Submodular maximization is a significant area of interest in combinatorial optimization.It has various real-world applications.In recent years,streaming algorithms for submodular maximization have gained attention,all...Submodular maximization is a significant area of interest in combinatorial optimization.It has various real-world applications.In recent years,streaming algorithms for submodular maximization have gained attention,allowing realtime processing of large data sets by examining each piece of data only once.However,most of the current state-of-the-art algorithms are only applicable to monotone submodular maximization.There are still significant gaps in the approximation ratios between monotone and non-monotone objective functions.In this paper,we propose a streaming algorithm framework for non-monotone submodular maximization and use this framework to design deterministic streaming algorithms for the d-knapsack constraint and the knapsack constraint.Our 1-pass streaming algorithm for the d-knapsack constraint has a 1/4(d+1)-∈approximation ratio,using O(BlogB/∈)memory,and O(logB/∈)query time per element,where B=MIN(n,b)is the maximum number of elements that the knapsack can store.As a special case of the d-knapsack constraint,we have the 1-pass streaming algorithm with a 1/8-∈approximation ratio to the knapsack constraint.To our knowledge,there is currently no streaming algorithm for this constraint when the objective function is non-monotone,even when d=1.In addition,we propose a multi-pass streaming algorithm with 1/6-∈approximation,which stores O(B)elements.展开更多
In this paper,we investigate the maximization of the differences between a nonnegative monotone diminishing return submodular(DR-submodular)function and a nonnegative linear function on the integer lattice.As it is al...In this paper,we investigate the maximization of the differences between a nonnegative monotone diminishing return submodular(DR-submodular)function and a nonnegative linear function on the integer lattice.As it is almost unapproximable for maximizing a submodular function without the condition of nonnegative,we provide weak(bifactor)approximation algorithms for this problem in two online settings,respectively.For the unconstrained online model,we combine the ideas of single-threshold greedy,binary search and function scaling to give an efficient algorithm with a 1/2 weak approximation ratio.For the online streaming model subject to a cardinality constraint,we provide a one-pass(3-√5)/2 weak approximation ratio streaming algorithm.Its memory complexity is(k log k/ε),and the update time for per element is(log^(2)k/ε).展开更多
In this work,we study a k-Cardinality Constrained Regularized Submodular Maximization(k-CCRSM)problem,in which the objective utility is expressed as the difference between a non-negative submodular and a modular funct...In this work,we study a k-Cardinality Constrained Regularized Submodular Maximization(k-CCRSM)problem,in which the objective utility is expressed as the difference between a non-negative submodular and a modular function.No multiplicative approximation algorithm exists for the regularized model,and most works have focused on designing weak approximation algorithms for this problem.In this study,we consider the k-CCRSM problem in a streaming fashion,wherein the elements are assumed to be visited individually and cannot be entirely stored in memory.We propose two multipass streaming algorithms with theoretical guarantees for the above problem,wherein submodular terms are monotonic and nonmonotonic.展开更多
We investigate the problem of maximizing the sum of submodular and supermodular functions under a fairness constraint.This sum function is non-submodular in general.For an offline model,we introduce two approximation ...We investigate the problem of maximizing the sum of submodular and supermodular functions under a fairness constraint.This sum function is non-submodular in general.For an offline model,we introduce two approximation algorithms:A greedy algorithm and a threshold greedy algorithm.For a streaming model,we propose a one-pass streaming algorithm.We also analyze the approximation ratios of these algorithms,which all depend on the total curvature of the supermodular function.The total curvature is computable in polynomial time and widely utilized in the literature.展开更多
Two-stage submodular maximization problem under cardinality constraint has been widely studied in machine learning and combinatorial optimization.In this paper,we consider knapsack constraint.In this problem,we give n...Two-stage submodular maximization problem under cardinality constraint has been widely studied in machine learning and combinatorial optimization.In this paper,we consider knapsack constraint.In this problem,we give n articles and m categories,and the goal is to select a subset of articles that can maximize the function F(S).Function F(S)consists of m monotone submodular functions fj,j=1,2,…,m,and each fj measures the similarity of each article in category j.We present a constant-approximation algorithm for this problem.展开更多
In this paper,we consider the parallel-machine customer order scheduling with delivery time and submodular rejection penalties.In this problem,we are given m dedicated machines in parallel and n customer orders.Each o...In this paper,we consider the parallel-machine customer order scheduling with delivery time and submodular rejection penalties.In this problem,we are given m dedicated machines in parallel and n customer orders.Each order has a delivery time and consists of m product types and each product type should be manufactured on a dedicated machine.An order is either rejected,in which case a rejection penalty has to be paid,or accepted and manufactured on the m dedicated machines.The objective is to find a solution to minimize the sum of the maximum delivery completion time of the accepted orders and the penalty of the rejected orders which is determined by a submodular function.We design an LP rounding algorithm with approximation ratio of n+1 for this problem.展开更多
近年来,无线能量传输技术(Wireless Power Transmission,WPT)快速发展.这促使在无线可充电传感器网络系统中可部署或调度充电器为可充电设备进行能量补充,以维持系统运行的持续性.基于此,研究者提出多种合作充电模型和相应的调度方法,...近年来,无线能量传输技术(Wireless Power Transmission,WPT)快速发展.这促使在无线可充电传感器网络系统中可部署或调度充电器为可充电设备进行能量补充,以维持系统运行的持续性.基于此,研究者提出多种合作充电模型和相应的调度方法,但是当前大部分部署方法仅考虑成本受限约束,而忽略了可充电设备可能具有空间占用的属性.因此,本文考虑了具有空间占用且充电成本受限的可移动传感器调度问题(Charging Cost-Constrained Scheduling,CCS).进一步地,本文以最大化充电效用为目的,提出了一个基于贪心的近似比为(1-1/e)的近似算法.大量仿真实验证明本文算法的优越性,该算法与传统算法对比充电效用提升30%,与粒子群算法对比充电效用提升5%.展开更多
The key issue in top-k retrieval, finding a set of k documents （from a large document collection） that can best answer a user＇s query, is to strike the optimal balance between relevance and diversity. In this paper...The key issue in top-k retrieval, finding a set of k documents （from a large document collection） that can best answer a user＇s query, is to strike the optimal balance between relevance and diversity. In this paper, we study the top-k re- trieval problem in the framework of facility location analysis and prove he submodularity of that objective function which provides a theoretical approximation guarantee of factor 1 -1/ε for the （best-first） greedy search algorithm. Furthermore, we propose a two-stage hybrid search strategy which first ob- tains a high-quality initial set of top-k documents via greedy search, and then refines that result set iteratively via local search. Experiments on two large TREC benchmark datasets show that our two-stage hybrid search strategy approach can supersede the existing ones effectively and efficiently.展开更多
In this paper, we study the dynamic facility location problem with submodular penalties （DFLPSP）. We present a combinatorial primal-dual 3-approximation algorithm for the DFLPSP.
In many kinds of games with economic significance,it is very important to study the submodularity of functions.In this paper,wemainly study the problem of maximizing a concave function over an intersection of two matr...In many kinds of games with economic significance,it is very important to study the submodularity of functions.In this paper,wemainly study the problem of maximizing a concave function over an intersection of two matroids.We obtain that the submod-ularity may not be preserved,but it involves one maximal submodular problem(or minimal supermodular problem)with some conditions.Moreover,we also present examples showing that these conditions can be satisfied.展开更多
Recent progress in maximizing submodular functions with a cardinality constraint through centralized and streaming modes has demonstrated a wide range of applications and also developed comprehensive theoretical guara...Recent progress in maximizing submodular functions with a cardinality constraint through centralized and streaming modes has demonstrated a wide range of applications and also developed comprehensive theoretical guarantees.The submodularity was investigated to capture the diversity and representativeness of the utilities,and the monotonicity has the advantage of improving the coverage.Regularized submodular optimization models were developed in the latest studies(such as a house on fire),which aimed to sieve subsets with constraints to optimize regularized utilities.This study is motivated by the setting in which the input stream is partitioned into several disjoint parts,and each part has a limited size constraint.A first threshold-based bicriteria(1/3,2/3/)-approximation for the problem is provided.展开更多
In this paper,we mainly investigate the optimization model that minimizes the cost function such that the cover function exceeds a required threshold in the set cover problem,where the cost function is additive linear...In this paper,we mainly investigate the optimization model that minimizes the cost function such that the cover function exceeds a required threshold in the set cover problem,where the cost function is additive linear,and the cover function is non-monotone approximately submodular.We study the problem under streaming model and propose three bicriteria approximation algorithms.Firstly,we provide an intuitive streaming algorithm under the assumption of known optimal objective value.The intuitive streaming algorithm returns a solution such that its cover function value is no less thanα(1−ϵ)times threshold,and the cost function is no more than(2+ϵ)^(2)/(ϵ^(2)ω^(2))⋅κ,whereκis a value that we suppose for the optimal solution andαis the approximation ratio of an algorithm for unconstrained maximization problem that we can call directly.Next we present a bicriteria streaming algorithm scanning the ground set multi-pass to weak the assumption that we guess the optimal objective value in advance,and maintain the same bicriteria approximation ratio.Finally we modify the multi-pass streaming algorithm to a single-pass one without compromising the performance ratio.Additionally,we also propose some numerical experiments to test our algorithm’s performance comparing with some existing methods.展开更多
Recently intensive interest has been raised on approximation of the NPhard submodular maximization problem due to their theoretical and practical significance.In this work,we extend this line of research by focusing o...Recently intensive interest has been raised on approximation of the NPhard submodular maximization problem due to their theoretical and practical significance.In this work,we extend this line of research by focusing on the simultaneous approximation of multiple submodular function maximization.We address the existence and nonexistence results for both deterministic and randomized approximation when the submodular functions are symmetric and asymmetric,respectively,along with algorithmic corollaries.We offer complete characterization of the symmetric case and partial results on the asymmetric case.展开更多
In this paper,we consider the generalized prize-collecting Steiner forest problem with submodular penalties(GPCSF-SP problem).In this problem,we are given an undirected connected graph G=(V,E)and a collection of disjo...In this paper,we consider the generalized prize-collecting Steiner forest problem with submodular penalties(GPCSF-SP problem).In this problem,we are given an undirected connected graph G=(V,E)and a collection of disjoint vertex subsets V={V_(1),V_(2),…,V_(l)}.Assume c:E→R_(+)is an edge cost function andπ:2^(V)→R_(+)is a submodular penalty function.The objective of the GPCSF-SP problem is to find an edge subset F such that the total cost including the edge cost in F and the penalty cost of the subcollection S containing these Vi not connected by F is minimized.By using the primal-dual technique,we give a 3-approximation algorithm for this problem.展开更多
It is shown that for a valid non-cooperative utility system,if the social utility function is submodular,then any Nash equilibrium achieves at least 1/2 of the optimal social utility,subject to a function-dependent ad...It is shown that for a valid non-cooperative utility system,if the social utility function is submodular,then any Nash equilibrium achieves at least 1/2 of the optimal social utility,subject to a function-dependent additive term.Moreover,if the social utility function is nondecreasing and submodular,then any Nash equilibrium achieves at least 1/(1+c)of the optimal social utility,where c is the curvature of the social utility function.In this paper,we consider variations of the utility system considered by Vetta,in which users are grouped together.Our aim is to establish how grouping and cooperation among users affect performance bounds.We consider two types of grouping.The first type is from a previous paper,where each user belongs to a group of users having social ties with it.For this type of utility system,each user’s strategy maximises its social group utility function,giving rise to the notion of social-aware Nash equilibrium.We prove that this social utility system yields to the bounding results of Vetta for non-cooperative system,thus establishing provable performance guarantees for the social-aware Nash equilibria.For the second type of grouping we consider,the set of users is partitioned into l disjoint groups,where the users within a group cooperate to maximise their group utility function,giving rise to the notion of group Nash equilibrium.In this case,each group can be viewed as a new user with vector-valued actions,and a 1/2 bound for the performance of group Nash equilibria follows from the result of Vetta.But as we show tighter bounds involving curvature can be established.By defining the group curvature cki associated with group i with ki users,we show that if the social utility function is nondecreasing and submodular,then any group Nash equilibrium achieves at least 1/(1+max1≤i≤l cki)of the optimal social utility,which is tighter than that for the case without grouping.As a special case,if each user has the same action space,then we have that any group Nash equilibrium achieves at least 1/(1+ck∗)of the optimal social utility,where k∗is the least number of users among the l groups.Finally,we present an example of a utility system for database-assisted spectrum access to illustrate our results.展开更多
在单无人机辅助的移动边缘计算系统中,为使无人机能服务于大区域中的所有用户设备,可将大区域分成多个子区域,并设定无人机以固定路线在各个子区域间飞行来为用户设备提供计算服务。考虑到用户设备计算资源较匮乏且无人机覆盖区域外的...在单无人机辅助的移动边缘计算系统中,为使无人机能服务于大区域中的所有用户设备,可将大区域分成多个子区域,并设定无人机以固定路线在各个子区域间飞行来为用户设备提供计算服务。考虑到用户设备计算资源较匮乏且无人机覆盖区域外的用户可选择移动至覆盖区域内进行任务卸载以最大化自身效用,可将用户设备的部分卸载问题转化为每个用户设备的效用最大化问题,并利用混合策略博弈和子模博弈来分别确定用户设备的移动概率和卸载数据量,从而得出最优卸载策略,且分别证明了混合策略纳什均衡和纯策略纳什均衡的存在性。仿真结果表明,所提方案与MBO(Binary Offloading Based on Mixed Strategy Game)等经典方案相比可有效提高用户设备的效用,并验证了其收敛性和稳定性。展开更多
基金supported in part by the National Natural Science Foundation of China(Grant Nos.62325210 and 62272441).
文摘Submodular maximization is a significant area of interest in combinatorial optimization.It has various real-world applications.In recent years,streaming algorithms for submodular maximization have gained attention,allowing realtime processing of large data sets by examining each piece of data only once.However,most of the current state-of-the-art algorithms are only applicable to monotone submodular maximization.There are still significant gaps in the approximation ratios between monotone and non-monotone objective functions.In this paper,we propose a streaming algorithm framework for non-monotone submodular maximization and use this framework to design deterministic streaming algorithms for the d-knapsack constraint and the knapsack constraint.Our 1-pass streaming algorithm for the d-knapsack constraint has a 1/4(d+1)-∈approximation ratio,using O(BlogB/∈)memory,and O(logB/∈)query time per element,where B=MIN(n,b)is the maximum number of elements that the knapsack can store.As a special case of the d-knapsack constraint,we have the 1-pass streaming algorithm with a 1/8-∈approximation ratio to the knapsack constraint.To our knowledge,there is currently no streaming algorithm for this constraint when the objective function is non-monotone,even when d=1.In addition,we propose a multi-pass streaming algorithm with 1/6-∈approximation,which stores O(B)elements.
基金supported by the National Natural Science Foundation of China(Nos.12001025 and 12131003)The second author is supported by the Natural Sciences and Engineering Research Council(No.06446),and the National Natural Science Foundation of China(Nos.11771386 and 11728104)+2 种基金The third author is supported by the National Natural Science Foundation of China(Nos.11501171 and 11771251)the Province Natural Science Foundation of Shandong(No.ZR2020MA028)The fourth author is supported by the National Natural Science Foundation of China(No.11701150)。
文摘In this paper,we investigate the maximization of the differences between a nonnegative monotone diminishing return submodular(DR-submodular)function and a nonnegative linear function on the integer lattice.As it is almost unapproximable for maximizing a submodular function without the condition of nonnegative,we provide weak(bifactor)approximation algorithms for this problem in two online settings,respectively.For the unconstrained online model,we combine the ideas of single-threshold greedy,binary search and function scaling to give an efficient algorithm with a 1/2 weak approximation ratio.For the online streaming model subject to a cardinality constraint,we provide a one-pass(3-√5)/2 weak approximation ratio streaming algorithm.Its memory complexity is(k log k/ε),and the update time for per element is(log^(2)k/ε).
基金This work was supported by the Beijing Natural Science Foundation Project(No.Z220004)the National Natural Science Foundation of China(Nos.11901544 and 12101587)the China Postdoctoral Science Foundation(No.2022M720329).
文摘In this work,we study a k-Cardinality Constrained Regularized Submodular Maximization(k-CCRSM)problem,in which the objective utility is expressed as the difference between a non-negative submodular and a modular function.No multiplicative approximation algorithm exists for the regularized model,and most works have focused on designing weak approximation algorithms for this problem.In this study,we consider the k-CCRSM problem in a streaming fashion,wherein the elements are assumed to be visited individually and cannot be entirely stored in memory.We propose two multipass streaming algorithms with theoretical guarantees for the above problem,wherein submodular terms are monotonic and nonmonotonic.
基金The first author was supported by the National Natural Science Foundation of China(Nos.12001025 and 12131003)The second author was supported by the Spark Fund of Beijing University of Technology(No.XH-2021-06-03)+2 种基金The third author was supported by the Natural Sciences and Engineering Research Council of Canada(No.283106)the Natural Science Foundation of China(Nos.11771386 and 11728104)The fourth author is supported by the National Natural Science Foundation of China(No.12001335).
文摘We investigate the problem of maximizing the sum of submodular and supermodular functions under a fairness constraint.This sum function is non-submodular in general.For an offline model,we introduce two approximation algorithms:A greedy algorithm and a threshold greedy algorithm.For a streaming model,we propose a one-pass streaming algorithm.We also analyze the approximation ratios of these algorithms,which all depend on the total curvature of the supermodular function.The total curvature is computable in polynomial time and widely utilized in the literature.
基金supported by the National Natural Science Foundation of China(Nos.12131003,12271259,11371001,11771386,and 11728104)the Natural Sciences and Engineering Research Council of Canada(NSERC)(No.06446)+1 种基金the Natural Science Foundation of Jiangsu Province(No.BK20200267)Qinglan Project.
文摘Two-stage submodular maximization problem under cardinality constraint has been widely studied in machine learning and combinatorial optimization.In this paper,we consider knapsack constraint.In this problem,we give n articles and m categories,and the goal is to select a subset of articles that can maximize the function F(S).Function F(S)consists of m monotone submodular functions fj,j=1,2,…,m,and each fj measures the similarity of each article in category j.We present a constant-approximation algorithm for this problem.
基金the National Natural Science Foundation of China(No.11971146)the Natural Science Foundation of Hebei Province of China(Nos.A2019205089 and A2019205092)+1 种基金Hebei Province Foundation for Returnees(No.CL201714)the Graduate Innovation Grant Program of Hebei Normal University(No.CXZZSS2022053).
文摘In this paper,we consider the parallel-machine customer order scheduling with delivery time and submodular rejection penalties.In this problem,we are given m dedicated machines in parallel and n customer orders.Each order has a delivery time and consists of m product types and each product type should be manufactured on a dedicated machine.An order is either rejected,in which case a rejection penalty has to be paid,or accepted and manufactured on the m dedicated machines.The objective is to find a solution to minimize the sum of the maximum delivery completion time of the accepted orders and the penalty of the rejected orders which is determined by a submodular function.We design an LP rounding algorithm with approximation ratio of n+1 for this problem.
文摘近年来,无线能量传输技术(Wireless Power Transmission,WPT)快速发展.这促使在无线可充电传感器网络系统中可部署或调度充电器为可充电设备进行能量补充,以维持系统运行的持续性.基于此,研究者提出多种合作充电模型和相应的调度方法,但是当前大部分部署方法仅考虑成本受限约束,而忽略了可充电设备可能具有空间占用的属性.因此,本文考虑了具有空间占用且充电成本受限的可移动传感器调度问题(Charging Cost-Constrained Scheduling,CCS).进一步地,本文以最大化充电效用为目的,提出了一个基于贪心的近似比为(1-1/e)的近似算法.大量仿真实验证明本文算法的优越性,该算法与传统算法对比充电效用提升30%,与粒子群算法对比充电效用提升5%.
基金This work was supported by the National Natural Science Foundation of China （Grant Nos. 61572135 and 61170085）, 973 project （2010CB328106）, Program for New Century Excellent Talents in China （NCET-10-0388）.
文摘The key issue in top-k retrieval, finding a set of k documents （from a large document collection） that can best answer a user＇s query, is to strike the optimal balance between relevance and diversity. In this paper, we study the top-k re- trieval problem in the framework of facility location analysis and prove he submodularity of that objective function which provides a theoretical approximation guarantee of factor 1 -1/ε for the （best-first） greedy search algorithm. Furthermore, we propose a two-stage hybrid search strategy which first ob- tains a high-quality initial set of top-k documents via greedy search, and then refines that result set iteratively via local search. Experiments on two large TREC benchmark datasets show that our two-stage hybrid search strategy approach can supersede the existing ones effectively and efficiently.
基金Supported in part by Hebei Province Department of Education Fund under Grant No.Z2012017the National Natural Science Foundation of China under Grant No.11371001 and 11201013
文摘In this paper, we study the dynamic facility location problem with submodular penalties （DFLPSP）. We present a combinatorial primal-dual 3-approximation algorithm for the DFLPSP.
基金supported by Higher Educational Science and Technology Program of Shandong Province(No.J17KA171)Natural Science and Engineering Research Council of Canada(No.06446)+1 种基金the National Natural Science Foundation of China(No.11871081)Science and Technology Program of Beijing Education Commission(No.KM201810005006).
文摘In many kinds of games with economic significance,it is very important to study the submodularity of functions.In this paper,wemainly study the problem of maximizing a concave function over an intersection of two matroids.We obtain that the submod-ularity may not be preserved,but it involves one maximal submodular problem(or minimal supermodular problem)with some conditions.Moreover,we also present examples showing that these conditions can be satisfied.
基金This work was supported by the Beijing Natural Science Foundation Project(No.Z200002)the National Natural Science Foundation of China(Nos.12001523,12131003,and 12101587)+1 种基金the National Innovation and Entrepreneurship Training Program for College Students of Beijing University of Technology(No.GJDC-2022-01-39)the China Postdoctoral Science Foundation(No.2022M720329).
文摘Recent progress in maximizing submodular functions with a cardinality constraint through centralized and streaming modes has demonstrated a wide range of applications and also developed comprehensive theoretical guarantees.The submodularity was investigated to capture the diversity and representativeness of the utilities,and the monotonicity has the advantage of improving the coverage.Regularized submodular optimization models were developed in the latest studies(such as a house on fire),which aimed to sieve subsets with constraints to optimize regularized utilities.This study is motivated by the setting in which the input stream is partitioned into several disjoint parts,and each part has a limited size constraint.A first threshold-based bicriteria(1/3,2/3/)-approximation for the problem is provided.
基金This work was supported by the National Natural Science Foundation of China(Nos.72192804,72192800,and 12201619)the China Postdoctoral Science Foundation(No.2022M723333).
文摘In this paper,we mainly investigate the optimization model that minimizes the cost function such that the cover function exceeds a required threshold in the set cover problem,where the cost function is additive linear,and the cover function is non-monotone approximately submodular.We study the problem under streaming model and propose three bicriteria approximation algorithms.Firstly,we provide an intuitive streaming algorithm under the assumption of known optimal objective value.The intuitive streaming algorithm returns a solution such that its cover function value is no less thanα(1−ϵ)times threshold,and the cost function is no more than(2+ϵ)^(2)/(ϵ^(2)ω^(2))⋅κ,whereκis a value that we suppose for the optimal solution andαis the approximation ratio of an algorithm for unconstrained maximization problem that we can call directly.Next we present a bicriteria streaming algorithm scanning the ground set multi-pass to weak the assumption that we guess the optimal objective value in advance,and maintain the same bicriteria approximation ratio.Finally we modify the multi-pass streaming algorithm to a single-pass one without compromising the performance ratio.Additionally,we also propose some numerical experiments to test our algorithm’s performance comparing with some existing methods.
基金supported by the Natural Sciences and Engineering Research Council of Canada(NSERC,No.283103)This work was partially done while the second author was a visiting doctorate student at the Faculty of Business Administration,University of New Brunswick and supported in part by NSERC(No.283103)+2 种基金The research of the third author is supported by the National Basic Research Program of China(No.2010CB732501)The fourth author’s research is supported by National Natural Science Foundation of China(No.11371001)Scientific Research Common Program of Beijing Municipal Commission of Education(No.KM201210005033).
文摘Recently intensive interest has been raised on approximation of the NPhard submodular maximization problem due to their theoretical and practical significance.In this work,we extend this line of research by focusing on the simultaneous approximation of multiple submodular function maximization.We address the existence and nonexistence results for both deterministic and randomized approximation when the submodular functions are symmetric and asymmetric,respectively,along with algorithmic corollaries.We offer complete characterization of the symmetric case and partial results on the asymmetric case.
基金This work is supported by the National Natural Science Foundation of China(No.11971146)the Natural Science Foundation of Hebei Province(Nos.A2019205089 and A2019205092)+1 种基金Hebei Province Foundation for Returnees(No.CL201714)Overseas Expertise Introduction Program of Hebei Auspices(No.25305008).
文摘In this paper,we consider the generalized prize-collecting Steiner forest problem with submodular penalties(GPCSF-SP problem).In this problem,we are given an undirected connected graph G=(V,E)and a collection of disjoint vertex subsets V={V_(1),V_(2),…,V_(l)}.Assume c:E→R_(+)is an edge cost function andπ:2^(V)→R_(+)is a submodular penalty function.The objective of the GPCSF-SP problem is to find an edge subset F such that the total cost including the edge cost in F and the penalty cost of the subcollection S containing these Vi not connected by F is minimized.By using the primal-dual technique,we give a 3-approximation algorithm for this problem.
基金NSF and Division of Computing and Communication Foundations[grant number CCF-1422658]the CSU Information Science and Technology Center(ISTeC)。
文摘It is shown that for a valid non-cooperative utility system,if the social utility function is submodular,then any Nash equilibrium achieves at least 1/2 of the optimal social utility,subject to a function-dependent additive term.Moreover,if the social utility function is nondecreasing and submodular,then any Nash equilibrium achieves at least 1/(1+c)of the optimal social utility,where c is the curvature of the social utility function.In this paper,we consider variations of the utility system considered by Vetta,in which users are grouped together.Our aim is to establish how grouping and cooperation among users affect performance bounds.We consider two types of grouping.The first type is from a previous paper,where each user belongs to a group of users having social ties with it.For this type of utility system,each user’s strategy maximises its social group utility function,giving rise to the notion of social-aware Nash equilibrium.We prove that this social utility system yields to the bounding results of Vetta for non-cooperative system,thus establishing provable performance guarantees for the social-aware Nash equilibria.For the second type of grouping we consider,the set of users is partitioned into l disjoint groups,where the users within a group cooperate to maximise their group utility function,giving rise to the notion of group Nash equilibrium.In this case,each group can be viewed as a new user with vector-valued actions,and a 1/2 bound for the performance of group Nash equilibria follows from the result of Vetta.But as we show tighter bounds involving curvature can be established.By defining the group curvature cki associated with group i with ki users,we show that if the social utility function is nondecreasing and submodular,then any group Nash equilibrium achieves at least 1/(1+max1≤i≤l cki)of the optimal social utility,which is tighter than that for the case without grouping.As a special case,if each user has the same action space,then we have that any group Nash equilibrium achieves at least 1/(1+ck∗)of the optimal social utility,where k∗is the least number of users among the l groups.Finally,we present an example of a utility system for database-assisted spectrum access to illustrate our results.
文摘在单无人机辅助的移动边缘计算系统中,为使无人机能服务于大区域中的所有用户设备,可将大区域分成多个子区域,并设定无人机以固定路线在各个子区域间飞行来为用户设备提供计算服务。考虑到用户设备计算资源较匮乏且无人机覆盖区域外的用户可选择移动至覆盖区域内进行任务卸载以最大化自身效用,可将用户设备的部分卸载问题转化为每个用户设备的效用最大化问题,并利用混合策略博弈和子模博弈来分别确定用户设备的移动概率和卸载数据量,从而得出最优卸载策略,且分别证明了混合策略纳什均衡和纯策略纳什均衡的存在性。仿真结果表明,所提方案与MBO(Binary Offloading Based on Mixed Strategy Game)等经典方案相比可有效提高用户设备的效用,并验证了其收敛性和稳定性。
