In this paper, the Minimum Cost Submodular Cover problem is studied, which is to minimize a modular cost function such that the monotone submodular benefit function is above a threshold. For this problem, an evolutionary algorithm EASC is introduced that achieves a constant, bicriteria approximation in expected polynomial time; this is the first polynomial-time evolutionary approximation algorithm for Minimum Cost Submodular Cover. To achieve this running time, ideas motivated by submodularity and monotonicity are incorporated into the evolutionary process, which likely will extend to other submodular optimization problems. In a practical application, EASC is demonstrated to outperform the greedy algorithm and converge faster than competing evolutionary algorithms for this problem.
In this paper, the monotone submodular maximization problem (SM) is studied. SM is to find a subset of size $κ$ from a universe of size $n$ that maximizes a monotone submodular objective function $f$. We show using a novel analysis that the Pareto optimization algorithm achieves a worst-case ratio of $(1-ε)(1-1/e)$ in expectation for every cardinality constraint $κ< P$, where $P\leq n+1$ is an input, in $O(nP\ln(1/ε))$ queries of $f$. In addition, a novel evolutionary algorithm called the biased Pareto optimization algorithm, is proposed that achieves a worst-case ratio of $(1-ε)(1-1/e)$ in ex…
A core feature of evolutionary algorithms is their mutation operator. Recently, much attention has been devoted to the study of mutation operators with dynamic and non-uniform mutation rates. Following up on this line of work, we propose a new mutation operator and analyze its performance on the (1+1) Evolutionary Algorithm (EA). Our analyses show that this mutation operator competes with pre-existing ones, when used by the (1+1) EA on classes of problems for which results on the other mutation operators are available. We show that the (1+1) EA using our mutation operator finds a (1/3)-approxi…
In this work, we study the Submodular Cost Submodular Cover problem, which is to minimize the submodular cost required to ensure that the submodular benefit function exceeds a given threshold. Existing approximation ratios for the greedy algorithm assume a value oracle to the benefit function. However, access to a value oracle is not a realistic assumption for many applications of this problem, where the benefit function is difficult to compute. We present two incomparable approximation ratios for this problem with an approximate value oracle and demonstrate that the ratios take on empirically…
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…
Nanjing University（南京大学）；Southern University of Science and Technology（南方科技大学）
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 …
We study the Minimum Latency Submodular Cover problem (MLSC), which consists of a metric $(V,d)$ with source $r\in V$ and $m$ monotone submodular functions $f_1, f_2, ..., f_m: 2^V \rightarrow [0,1]$. The goal is to find a path originating at $r$ that minimizes the total cover time of all functions. This generalizes well-studied problems, such as Submodular Ranking [AzarG11] and Group Steiner Tree [GKR00]. We give a polynomial time $O(\log \frac{1}{\eps} \cdot \log^{2+δ} |V|)$-approximation algorithm for MLSC, where $ε>0$ is the smallest non-zero marginal increase of any $\{f_i\}_{i=1}^m$ and …
In the stochastic submodular cover problem, the goal is to select a subset of stochastic items of minimum expected cost to cover a submodular function. Solutions in this setting correspond to sequential decision processes that select items one by one "adaptively" (depending on prior observations). While such adaptive solutions achieve the best objective, the inherently sequential nature makes them undesirable in many applications. We ask: how well can solutions with only a few adaptive rounds approximate fully-adaptive solutions? We give nearly tight answers for both independent and correlated…
In this paper, we study a novel problem, Minimum Robust Multi-Submodular Cover for Fairness (MinRF), as follows: given a ground set $V$; $m$ monotone submodular functions $f_1,...,f_m$; $m$ thresholds $T_1,...,T_m$ and a non-negative integer $r$, MinRF asks for the smallest set $S$ such that for all $i \in [m]$, $\min_{|X| \leq r} f_i(S \setminus X) \geq T_i$. We prove that MinRF is inapproximable within $(1-ε)\ln m$; and no algorithm, taking fewer than exponential number of queries in term of $r$, is able to output a feasible set to MinRF with high certainty. Three bicriteria approximation al…
In the submodular cover problem, we are given a non-negative monotone submodular function $f$ over a ground set $E$ of items, and the goal is to choose a smallest subset $S \subseteq E$ such that $f(S) = Q$ where $Q = f(E)$. In the stochastic version of the problem, we are given $m$ stochastic items which are different random variables that independently realize to some item in $E$, and the goal is to find a smallest set of stochastic items whose realization $R$ satisfies $f(R) = Q$. The problem captures as a special case the stochastic set cover problem and more generally, stochastic covering…
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…
This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.
We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between…
In this paper, we consider the optimization problem \scpl (\scp), which is to find a minimum cost subset of a ground set $U$ such that the value of a submodular function $f$ is above a threshold $τ$. In contrast to most existing work on \scp, it is not assumed that $f$ is monotone. Two bicriteria approximation algorithms are presented for \scp that, for input parameter $0 < ε< 1$, give $O( 1 / ε^2 )$ ratio to the optimal cost and ensures the function $f$ is at least $τ(1 - ε)/2$. A lower bound shows that under the value query model shows that no polynomial-time algorithm can ensure that $f$ is…
NYU Tandon School of Engineering（纽约大学坦登工程学院）；Brooklyn College, City University of New York（纽约城市大学布鲁克林学院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Many problems in Machine Learning can be modeled as submodular optimization problems. Recent work has focused on stochastic or adaptive versions of these problems. We consider the Scenario Submodular Cover problem, which is a counterpart to the Stochastic Submodular Cover problem studied by Golovin and Krause. In Scenario Submodular Cover, the goal is to produce a cover with minimum expected cost, where the expectation is with respect to an empirical joint distribution, given as input by a weighted sample of realizations. In contrast, in Stochastic Submodular Cover, the variables of the input …
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…
This paper studies randomized approximation algorithm for a variant of the set cover problem called minimum submodular cost partial multi-cover (SCPMC), in which each element $e$ has a covering requirement $r_e$ and a profit $p_e$, and the cost function on sub-collection of sets is submodular, the goal is to find a minimum cost sub-collection of sets which fully covers at least $q$-percentage of total profit, where an element $e$ is fully covered by sub-collection $S'$ if and only if it belongs to at least $r_e$ sets of $\mathcal S'$. Previous work shows that such a combination enormously incr…
Fast Stochastic Greedy Algorithm for $k$-Submodular Cover Problem
Graduate University of Science and Technology（科学技术研究生大学）；Vietnam Academy of Science and Technology（越南科学技术学院）；Faculty of Information Technology（信息科技学院）；Hanoi Architecture University（河内建筑大学）；ORLab（OR实验室）；Phenikaa University（Phenikaa大学）
In the minimum cost submodular cover problem (MinSMC), we are given a monotone nondecreasing submodular function $f\colon 2^V \rightarrow \mathbb{Z}^+$, a linear cost function $c: V\rightarrow \mathbb R^{+}$, and an integer $k\leq f(V)$, the goal is to find a subset $A\subseteq V$ with the minimum cost such that $f(A)\geq k$. The MinSMC can be found at the heart of many machine learning and data mining applications. In this paper, we design a parallel algorithm for the MinSMC that takes at most $O(\frac{\log km\log k(\log m+\log\log mk)}{\varepsilon^4})$ adaptive rounds, and it achieves an app…
We study the problem of ranking with submodular valuations. An instance of this problem consists of a ground set $[m]$, and a collection of $n$ monotone submodular set functions $f^1, \ldots, f^n$, where each $f^i: 2^{[m]} \to R_+$. An additional ingredient of the input is a weight vector $w \in R_+^n$. The objective is to find a linear ordering of the ground set elements that minimizes the weighted cover time of the functions. The cover time of a function is the minimal number of elements in the prefix of the linear ordering that form a set whose corresponding function value is greater than a…
Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant $ε>0$ achieves a guarantee of $1-\frac{1}{\mathrm{e}}-ε$ while violating only the covering constraints by a multiplicative factor of $1-ε$. Our algorithm is based on a novel enumeration method, which unlike previous known enumeration techniques, can handle both packing and covering constraints. We extend the above mai…
The University of North Texas（北德克萨斯大学）；The University of Texas at Dallas（德克萨斯大学达拉斯分校）
In this paper, we study the adaptive submodular cover problem under the worst-case setting. This problem generalizes many previously studied problems, namely, the pool-based active learning and the stochastic submodular set cover. The input of our problem is a set of items (e.g., medical tests) and each item has a random state (e.g., the outcome of a medical test), whose realization is initially unknown. One must select an item at a fixed cost in order to observe its realization. There is an utility function which maps a subset of items and their states to a non-negative real number. We aim to…
Submodularity is a fundamental phenomenon in combinatorial optimization. Submodular functions occur in a variety of combinatorial settings such as coverage problems, cut problems, welfare maximization, and many more. Therefore, a lot of work has been concerned with maximizing or minimizing a submodular function, often subject to combinatorial constraints. Many of these algorithmic results exhibit a common structure. Namely, the function is extended to a continuous, usually non-linear, function on a convex domain. Then, this relaxation is solved, and the fractional solution rounded to yield an …
We study minimum entropy submodular optimization, a common generalization of the minimum entropy set cover problem, studied earlier by Cardinal et al., and the submodular set cover problem. We give a general bound of the approximation performance of the greedy algorithm using an approach that can be interpreted in terms of a particular type of biased network flows. As an application we rederive known results for the Minimum Entropy Set Cover and Minimum Entropy Orientation problems, and obtain a nontrivial bound for a new problem called the Minimum Entropy Spanning Tree problem. The problem ca…
Cardinality constrained submodular function maximization, which aims to select a subset of size at most $k$ to maximize a monotone submodular utility function, is the key in many data mining and machine learning applications such as data summarization and maximum coverage problems. When data is given as a stream, streaming submodular optimization (SSO) techniques are desired. Existing SSO techniques can only apply to insertion-only streams where each element has an infinite lifespan, and sliding-window streams where each element has a same lifespan (i.e., window size). However, elements in som…
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…
NYU Tandon School of Engineering（纽约大学坦登工程学院）；Brooklyn College, City University of New York（纽约城市大学布鲁克林学院）；IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）
We show that the Adaptive Greedy algorithm of Golovin and Krause (2011) achieves an approximation bound of $(\ln (Q/η)+1)$ for Stochastic Submodular Cover: here $Q$ is the "goal value" and $η$ is the smallest non-zero marginal increase in utility deliverable by an item. (For integer-valued utility functions, we show a bound of $H(Q)$, where $H(Q)$ is the $Q^{th}$ Harmonic number.) Although this bound was claimed by Golovin and Krause in the original version of their paper, the proof was later shown to be incorrect by Nan and Saligrama (2017). The subsequent corrected proof of Golovin and Kraus…
University of Michigan（密歇根大学）；Bloomberg LP（彭博有限合伙公司）
We study a general stochastic ranking problem where an algorithm needs to adaptively select a sequence of elements so as to "cover" a random scenario (drawn from a known distribution) at minimum expected cost. The coverage of each scenario is captured by an individual submodular function, where the scenario is said to be covered when its function value goes above a given threshold. We obtain a logarithmic factor approximation algorithm for this adaptive ranking problem, which is the best possible (unless P=NP). This problem unifies and generalizes many previously studied problems with applicat…
Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is g…
We consider a variant of the prize collecting Steiner tree problem in which we are given a \emph{directed graph} $D=(V,A)$, a monotone submodular prize function $p:2^V \rightarrow \mathbb{R}^+ \cup \{0\}$, a cost function $c:V \rightarrow \mathbb{Z}^{+}$, a root vertex $r \in V$, and a budget $B$. The aim is to find an out-subtree $T$ of $D$ rooted at $r$ that costs at most $B$ and maximizes the prize function. We call this problem \emph{Directed Rooted Submodular Tree} (\textbf{DRSO}). Very recently, Ghuge and Nagarajan [SODA\ 2020] gave an optimal quasi-polynomial-time $O\left(\frac{\log n'}…
An effective technique for solving optimization problems over massive data sets is to partition the data into smaller pieces, solve the problem on each piece and compute a representative solution from it, and finally obtain a solution inside the union of the representative solutions for all pieces. This technique can be captured via the concept of {\em composable core-sets}, and has been recently applied to solve diversity maximization problems as well as several clustering problems. However, for coverage and submodular maximization problems, impossibility bounds are known for this technique \…
In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of $n$ items, each of which must be allocated to one of $m$ agents. Each agent $\ell$ has a valuation function $v_\ell$, where $v_\ell(S)$ denotes the welfare obtained by this agent if she receives the set of items $S$. The functions $v_\ell$ are all submodular; as is standard, we assume that they are monotone and $v_\ell(\emptyset) = 0$. The goal is to partition the items into $m$ disjoint subsets $S_1, S_2, \ldots S_m$ in order to maximize the social welfare, defined as $\sum_{\ell = 1}^m v_\ell(S_\ell)$. In th…
In bipartite matching problems, vertices on one side of a bipartite graph are paired with those on the other. In its online variant, one side of the graph is available offline, while the vertices on the other side arrive online. When a vertex arrives, an irrevocable and immediate decision should be made by the algorithm; either match it to an available vertex or drop it. Examples of such problems include matching workers to firms, advertisers to keywords, organs to patients, and so on. Much of the literature focuses on maximizing the total relevance---modeled via total weight---of the matching…
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；Google Zurich（谷歌苏黎世）；EPFL（洛桑联邦理工学院）；Microsoft Research（微软研究院）
Submodular maximization has become established as the method of choice for the task of selecting representative and diverse summaries of data. However, if datapoints have sensitive attributes such as gender or age, such machine learning algorithms, left unchecked, are known to exhibit bias: under- or over-representation of particular groups. This has made the design of fair machine learning algorithms increasingly important. In this work we address the question: Is it possible to create fair summaries for massive datasets? To this end, we develop the first streaming approximation algorithms fo…
Maximizing submodular functions under cardinality constraints lies at the core of numerous data mining and machine learning applications, including data diversification, data summarization, and coverage problems. In this work, we study this question in the context of data streams, where elements arrive one at a time, and we want to design low-memory and fast update-time algorithms that maintain a good solution. Specifically, we focus on the sliding window model, where we are asked to maintain a solution that considers only the last $W$ items. In this context, we provide the first non-trivial a…
In this paper we consider a generalization of the well-known budgeted maximum coverage problem. We are given a ground set of elements and a set of bins. The goal is to find a subset of elements along with an associated set of bins, such that the overall cost is at most a given budget, and the profit is maximized. Each bin has its own cost and the cost of each element depends on its associated bin. The profit is measured by a monotone submodular function over the elements. We first present an algorithm that guarantees an approximation factor of $\frac{1}{2}\left(1-\frac{1}{e^α}\right)$, where $…
