Lisa Hellerstein, Thomas F. Lidbetter, R. Teal Witter · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2022
We consider the problem of solving the Min-Sum Submodular Cover problem using local search. The Min-Sum Submodular Cover problem generalizes the NP-complete Min-Sum Set Cover problem, replacing the input set cover instance with a monotone submodular set function. A simple greedy algorithm achieves an approximation factor of 4, which is tight unless P=NP [Streeter and Golovin, NeurIPS, 2008]. We complement the greedy algorithm with analysis of a local search algorithm. Building on work of Munagala et al. [ICDT, 2005], we show that, using simple initialization, a straightforward local search algorithm achieves a $(4+ε)$-approximate solution in time $O(n^3\log(n/ε))$, provided that the monotone submodular set function is also second-order supermodular. Second-order supermodularity has been shown to hold for a number of submodular functions of practical interest, including functions associated with set cover, matching, and facility location. We present experiments on two special cases of Min-Sum Submodular Cover and find that the local search algorithm can outperform the greedy algorithm on small data sets.
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Abir De, Soumen Chakrabarti · arXiv (Cornell University) · 2022
Submodular functions and variants, through their ability to characterize diversity and coverage, have emerged as a key tool for data selection and summarization. Many recent approaches to learn submodular functions suffer from limited expressiveness. In this work, we propose FLEXSUBNET, a family of flexible neural models for both monotone and non-monotone submodular functions. To fit a latent submodular function from (set, value) observations, FLEXSUBNET applies a concave function on modular functions in a recursive manner. We do not draw the concave function from a restricted family, but rather learn from data using a highly expressive neural network that implements a differentiable quadrature procedure. Such an expressive neural model for concave functions may be of independent interest. Next, we extend this setup to provide a novel characterization of monotone α-submodular functions, a recently introduced notion of approximate submodular functions. We then use this characterization to design a novel neural model for such functions. Finally, we consider learning submodular set functions under distant supervision in the form of (perimeter-set, high-value-subset) pairs. This yields a novel subset selection method based on an order-invariant, yet greedy sampler built around the above neural set functions. Our experiments on synthetic and real data show that FLEXSUBNET outperforms several baselines.
Cited by 2Open access
Soto San Martín, José · arXiv (Cornell University) · 2014
We present an efficient algorithm to find non-empty minimizers of a symmetric submodular function over any family of sets closed under inclusion. This for example includes families defined by a cardinality constraint, a knapsack constraint, a matroid independence constraint, or any combination of such constraints. Our algorithm make $O(n^3)$ oracle calls to the submodular function where $n$ is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within $o(\sqrt{n/\log n})$ (Svitkina and Fleischer [2008]). The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to find all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of cardinality $n$ using $O(n^3)$ oracle calls. Their procedure in turn is based on Queyranne's algorithm [1998] to minimize a symmetric submodular
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Rohit Gurjar, Rajat Rathi · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2020
A set function f : 2^E → ℝ on the subsets of a set E is called submodular if it satisfies a natural diminishing returns property: for any S ⊆ E and x,y ∉ S, we have f(S ∪ {x,y}) - f(S ∪ {y}) ≤ f(S ∪ {x}) - f(S). Submodular minimization problem asks for finding the minimum value a given submodular function takes. We give an algebraic algorithm for this problem for a special class of submodular functions that are "linearly representable". It is known that every submodular function f can be decomposed into a sum of two monotone submodular functions, i.e., there exist two non-decreasing submodular functions f₁,f₂ such that f(S) = f₁(S) + f₂(E ⧵ S) for each S ⊆ E. Our class consists of those submodular functions f, for which each of f₁ and f₂ is a sum of k rank functions on families of subspaces of 𝔽ⁿ, for some field 𝔽. Our algebraic algorithm for this class of functions can be parallelized, and thus, puts the problem of finding the minimizing set in the complexity class randomized NC. Further, we derandomize our algorithm so that it needs only O(log²(kn|E|)) many random bits. We also give reductions from two combinatorial optimization problems to linearly representable submodular minimization, and thus, get such parallel algorithms for these problems. These problems are (i) covering a directed graph by k a-arborescences and (ii) packing k branchings with given root sets in a directed graph.
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Hongyang Zhang, Wenchang Luo · arXiv (Cornell University) · 2023
A $k$-submodular function is a generalization of the submodular set function. Many practical applications can be modeled as maximizing a $k$-submodular function, such as multi-cooperative games, sensor placement with $k$ type sensors, influence maximization with $k$ topics, and feature selection with $k$ partitions. In this paper, we provide a unified continuous greedy algorithm for $k$-submodular maximization problem under a down-monotone constraint. Our technique involves relaxing the discrete variables in a continuous space by using the multilinear extension of $k$-submodular function to find a fractional solution, and then rounding it to obtain the feasible solution. Our proposed algorithm runs in polynomial time and can be applied to both the non-monotone and monotone cases. When the objective function is non-monotone, our algorithm achieves an approximation ratio of $(1/e-o(1))$; for a monotone $k$-submodular objective function, it achieves an approximation ratio of $(1-1/e-o(1))$.
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Kazuo Murota · 2008
This paper sheds a new light on submodular function minimization and maximization from the viewpoint of discrete convex analysis. L♮-convex functions and M♮-concave func-tions constitute subclasses of submodular functions on an integer interval. Whereas L♮-convex functions can be minimized efficiently on the basis of submodular (set) function minimization algorithms, M♮-concave functions are identified as a computationally tractable subclass for maximization. x
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Alberto Schiabel, Vyacheslav Kungurtsev, Jakub Mareček · arXiv (Cornell University) · 2021
Optimization problems with set submodular objective functions have many real-world applications. In discrete scenarios, where the same item can be selected more than once, the domain is generalized from a 2-element set to a bounded integer lattice. In this work, we consider the problem of maximizing a monotone submodular function on the bounded integer lattice subject to a cardinality constraint. In particular, we focus on maximizing DR-submodular functions, i.e., functions defined on the integer lattice that exhibit the diminishing returns property. Given any epsilon > 0, we present a randomized algorithm with probabilistic guarantees of O(1 - 1/e - epsilon) approximation, using a framework inspired by a Stochastic Greedy algorithm developed for set submodular functions by Mirzasoleiman et al. We then show that, on synthetic DR-submodular functions, applying our proposed algorithm on the integer lattice is faster than the alternatives, including reducing a target problem to the set domain and then applying the fastest known set submodular maximization algorithm.
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Bryan Wilder · Proceedings of the AAAI Conference on Artificial Intelligence · 2018
The conditional value at risk (CVaR) is a popular risk measure which enables risk-averse decision making under uncertainty. We consider maximizing the CVaR of a continuous submodular function, an extension of submodular set functions to a continuous domain. One example application is allocating a continuous amount of energy to each sensor in a network, with the goal of detecting intrusion or contamination. Previous work allows maximization of the CVaR of a linear or concave function. Continuous submodularity represents a natural set of nonconcave functions with diminishing returns, to which existing techniques do not apply. We give a (1 - 1/e)-approximation algorithm for maximizing the CVaR of a monotone continuous submodular function. This also yields an algorithm for submodular set functions which produces a distribution over feasible sets with guaranteed CVaR. Experimental results in two sensor placement domains confirm that our algorithm substantially outperforms competitive baselines.
Cited by 21Open access
Shabbir Ahmed, Alper Atamtürk · Mathematical Programming · 2009
Given a finite ground set N and a value vector $${a \in \mathbb{R}^N}$$ , we consider optimization problems involving maximization of a submodular set utility function of the form $${h(S)= f \left(\sum_{i \in S} a_i \right ), S \subseteq N}$$ , where f is a strictly concave, increasing, differentiable function. This utility function appears frequently in combinatorial optimization problems when modeling risk aversion and decreasing marginal preferences, for instance, in risk-averse capital budgeting under uncertainty, competitive facility location, and combinatorial auctions. These problems can be formulated as linear mixed 0-1 programs. However, the standard formulation of these problems using submodular inequalities is ineffective for their solution, except for very small instances. In this paper, we perform a polyhedral analysis of a relevant mixed-integer set and, by exploiting the structure of the utility function h, strengthen the standard submodular formulation significantly. We show the lifting problem of the submodular inequalities to be a submodular maximization problem with a special structure solvable by a greedy algorithm, which leads to an easily-computable strengthening by subadditive lifting of the inequalities. Computational experiments on expected utility maximization in capital budgeting show the effectiveness of the new formulation.
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Anupam Gupta, Roie Levin · ArXiv.org · 2025
In the submodular cover problem, we are given a monotone submodular function $f$, and we want to pick the min-cost set $S$ such that $f(S) = f(N)$. Motivated by problems in network monitoring and resource allocation, we consider the submodular cover problem in an online setting. As a concrete example, suppose at each time $t$, a nonnegative monotone submodular function $g_t$ is given to us. We define $f^{(t)} = \sum_{s \leq t} g_s$ as the sum of all functions seen so far. We need to maintain a submodular cover of these submodular functions $f^{(1)}, f^{(2)}, \ldots f^{(T)}$ in an online fashion; i.e., we cannot revoke previous choices. Formally, at each time $t$ we produce a set $S_t \subseteq N$ such that $f^{(t)}(S_t) = f^{(t)}(N)$ -- i.e., this set $S_t$ is a cover -- such that $S_{t-1} \subseteq S_t$, so previously decisions to pick elements cannot be revoked. (We actually allow more general sequences $\{f^{(t)}\}$ of submodular functions, but this sum-of-simpler-submodular-functions case is useful for concreteness.) We give polylogarithmic competitive algorithms for this online submodular cover problem. The competitive ratio on an input sequence of length $T$ is $O(\ln n \ln (T \cdot f(N) / f_{\text{min}}))$, where $f_{\text{min}}$ is the smallest nonzero marginal for functions $f^{(t)}$, and $|N| = n$. For the special case of online set cover, our competitive ratio matches that of Alon et al. [SIAM J. Comp. 03], which are best possible for polynomial-time online algorithms unless $NP \subseteq BPP$ (see Korman 04). Since existing offline algorithms for submodular cover are based on greedy approaches which seem difficult to implement online, the technical challenge is to (approximately) solve the exponential-sized linear programming relaxation for submodular cover, and to round it, both in the online setting.
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Francis Bach · now publishers, Inc. eBooks · 2013
Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions, and (2) the Lovász extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In Learning with Submodular Functions: A Convex Optimization Perspective, the theory of submodular functions is presented in a self-contained way from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, it describes how submodular function minimization is equivalent to solving a wide variety of convex optimization problems. This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance. By listing many examples of submodular functions, it reviews various applications to machine learning, such as clustering, experimental design, sensor placement, graphical model structure learning or subset selection, as well as a family of structured sparsity-inducing norms that can be derived and used from submodular functions. Learning with Submodular Functions: A Convex Optimization Perspective is an ideal reference for researchers, scientists, or engineers with an interest in applying submodular functions to machine learning problems.
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Deza, Anna, Hall, Georgina · arXiv (Cornell University) · 2025
We introduce the notion of $t$-sum of squares (sos) submodularity, which is a hierarchy, indexed by $t$, of sufficient algebraic conditions for certifying submodularity of set functions. We show that, for fixed $t$, each level of the hierarchy can be verified via a semidefinite program of size polynomial in $n$, the size of the ground set of the set function. This is particularly relevant given existing hardness results around testing whether a set function is submodular (Crama, 1989). We derive several equivalent algebraic characterizations of $t$-sos submodularity and identify submodularity-preserving operations that also preserve $t$-sos submodularity. We further present a complete classification of the cases for which submodularity and $t$-sos submodularity coincide, as well as examples of $t$-sos-submodular functions. We demonstrate the usefulness of $t$-sos submodularity through three applications: (i) a new convex approach to submodular regression, involving minimal manual tuning; (ii) a systematic procedure to derive lower bounds on the submodularity ratio in approximate submodular maximization, and (iii) improved difference-of-submodular decompositions for difference-of-submodular optimization. Overall, our work builds a new bridge between discrete optimization and real algebraic geometry by connecting sum of squares-based algebraic certificates to a fundamental discrete structure, submodularity.
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Deza, Anna, Hall, Georgina · ArXiv.org · 2025
We introduce the notion of $t$-sum of squares (sos) submodularity, which is a hierarchy, indexed by $t$, of sufficient algebraic conditions for certifying submodularity of set functions. We show that, for fixed $t$, each level of the hierarchy can be verified via a semidefinite program of size polynomial in $n$, the size of the ground set of the set function. This is particularly relevant given existing hardness results around testing whether a set function is submodular (Crama, 1989). We derive several equivalent algebraic characterizations of $t$-sos submodularity and identify submodularity-preserving operations that also preserve $t$-sos submodularity. We further present a complete classification of the cases for which submodularity and $t$-sos submodularity coincide, as well as examples of $t$-sos-submodular functions. We demonstrate the usefulness of $t$-sos submodularity through three applications: (i) a new convex approach to submodular regression, involving minimal manual tuning; (ii) a systematic procedure to derive lower bounds on the submodularity ratio in approximate submodular maximization, and (iii) improved difference-of-submodular decompositions for difference-of-submodular optimization. Overall, our work builds a new bridge between discrete optimization and real algebraic geometry by connecting sum of squares-based algebraic certificates to a fundamental discrete structure, submodularity.
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Bich-Ngan T. Nguyen, Phuong N. H. Phạm, Van-Vang Le et al. · Mathematics · 2022
In recent years, the issue of maximizing submodular functions has attracted much interest from research communities. However, most submodular functions are specified in a set function. Meanwhile, recent advancements have been studied for maximizing a diminishing return submodular (DR-submodular) function on the integer lattice. Because plenty of publications show that the DR-submodular function has wide applications in optimization problems such as sensor placement impose problems, optimal budget allocation, social network, and especially machine learning. In this research, we propose two main streaming algorithms for the problem of maximizing a monotone DR-submodular function under cardinality constraints. Our two algorithms, which are called StrDRS1 and StrDRS2, have (1/2−ϵ), (1−1/e−ϵ) of approximation ratios and O(nϵlog(logBϵ)logk), O(nϵlogB), respectively. We conducted several experiments to investigate the performance of our algorithms based on the budget allocation problem over the bipartite influence model, an instance of the monotone submodular function maximization problem over the integer lattice. The experimental results indicate that our proposed algorithms not only provide solutions with a high value of the objective function, but also outperform the state-of-the-art algorithms in terms of both the number of queries and the running time.
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Rishabh Iyer, Jeff Bilmes · arXiv (Cornell University) · 2020
Submodular Functions are a special class of Set Functions, which generalize several Information Theoretic quantities such as Entropy and Mutual Information [1]. Submodular functions have subgradients and subdifferentials [2] and admit polynomial time algorithms for minimization, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular function maximization, though NP hard, admits constant factor approximation guarantees and concave functions composed with modular functions are submodular. In this paper, we try to provide a more complete picture on the relationship between submodularity with concavity. We characterize the superdifferentials and polyhedra associated with upper bounds and provide optimality conditions for submodular maximization using the superdifferentials.
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Neal E. Young · 2016
entry editor: INDEX TERMS: dominating set, greedy algorithm, hitting set, set cover, minimizing a linear function subject to a submodular constraint
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Guangyi Zhang, Nikolaj Tatti, Aristides Gionis · Data Mining and Knowledge Discovery · 2022
Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a novel formulation for ranking items with submodular valuations and budget constraints. We refer to this problem as <i>max-submodular ranking</i> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>MSR</mml:mtext></mml:math> ). In more detail, given a set of items and a set of non-decreasing submodular functions, where each function is associated with a budget, we aim to find a ranking of the set of items that maximizes the sum of values achieved by all functions under the budget constraints. For the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>MSR</mml:mtext></mml:math> problem with cardinality- and knapsack-type budget constraints we propose practical algorithms with approximation guarantees. In addition, we perform an empirical evaluation, which demonstrates the superior performance of the proposed algorithms against strong baselines.
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Michel X. Goemans, Nicholas J. A. Harvey, Satoru Iwata et al. · Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms · 2009
Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization. In this paper, we consider the problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function such that, for every set S, (S) approximates f(S) within a factor α(n), where for rank functions of matroids and for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids. Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than , even for rank functions of a matroid.
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Alberto Schiabel, Vyacheslav Kungurtsev, Jakub Mareček · arXiv (Cornell University) · 2021
Optimization problems with set submodular objective functions have many\nreal-world applications. In discrete scenarios, where the same item can be\nselected more than once, the domain is generalized from a 2-element set to a\nbounded integer lattice. In this work, we consider the problem of maximizing a\nmonotone submodular function on the bounded integer lattice subject to a\ncardinality constraint. In particular, we focus on maximizing DR-submodular\nfunctions, i.e., functions defined on the integer lattice that exhibit the\ndiminishing returns property. Given any epsilon > 0, we present a randomized\nalgorithm with probabilistic guarantees of O(1 - 1/e - epsilon) approximation,\nusing a framework inspired by a Stochastic Greedy algorithm developed for set\nsubmodular functions by Mirzasoleiman et al. We then show that, on synthetic\nDR-submodular functions, applying our proposed algorithm on the integer lattice\nis faster than the alternatives, including reducing a target problem to the set\ndomain and then applying the fastest known set submodular maximization\nalgorithm.\n
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Bo Tang, Yajun Wang · arXiv (Cornell University) · 2011
We study the matroid secretary problem when the valuation is defined by a submodular function. In this problem, the elements of the ground set of a matroid are arriving in random order. When one element arrives, we have to make an immediate and irrevocable decision regarding whether or not to accept it. Our objective is to select an independent set under the given matroid, so that the valuation of the set is maximized with a submodular function. Our major result is a constant-competitive algorithm for the submodular matroid secretary problem under transversal matroids with monotonically increasing submodular valuation functions. In addressing it, we instead develop a constant competitive algorithm for a more general online bipartite graph matching problem. As a natural special case, we also consider the secretary problem under partition matroids and provide an extremely simple algorithm which can handle non-monotone submodular functions. This algorithm significantly improves the competitive ratio over the previous work. Shanghai Jiao Tong University. This work was done when the author was visiting Microsoft Research Asia. Microsoft Research Asia, yajunw@microsoft.com
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