Submodular Cover Problem Research Articles

Fast bicriteria streaming algorithms for submodular cover problem under noise models

The Submodular Cover (SC) problem has attracted the attention of researchers because of its wide variety of applications in many domains. Previous studies on this problem have focused on solving it under the assumption of a non-noise environment or using the greedy algorithm to solve it under noise. However, in some applications, the data is often large-scale and brings a noisy version, so the existing solutions are ineffective or not applicable to large and noisy data. Motivated by this phenomenon, we study the Submodular Cover under Noises (SCN) problem and propose two efficient streaming algorithms, which provide a solution with theoretical bounds under two common noise models, multiplicative and additive noises. The experimental results indicate that our proposed algorithms not only provide the solution with a high objective function value but also outperform the state-of-the-art algorithm in terms of both the number of queries and the running time.

Efficient Parallel Algorithm for Minimum Cost Submodular Cover Problem with Lower Adaptive Complexity

In this paper, we study the Minimum Cost Submodular Cover (MCSC) problem over the ground set of size [Formula: see text], which aims at finding a subset with the minimal cost required so that the utility submodular function exceeds a given threshold. The problem has recently attracted a lot of attention due to its applications in various domains of operations research and artificial intelligence. However, the existing algorithms for this problem may not be effectively parallelized because of their costly adaptive complexity. This paper proposes an efficient parallel algorithm that returns a [Formula: see text]-bicriteria approximation solution within [Formula: see text] adaptive complexity, where [Formula: see text] are fixed parameters. Our algorithm requires [Formula: see text] query complexity, however, it can reduce to [Formula: see text] instead while retaining a low adaptive complexity of [Formula: see text]. Therefore, our algorithm not only achieves the same approximation guarantees as the state of the art but also significantly improves the best-known low adaptive complexity algorithm for the above problem.

Online Multiset Submodular Cover

We study the Online Multiset Submodular Cover problem (OMSC), where we are given a universe U of elements and a collection of subsets S⊆2U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}\\subseteq 2^U$$\\end{document}. Each element uj∈U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_j \\in U$$\\end{document} is associated with a nonnegative, nondecreasing, submodular polynomially computable set function fj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_j$$\\end{document}. Initially, the elements are uncovered, and therefore we pay a penalty per each unit of uncovered element. Subsets with various coverage and cost arrive online. Upon arrival of a new subset, the online algorithm must decide how many copies of the arriving subset to add to the solution. This decision is irrevocable, in the sense that the algorithm will not be able to add more copies of this subset in the future. On the other hand, the algorithm can drop copies of a subset, but such copies cannot be retrieved later. The goal is to minimize the total cost of subsets taken plus penalties for uncovered elements. We present an O(ρmax)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\sqrt{\\rho _{\\max }})$$\\end{document}-competitive algorithm for OMSC that does not dismiss subset copies that were taken into the solution, but relies on prior knowledge of the value of ρmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _{\\max }$$\\end{document}, where ρmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _{\\max }$$\\end{document} is the maximum ratio, over all subsets, between the penalties covered by a subset and its cost. We provide an Olog(ρmax)ρmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O\\left( \\log (\\rho _{\\max }) \\sqrt{\\rho _{\\max }} \\right) $$\\end{document}-competitive algorithm for OMSC that does not rely on advance knowledge of ρmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _{\\max }$$\\end{document} but uses dismissals of previously taken subsets. Finally, for the capacitated versions of the Online Multiset Multicover problem, we obtain an O(ρmax′)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\sqrt{\\rho _{\\max }'})$$\\end{document}-competitive algorithm when ρmax′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _{\\max }'$$\\end{document} is known and an Olog(ρmax′)ρmax′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O\\left( \\log (\\rho _{\\max }') \\sqrt{\\rho _{\\max }'} \\right) $$\\end{document}-competitive algorithm when ρmax′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _{\\max }'$$\\end{document} is unknown, where ρmax′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _{\\max }'$$\\end{document} is the maximum ratio over all subset incarnations between the penalties covered by this incarnation and its cost.

Evolutionary Algorithm on General Cover with Theoretically Guaranteed Approximation Ratio

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently embedded in many evolutionary algorithms. In this paper, we identify such a relation by proposing a unified analysis framework for a global simple multiobjective evolutionary algorithm (GSEMO) and apply it on a minimum weight general cover problem, which is general enough to subsume many important problems including the minimum submodular cover problem in which the submodular function is real-valued, and the minimum connected dominating set problem for which the potential function is nonsubmodular. We show that GSEMO yields theoretically guaranteed approximation ratios matching those achievable by a greedy algorithm in expected polynomial time when the potential function g is polynomial in the input size and the minimum gap between different g-values is a constant. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by National Natural Science Foundation of China [11771013, U20A2068]; Zhejiang Provincial Natural Science Foundation of China [LD19A010001].

A note for approximating the submodular cover problem over integer lattice with low adaptive and query complexities

This paper studies the Diminishing Return Submodular Cover (DRSC) problem as follows: given a diminishing return submodular function f:Z+E↦R+ over the ground set E of size n, a positive integer B as the maximum value of a coordinate of a vector in Z+E and a threshold α>0, the problem asks to find the vector with the smallest size x so that f(x)≥α. This problem has been attracted by much research recently due to its importance in machine learning and combinatorial optimization. However, because of high query and adaptive complexities, the existing algorithms might still not be efficient enough when working with large input data. This paper proposes a randomized and bicriteria approximation algorithm that has O(log⁡n) adaptive complexity and O((n+log⁡nlog⁡B)log⁡(nB)) query complexity. Our algorithm significantly reduces both adaptive and query complexities compared to some state-of-the-art algorithms by factors of Ω(log⁡(m)log2⁡(mnlog⁡B)(1+log⁡(log⁡B)/log⁡(n))), Ω(min⁡{n/log⁡(n),log⁡(B)}) respectively, where m=f(B⋅1).

Minimum non-submodular cover problem with applications

Minimum Submodular Cover problem often occurs naturally in the context of combinatorial optimization. It is well-known that the greedy algorithm achieves an H(δ)-approximation guarantee for an integer-valued polymatroid potential function f, where δ is the maximum value of f over all singletons and H(δ) is the δ-th harmonic number. In this paper, we extend the setting into the non-submodular potential functions and investigate Minimum Non-submodular Cover problem with integer-valued and fraction-valued potential functions respectively, yielding similar performance results. In addition, we address several real-world applications which can be formulated as Minimum Non-submodular Cover problem.

Algorithms and Complexity for a Class of Combinatorial Optimization Problems with Labelling

In this paper, we propose to study a wide class of combinatorial optimization problems called combinatorial optimization problems with labelling. First, we give a combinatorial method to deal with the labelling version of some classical combinatorial optimization problems including minimum vertex cover, maximum independent set, minimum dominating set and minimum set cover, and convert the labelling problems into the original problems by polynomial-time reduction. We show that, although the labelling version of these problem seems more universal than their original counterparts, they are actually equivalent to the corresponding original problem from an algorithmic point of view. Moreover, we generalize the greedy approach for solving submodular cover problem to its labelling version, and as simple applications of our new method, we use it to solve the labelling versions of the minimum weighted spanning tree and connected vertex cover problem in a unified way.

Precautionary rumor containment via trustworthy people in social networks

In a social network, rumor containment is vital, as the diffusion of a rumor will bring terrible results. Precautionary measure can be used to control rumor propagation: Anticipating the spread of a rumor, one can (1) select a set of trustworthy people (TP) in the network, (2) alert the TP about the rumor, and (3) ask the TP to protect their neighbors by sending out alerts. In this paper, we study the problem of how to select the least number of TP, satisfying the requirement that the entire network is protected by the alerts that the TP send. We propose an asymmetric trust (AT) information propagation model. Under this model, we study the Least Number TP Selection (LNTS) problem, establish its NP-hardness and reformulate it as a minimum submodular cover problem. As a result, the Greedy Algorithm is a constant-factor approximation algorithm. Using real-world data, we evaluate the performance of the Greedy Algorithm, and compare it with other algorithms. Experimental results indicate that the Greedy Algorithm performs the best among its competitors.

Greedy approximations for minimum submodular cover with submodular cost

It is well-known that a greedy approximation with an integer-valued polymatroid potential function f is H(γ)-approximation of the minimum submodular cover problem with linear cost where γ is the maximum value of f over all singletons and H(γ) is the γ-th harmonic number. In this paper, we establish similar results for the minimum submodular cover problem with a submodular cost (possibly nonlinear) and/or fractional submodular potential function f.