Efficient Submodular Function Maximization under Linear Packing Constraints

In the paper, we propose a deterministic approximation algorithm for maximizing a generalized monotone submodular function subject to a matroid constraint. The function is generalized through a curvature parameter \(c\in [0,1]\), and essentially reduces to a submodular function when \(c=1\). Our algorithm employs the deterministic approximation devised by Buchbinder et al. [3] for the \(c=1\) case of the problem as a building block, and eventually attains an approximation ratio of \(\frac{1+g_c(x)+\Delta \cdot \left[ 3+c-(2+c)x-(1+c)g_c(x)\right] }{2+c+(1+c)(1-x)}\) for the curvature parameter \(c\in [0,1]\) and for a calibrating parameter that is any \(x\in [0,1]\). For \(c=1\), the ratio attains 0.5008 by setting \(x=0.9\), coinciding with the renowned performance guarantee of the problem. Moreover, when the submodular set function degenerates to a linear function, our generalized algorithm always produces optimum solutions and thus achieves an approximation ratio 1.

We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-ε$ approximation for monotone functions and a $1/e-ε$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/ε^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-ε$ approximation using $O(\log(n/ε) \log(1/ε) \log(n+m)/ ε^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-ε$ approximation in $O(\log(n/ε)\log(m)/ε^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.

Analyzing Residual Random Greedy for monotone submodular maximization

Let $f:2^{N} \rightarrow \cal R^{+}$ be a non-decreasing submodular set function, and let $(N,\cal I)$ be a matroid. We consider the problem $\max_{S \in \cal I} f(S)$. It is known that the greedy algorithm yields a 1/2-approximation [9] for this problem. It is also known, via a reduction from the max-k-cover problem, that there is no (1 i¾? 1/e+ i¾?)-approximation for any constant i¾?> 0, unless P= NP[6]. In this paper, we improve the 1/2-approximation to a (1 i¾? 1/e)-approximation, when fis a sum of weighted rank functions of matroids. This class of functions captures a number of interesting problems including set coverage type problems. Our main tools are the pipage rounding technique of Ageev and Sviridenko [1] and a probabilistic lemma on monotone submodular functions that might be of independent interest. We show that the generalized assignment problem (GAP) is a special case of our problem; although the reduction requires |N| to be exponential in the original problem size, we are able to interpret the recent (1 i¾? 1/e)-approximation for GAP by Fleischer et al.[10] in our framework. This enables us to obtain a (1 i¾? 1/e)-approximation for variants of GAP with more complex constraints.

Deterministic approximation algorithm for submodular maximization subject to a matroid constraint

We consider the problem of maximizing monotone submodular function over the bounded integer lattice with a cardinality constraint. Function [Formula: see text] is submodular over integer lattice if [Formula: see text], [Formula: see text], where ∨ and ∧ represent elementwise maximum and minimum, respectively. Let [Formula: see text], and [Formula: see text], we study the problem of maximizing submodular function [Formula: see text] with constraints [Formula: see text] and [Formula: see text]. A random greedy [Formula: see text]-approximation algorithm and a deterministic [Formula: see text]-approximation algorithm are proposed in this paper. Both algorithms work in value oracle model. In the random greedy algorithm, we assume the monotone submodular function satisfies diminishing return property, which is not an equivalent definition of submodularity on integer lattice. Additionally, our random greedy algorithm makes [Formula: see text] value oracle queries and deterministic algorithm makes [Formula: see text] value oracle queries.

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.

Novel algorithms for maximum DS decomposition

The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they become accessible in order to maximize a global objective function. More generally, adaptive seeding is a stochastic optimization framework where the choices in the first stage affect the realizations in the second stage, over which we aim to optimize.Our main result is a (1 -- 1/e)2-approximation for the adaptive seeding problem for any monotone submodular function. While adaptive policies are often approximated via non-adaptive policies, our algorithm is based on a novel method we call locally-adaptive policies. These policies combine a non-adaptive global structure, with local adaptive optimizations. This method enables the (1 -- 1/e)2-approximation for general monotone submodular functions and circumvents some of the impossibilities associated with non-adaptive policies.We also introduce a fundamental problem in submodular optimization that may be of independent interest: given a ground set of elements where every element appears with some small probability, find a set of expected size at most k that has the highest expected value over the realization of the elements. We show a surprising result: there are classes of monotone submodular functions (including coverage) that can be approximated almost optimally as the probability vanishes. For general monotone submodular functions we show via a reduction from Planted-Clique that approximations for this problem are not likely to be obtainable. This optimization problem is an important tool for adaptive seeding via non-adaptive policies, and its hardness motivates the introduction of locally-adaptive policies we use in the main result.

The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they become accessible in order to maximize a global objective function. More generally, adaptive seeding is a stochastic optimization framework where the choices in the first stage affect the realizations in the second stage, over which we aim to optimize. Our main result is a $(1-1/e)^2$-approximation for the adaptive seeding problem for any monotone submodular function. While adaptive policies are often approximated via non-adaptive policies, our algorithm is based on a novel method we call \emph{locally-adaptive} policies. These policies combine a non-adaptive global structure, with local adaptive optimizations. This method enables the $(1-1/e)^2$-approximation for general monotone submodular functions and circumvents some of the impossibilities associated with non-adaptive policies. We also introduce a fundamental problem in submodular optimization that may be of independent interest: given a ground set of elements where every element appears with some small probability, find a set of expected size at most $k$ that has the highest expected value over the realization of the elements. We show a surprising result: there are classes of monotone submodular functions (including coverage) that can be approximated almost optimally as the probability vanishes. For general monotone submodular functions we show via a reduction from \textsc{Planted-Clique} that approximations for this problem are not likely to be obtainable. This optimization problem is an important tool for adaptive seeding via non-adaptive policies, and its hardness motivates the introduction of \emph{locally-adaptive} policies we use in the main result.

This paper addresses the Minimum Linear Ordering Problem (MLOP): Given a nonnegative set function f on a finite set V, find a linear ordering on V such that the sum of the function values for all the suffixes is minimized. This problem generalizes well-known problems such as the Minimum Linear Arrangement, Min Sum Set Cover, Minimum Latency Set Cover, and Multiple Intents Ranking. Extending a result of Feige, Lovasz, and Tetali (2004) on Min Sum Set Cover, we show that the greedy algorithm provides a factor 4 approximate optimal solution when the cost function f is supermodular. We also present a factor 2 rounding algorithm for MLOP with a monotone submodular cost function, using the convexity of the Lovasz extension. These are among very few constant factor approximation algorithms for NP-hard minimization problems formulated in terms of submodular/supermodular functions. In contrast, when f is a symmetric submodular function, the problem has an information theoretic lower bound of 2 on the approximability.

Submodular maximization problems belong to the family of combinatorial optimization problems and enjoy wide applications. In this paper, we focus on the problem of maximizing a monotone submodular function subject to a d-knapsack constraint, for which we propose a streaming algorithm that achieves a (1/1+2D − e)-approximation of the optimal value, while it only needs one single pass through the dataset without storing all the data in the memory. In our experiments, we extensively evaluate the effectiveness of our proposed algorithm via an application in scientific literature recommendation. It is observed that the proposed streaming algorithm achieves both execution speedup and memory saving by several orders of magnitude, compared with existing approaches.

The multilinear framework for submodular maximization was developed to achieve a tight $$1-1/e$$ 1 - 1 / e approximation for maximizing a monotone submodular function subject to a matroid constraint, including as special case the submodular welfare problem. The framework has a continuous optimization step (solving the multilinear extension of a submodular function) and a rounding part (rounding a fractional solution to an integral one). We extend both parts to provide a framework for a wider array of applications. The continuous part works for a more general class of continuous functions parameterized by a new smoothness parameter $$\sigma $$ σ . A twice differential function F is called $$\sigma $$ σ -one-sided-smooth ( $$\sigma $$ σ -OSS) if its second derivatives are bounded as follows: $$\frac{1}{2}u^T\nabla ^2 F(x) u \le \sigma \cdot \frac{\Vert u\Vert _1}{\Vert x\Vert _1} u^T \nabla F(x)$$ 1 2 u T ∇ 2 F ( x ) u ≤ σ · ‖ u ‖ 1 ‖ x ‖ 1 u T ∇ F ( x ) for all $$u,x\ge 0$$ u , x ≥ 0 , $$x\ne 0$$ x ≠ 0 . For $$\sigma =0$$ σ = 0 this includes previously studied continuous DR-Submodular functions as well as quadratics defined by copositive matrices. We give a modification of the continuous greedy algorithm which finds a solution for maximizing a monotone $$\sigma $$ σ -OSS F over a polytope in the non-negative orthant; the solution approximates the optimum to within factors which are functions of $$\sigma $$ σ which depend on additional properties. Interestingly, $$\sigma $$ σ -OSS functions arise as the multilinear extensions of set functions associated with several well-studied diversity maximization problems: $$\max f(S) = \sum _{i,j \in S} A_{ij} : |S| \le k$$ max f ( S ) = ∑ i , j ∈ S A ij : | S | ≤ k . For instance, when $$A_{ij}$$ A ij defines a $$\sigma $$ σ -semi-metric, its extension is $$\sigma $$ σ -OSS. In these settings, we also develop rounding schemes to approximate the discrete problem.

Collaborative work often benefits from having teams or organizations with heterogeneous members. In this paper, we present a method to form such diverse teams from people arriving sequentially over time. We define a monotone submodular objective function that combines the diversity and quality of a team and proposes an algorithm to maximize the objective while satisfying multiple constraints. This allows us to balance both how diverse the team is and how well it can perform the task at hand. Using crowd experiments, we show that, in practice, the algorithm leads to large gains in team diversity. Using simulations, we show how to quantify the additional cost of forming diverse teams and how to address the problem of simultaneously maximizing diversity for several attributes (e.g., country of origin and gender). Our method has applications in collaborative work ranging from team formation, the assignment of workers to teams in crowdsourcing, and reviewer allocation to journal papers arriving sequentially. Our code is publicly accessible for further research.

Maximization of non-negative monotone submodular set functions under a knapsack constraint have been extensively studied in the last decade. Here, we consider the streaming algorithms of this problem on the integer lattice, or on a multi-set equivalently. This is more realistic for many practical problems such as sensor location and influence maximization. It is well known that submodularity and diminishing return submodularity are not equivalent on the integer lattice. We mainly focus on maximizing the diminishing return submodular (DR-submodular) functions with knapsack constraint on the integer lattice. Finally, by utilizing the binary search algorithm as a subroutine, we design an online streaming algorithm called DynamicMRT. Furthermore, we prove that it is a \((1/3-\varepsilon )\)-approximation algorithm with \(O(K\log K)\) memory complexity and \(O(\log K)\) query complexity per element.