finding a $b$-matching that maximizes a submodular function. When the extra constraint…
This paper presents a polynomial-time $1/2$-approximation algorithm for maximizing nonnegative $k$-submodular functions. This improves upon the previous $\max\{1/3, 1/(1+a)\}$-approximation by Ward and Živný~(SODA'14), where $a=\max\{1, \sqrt{(k-1)/4}\}$. We also show that for monotone $k$-submodular functions there is a polynomial-time $k/(2k-1)$-approximation algorithm while for any $\varepsilon>0$ a $((k+1)/2k+\varepsilon)$-approximation algorithm for maximizing monotone $k$-submodular functions would require exponentially many queries. In particular, our hardness result implies that our al…
As the scales of data sets expand rapidly in some application scenarios, increasing efforts have been made to develop fast submodular maximization algorithms. This paper presents a currently the most efficient algorithm for maximizing general non-negative submodular objective functions subject to $k$-extendible system constraints. Combining the sampling process and the decreasing threshold strategy, our algorithm Sample Decreasing Threshold Greedy Algorithm (SDTGA) obtains an expected approximation guarantee of ($p-ε$) for monotone submodular functions and of ($p(1-p)-ε$) for non-monotone case…
The need for real time analysis of rapidly producing data streams (e.g., video and image streams) motivated the design of streaming algorithms that can efficiently extract and summarize useful information from massive data "on the fly". Such problems can often be reduced to maximizing a submodular set function subject to various constraints. While efficient streaming methods have been recently developed for monotone submodular maximization, in a wide range of applications, such as video summarization, the underlying utility function is non-monotone, and there are often various constraints impo…
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 investigate the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of Greedy for maximizing submodular functions, there are few guarantees for non-submodular ones. However, Greedy enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by a combination of …
University of Washington（华盛顿大学）；University of California, Berkeley（加州大学伯克利分校）
We present a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidifferentials (sub- and super-differentials). The resulting algorithms, which repeatedly compute and then efficiently optimize submodular semigradients, offer new and generalize many old methods for submodular optimization. Our approach, moreover, takes steps towards providing a unifying paradigm applicable to both submodular min- imization and maximization, problems that historically have been treated quite distinctly. The practicality of our algorith…
We present a polynomial-time online algorithm for maximizing the conditional value at risk (CVaR) of a monotone stochastic submodular function. Given $T$ i.i.d. samples from an underlying distribution arriving online, our algorithm produces a sequence of solutions that converges to a ($1-1/e$)-approximate solution with a convergence rate of $O(T^{-1/4})$ for monotone continuous DR-submodular functions. Compared with previous offline algorithms, which require $Ω(T)$ space, our online algorithm only requires $O(\sqrt{T})$ space. We extend our online algorithm to portfolio optimization for monoto…
In this paper we study submodular maximization under a matroid constraint in the adaptive complexity model. This model was recently introduced in the context of submodular optimization in [BS18a] to quantify the information theoretic complexity of black-box optimization in a parallel computation model. Informally, the adaptivity of an algorithm is the number of sequential rounds it makes when each round can execute polynomially-many function evaluations in parallel. Since submodular optimization is regularly applied on large datasets we seek algorithms with low adaptivity to enable speedups vi…
In machine learning and big data, the optimization objectives based on set-cover, entropy, diversity, influence, feature selection, etc. are commonly modeled as submodular functions. Submodular (function) maximization is generally NP-hard, even in the absence of constraints. Recently, submodular maximization has been successfully investigated for the settings where the objective function is monotone or the constraint is computation-tractable. However, maximizing nonmonotone submodular function with complex constraints is not yet well-understood. In this paper, we consider the nonmonotone submo…
We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. (2017) described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same ap…
In this paper, we present a thorough study of maximizing a regularized non-monotone submodular function subject to various constraints, i.e., $\max \{ g(A) - \ell(A) : A \in \mathcal{F} \}$, where $g \colon 2^Ω\to \mathbb{R}_+$ is a non-monotone submodular function, $\ell \colon 2^Ω\to \mathbb{R}_+$ is a normalized modular function and $\mathcal{F}$ is the constraint set. Though the objective function $f := g - \ell$ is still submodular, the fact that $f$ could potentially take on negative values prevents the existing methods for submodular maximization from providing a constant approximation …
Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. For the problem of maximizing a non-monotone submodular function, Feige, Mirrokni, and Vondrák recently developed a $2\over 5$-approximation algorithm \cite{FMV07}, however, their algorithms do not handle side constraints.} In this pa…
