Submodular maximization with matroid and packing constraints in parallel

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.

A number of recent results on optimization problems involving submodular functions have made use of the "multilinear relaxation" of the problem. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of "symmetry gap". Our main result is that for any fixed instance that exhibits a certain "symmetry gap" in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1-1/e)-approximation for monotone submodular maximization under a cardinality constraint, and the impossibility of (1/2+epsilon)-approximation for unconstrained (non-monotone) submodular maximization. It follows from our result that (1/2+epsilon)-approximation is also impossible for non-monotone submodular maximization subject to a (non-trivial) matroid constraint. On the algorithmic side, we present a 0.309-approximation for this problem, improving the previously known factor of 1/4-o(1).As another application, we consider the problem of maximizing a non-monotone submodular function over the bases of a matroid. A (1/6-o(1))-approximation has been developed for this problem, assuming that the matroid contains two disjoint bases. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k≫=2, there is a class of matroids of fractional base packing number nu = k/(k-1), such that any algorithm achieving a better than (1-1/nu)-approximation for this class would require exponentially many value queries. On the positive side, we present a 1/2 (1-1/nu-o(1))-approximation algorithm for the same problem. Our hardness results hold in fact for very special symmetric instances. For such symmetric instances, we show that the approximation factors of 1/2 (for submodular maximization subject to a matroid constraint) and 1-1/nu (for a matroid base constraint) can be achieved algorithmically and hence are optimal.

Submodular maximization has wide applications in machine learning and data mining, where massive datasets have brought the great need for designing efficient and parallelizable algorithms. One measure of the parallelizability of a submodular maximization algorithm is its adaptivity complexity, which indicates the number of sequential rounds where a polynomial number of queries to the objective function can be executed in parallel. In this paper, we study the problem of non-monotone submodular maximization subject to a knapsack constraint, and propose the first combinatorial algorithm achieving an (8+epsilon)-approximation under O(log n) adaptive complexity, which is optimal up to a factor of O(loglog n). Moreover, under slightly larger adaptivity, we also propose approximation algorithms with nearly optimal query complexity of O(n), while achieving better approximation ratios. We show that our algorithms can also be applied to the special case of submodular maximization subject to a cardinality constraint, and achieve performance bounds comparable with those of state-of-the-art algorithms. Finally, the effectiveness of our approach is demonstrated by extensive experiments on real-world applications.

We consider fast algorithms for monotone submodular maximization subject to a\nmatroid constraint. We assume that the matroid is given as input in an explicit\nform, and the goal is to obtain the best possible running times for important\nmatroids. We develop a new algorithm for a \\emph{general matroid constraint}\nwith a $1 - 1/e - \\epsilon$ approximation that achieves a fast running time\nprovided we have a fast data structure for maintaining a maximum weight base in\nthe matroid through a sequence of decrease weight operations. We construct such\ndata structures for graphic matroids and partition matroids, and we obtain the\n\\emph{first algorithms} for these classes of matroids that achieve a\nnearly-optimal, $1 - 1/e - \\epsilon$ approximation, using a nearly-linear\nnumber of function evaluations and arithmetic operations.\n

We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A∈[0,1]m ×n, a vector b∈[1,∞)m, and a monotone submodular set function f: 2[n]→ℝ+. The objective is to find a set S that maximizes f(S) subject to AxS≤b, where xS stands for the characteristic vector of the set S. A well-studied special case of this problem is when f is linear. This special linear case captures the class of packing integer programs. Our main contribution is an efficient combinatorial algorithm that achieves an approximation ratio of Ω(1 / m1/W), where W= min {bi / Aij : Aij>0} is the width of the packing constraints. This result matches the best known performance guarantee for the linear case. One immediate corollary of this result is that the algorithm under consideration achieves constant factor approximation when the number of constraints is constant or when the width of the constraints is sufficiently large. This motivates us to study the large width setting, trying to determine its exact approximability. We develop an algorithm that has an approximation ratio of (1−e)(1−1/e) when W=Ω(ln m / e2). This result essentially matches the theoretical lower bound of 1−1/e. We also study the special setting in which the matrix A is binary and k-column sparse. A k-column sparse matrix has at most k non-zero entries in each of its column. We design a fast combinatorial algorithm that achieves an approximation ratio of Ω(1 / (Wk1/W)), that is, its performance guarantee only depends on the sparsity and width parameters.

On submodular prophet inequalities and correlation gap

The theory of evolutionary computation (EC) has experienced rapid and productive growth in recent years. New proof techniques and novel theoretical frameworks have allowed advances in our understanding of the processes and structures inherent in evolutionary optimization. As a result, the frontiers of our knowledge have been expanded further than ever before. Some recent trends in this field, which are covered in this issue, include developments in the understanding of the behavior of evolutionary algorithms (EAs) in dynamic environments rather than just static settings, a theoretical appreciation of the advantages arising from the parallelization of evolutionary algorithms through a greater comprehension of the underlying dynamics, and an understanding of algorithm behavior on broad function classes, including -hard problems.The primary goal of this special issue is to provide extended and polished versions of diverse examples of the best theoretical work presented at conferences in 2014, and to serve as a forum for researchers to advance the theoretical understanding of evolutionary computation methods. The papers included in this special issue span a plurality of topics and offer the reader a cross section of recent outstanding work in EC theory.In dynamic optimization the objective function changes over time, and optimization algorithms face the additional challenge of tracking these changes to be successful. The article “Analysis of Randomised Search Heuristics for Dynamic Optimisation,” by Thomas Jansen and Christine Zarges, presents a novel analytical framework for the analysis of randomized search heuristics on dynamic problems inspired by the fixed-budget computations perspective. The authors introduce a new interesting class of bi-stable dynamic functions where the optimum oscillates between two complementary strings, and apply the framework to analyze and compare the performance of evolutionary algorithms and artificial immune systems on the novel class of functions.Over three decades ago, László Lovász observed that in discrete optimization, submodularity is the counterpart to convexity. However, in contrast to the focus on convex functions in continuous evolutionary optimization, so far submodular functions have received comparatively little attention from EC theoreticians studying discrete functions. The article “Maximizing Submodular Functions under Matroid Constraints by Evolutionary Algorithms,” by Tobias Friedrich and Frank Neumann, addresses this gap by analyzing the performance of evolutionary algorithms on different classes of submodular functions. The maximization of submodular functions is -hard in general, and the authors present several approximation results for monotone submodular and nonmonotone symmetric submodular functions under different kinds of matroid constraints.The idea behind parallel evolutionary algorithms is to evolve multiple subpopulations in parallel and allow interprocess communication at given time intervals. During these migration phases, fractions of each subpopulation can be shared among the subpopulations. There is very little understanding of how the migration frequency affects algorithmic performance, so setting the migration interval parameter appropriately may be difficult. In the article “Design and Analysis of Schemes for Adapting Migration Intervals in Parallel Evolutionary Algorithms,” Andrea Mambrini and Dirk Sudholt propose two schemes to automatically adapt the migration interval of parallel EAs during execution and provide a rigorous analytical framework that yields upper bounds on the expected runtime and expected communication effort of the parallel EAs with different migration topologies for various function classes.In the field of genetic programming, a long-standing open problem is how to address the issue of bloat, the emergence during evolution of solution elements that do not contribute significantly or at all to program fitness or semantics but increase program complexity. The article “On the Performance of Different Genetic Programming Approaches for the SORTING Problem,” by Markus Wagner, Frank Neumann, and Tommaso Urli, tackles the issue of bloat control in the context of sorting. As a basis for their study, they consider program trees and use some measure of sortedness of an in-order traversal to evaluate their fitness. The authors investigate single- and multiobjective variants of genetic programming algorithms with and without bloat control mechanisms, give rigorous upper bounds on their running times, and complement the study with experiments.The topic of constraint handling has recently gained traction in the continuous domain. The article “Markov Chain Analysis of Cumulative Step-Size Adaptation on a Linear Constrained Problem,” by Alexandre Chotard, Anne Auger, and Nikolaus Hansen, presents a rigorous analysis of a (1,)-Evolution Strategy using resampling on a linear function with a linear constraint. The authors prove the previously assumed property that a Markov chain, describing the behavior of the algorithm, exhibits stability in cases with constant step-size and with cumulative step-size adaptation with cumulation parameter equal to 1. This property characterizes the divergence of the algorithm with constant step-size and the geometric divergence or convergence with step-size adaptation, implying fast convergence of Monte Carlo simulations of the divergence rate.In their seminal 2006 paper, Droste, Jansen, and Wegener introduced the concept of black box complexity in order to establish a complexity theory for general-purpose randomized search heuristics. Generally speaking, the black box complexity of a problem is a lower bound on the number of function evaluations needed by any black box algorithm to solve it. Recently, black box models have been refined and developed extensively, allowing the hardness of objective function classes to be more precisely understood. The article “Unbiased Black Box Complexities of Jump Functions,” by Benjamin Doerr, Carola Doerr, and Timo Kötzing, analyzes the unbiased black box complexity of a Jump function class where in each function a local optimum is k bits in distance from the global optimum. The authors provide polynomial upper bounds on the black box complexity of Jump for different sizes of the gap. In particular, they show that an unbiased polynomial-time black box algorithm exists even when almost all of the search space is a plateau of constant fitness.The guest editors would like to thank the authors for their contributions, the referees for their careful reviewing and constructive comments, and the editor-in-chief, Hans-Georg Beyer, for his support in preparing this special issue.

Submodular optimization generalizes many classic problems in combinatorial\noptimization and has recently found a wide range of applications in machine\nlearning (e.g., feature engineering and active learning). For many large-scale\noptimization problems, we are often concerned with the adaptivity complexity of\nan algorithm, which quantifies the number of sequential rounds where\npolynomially-many independent function evaluations can be executed in parallel.\nWhile low adaptivity is ideal, it is not sufficient for a distributed algorithm\nto be efficient, since in many practical applications of submodular\noptimization the number of function evaluations becomes prohibitively\nexpensive. Motivated by these applications, we study the adaptivity and query\ncomplexity of adaptive submodular optimization.\n Our main result is a distributed algorithm for maximizing a monotone\nsubmodular function with cardinality constraint $k$ that achieves a\n$(1-1/e-\\varepsilon)$-approximation in expectation. This algorithm runs in\n$O(\\log(n))$ adaptive rounds and makes $O(n)$ calls to the function evaluation\noracle in expectation. The approximation guarantee and query complexity are\noptimal, and the adaptivity is nearly optimal. Moreover, the number of queries\nis substantially less than in previous works. Last, we extend our results to\nthe submodular cover problem to demonstrate the generality of our algorithm and\ntechniques.\n

Online auction is the essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic secretary problem, is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the multiple-choice secretary problem were under a thorough study in the literature. In this paper, we consider a very general setting of the latter problem called the submodular secretary problem, in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constant-competitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log2 r)-competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints (i.e., a knapsack constraint assigns a nonnegative cost to each secretary, and requires that the total cost of all the secretaries employed be no more than a budget value) instead of the matroid constraints, for which we present an O(l)-competitive algorithm. In a sharp contrast, we show for a more general setting of subadditive secretary problem, there is no O(√n)-competitive algorithm and thus submodular functions are the most general functions to consider for constant-competitiveness in our setting. We complement this result by giving a matching O(√n)-competitive algorithm for the subadditive case.

In this paper, we consider the problem of maximizing a non-negative submodular function f , defined on a (finite) ground set N , subject to matroid constraints. A function $f: 2^N \rightarrow {\mathbb R}$ is submodular if for all S , T *** N , f (S *** T ) + f (S *** T ) ≤ f (S ) + f (T ). Furthermore, all submodular functions that we deal with are assumed to be non-negative. Throughout, we assume that our submodular function f is given by a value oracle ; i.e., for a given set S *** N , an algorithm can query an oracle to find the value f (S ). Without loss of generality, we take the ground set N to be [n ] = {1,2,...,n }.

Analyzing Residual Random Greedy for monotone submodular maximization

Efficient BSP/CGM Algorithms for the Maximum Subsequence Sum and Related Problems 1

One of the recent thrust areas in research on hyperelliptic curve cryptography has been to obtain explicit formulae for performing arithmetic in the Jacobian of such curves. We continue this line of research by obtaining parallel versions of such formulae. Our first contribution is to develop a general methodology for obtaining parallel algorithm of any explicit formula. Any parallel algorithm obtained using our methodology is provably optimal in the number of multiplication rounds. We next apply this methodology to Lange’s explicit formula for arithmetic in genus 2 hyperelliptic curve – both for the affine coordinate and inversion free arithmetic versions. Since encapsulated add-and-double algorithm is an important countermeasure against side channel attacks, we develop parallel algorithms for encapsulated add-and-double for both of Lange’s versions of explicit formula. For the case of inversion free arithmetic, we present parallel algorithms using 4, 8 and 12 multipliers. All parallel algorithms described in this paper are optimal in the number of parallel rounds. One of the conclusions from our work is the fact that the parallel version of inversion free arithmetic is more efficient than the parallel version of arithmetic using affine coordinates.Keywordshyperelliptic curve cryptographyexplicit formulaparallel algorithmJacobianencapsulated add-and-double

There are quantum algorithms for finding a function f satisfying a set of conditions, such as solving partial differential equations, and these achieve exponential quantum speedup compared to existing classical methods, especially when the number d of the variables of f is large. In general, however, these algorithms output the quantum state which encodes f in the amplitudes, and reading out the values of f as classical data from such a state can be so time-consuming that the quantum speedup is ruined. In this study, we propose a general method for this function readout task. Based on the function approximation by a combination of tensor network and orthogonal function expansion, we present a quantum circuit and its optimization procedure to obtain an approximating function of f that has a polynomial number of degrees of freedom with respect to d and is efficiently evaluable on a classical computer. We also conducted a numerical experiment to approximate a finance-motivated function to demonstrate that our method works.

The complex geographical environment and electromagnetic interference environment within the radar deployment position area will seriously degrade the radar detection performance. Focusing on the urgent requirement of optimizing radar deployment in complex geographic and electromagnetic environments, the environment-constrained radar position positioning optimization method is investigated. Firstly, based on the terrain and electromagnetic interference environment of radar positions, a radar position selection method under both geographic and electromagnetic environment constraints is established. Then the propagation loss under complex geographical terrain is calculated by combining the radar equations, and the detection power of target height layer and the coverage power of the focused area are used as the evaluation indicators for the superiority of radar sites. Finally, the detection range of the radar under multiple environmental constraints is visualized. The problem of complex radar operational scenarios in real systems and the single constraint considered in traditional radar equipment deployment methods is addressed. By calculating the propagation loss, terrain loss, electromagnetic loss under multiple environmental and electromagnetic constraints, and adding actual elevation and terrain data as reference, a more realistic and objective analysis of the impact of the environment around the position on the performance of radar equipment is provided, and finally the evaluation indexes and selection methods is proposed in this paper. The proposed evaluation indexes and selection methods and visualization can provide accurate, reliable and intuitive strategies for optimizing the deployment of radar equipment under multiple constraints.