In this work,we focus on maximizing the ratio of two monotone DR-submodular functions on the integer lattice.It is neither submodular nor supermodular.We prove that the Threshold Decrease Algorithm is a 1-e^(1-kg)-εa...In this work,we focus on maximizing the ratio of two monotone DR-submodular functions on the integer lattice.It is neither submodular nor supermodular.We prove that the Threshold Decrease Algorithm is a 1-e^(1-kg)-εapproximation ratio algorithm.Additionally,we construct the relationship between maximizing the ratio of two monotone DR-submodular functions and maximizing the difference of two monotone DR-submodular functions on the integer lattice.Based on this relationship,we combine the dichotomy technique and Threshold Decrease Algorithm to solve maximizing the difference of two monotone DR-submodular functions on the integer lattice and prove its approximation ratio is f(x)-g(x)≥1-e^(1-kg)f(X^(*))-g(X^(*)).To the best of our knowledge,before us,there was no work to focus on maximizing the ratio of two monotone DR-submodular functions on integer lattice by using the Threshold Decrease Algorithm.展开更多
In this paper,we study a class of online continuous optimization problems.At each round,the utility function is the sum of a weakly diminishing-returns(DR)submodular function and a concave function,certain cost associ...In this paper,we study a class of online continuous optimization problems.At each round,the utility function is the sum of a weakly diminishing-returns(DR)submodular function and a concave function,certain cost associated with the action will occur,and the problem has total limited budget.Combining the two methods,the penalty function and Frank-Wolfe strategies,we present an online method to solve the considered problem.Choosing appropriate stepsize and penalty parameters,the performance of the online algorithm is guaranteed,that is,it achieves sub-linear regret bound and certain mild constraint violation bound in expectation.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
基金upported by the China Scholarship Council(No.202004910755)supported by the National Natural Science Foundation of China(Nos.12071459 and 11991022)+2 种基金the Fundamental Research Funds for Central Universities(No.E1E40107X2)supported by the Natural Sciences and Engineering Research Council of Canada(No.283106)the National Natural Science Foundation of China(Nos.11771386 and 11728104)。
文摘In this work,we focus on maximizing the ratio of two monotone DR-submodular functions on the integer lattice.It is neither submodular nor supermodular.We prove that the Threshold Decrease Algorithm is a 1-e^(1-kg)-εapproximation ratio algorithm.Additionally,we construct the relationship between maximizing the ratio of two monotone DR-submodular functions and maximizing the difference of two monotone DR-submodular functions on the integer lattice.Based on this relationship,we combine the dichotomy technique and Threshold Decrease Algorithm to solve maximizing the difference of two monotone DR-submodular functions on the integer lattice and prove its approximation ratio is f(x)-g(x)≥1-e^(1-kg)f(X^(*))-g(X^(*)).To the best of our knowledge,before us,there was no work to focus on maximizing the ratio of two monotone DR-submodular functions on integer lattice by using the Threshold Decrease Algorithm.
基金supported by the National Natural Science Foundation of China(Nos.12101587,12001025,and 12131003)the China Postdoctoral Science Foundation(No.2022M720329)the Postdoctoral Research Foundation of Chaoyang District(Nos.Q1006E11202301 and Q1006E11202302).
文摘In this paper,we study a class of online continuous optimization problems.At each round,the utility function is the sum of a weakly diminishing-returns(DR)submodular function and a concave function,certain cost associated with the action will occur,and the problem has total limited budget.Combining the two methods,the penalty function and Frank-Wolfe strategies,we present an online method to solve the considered problem.Choosing appropriate stepsize and penalty parameters,the performance of the online algorithm is guaranteed,that is,it achieves sub-linear regret bound and certain mild constraint violation bound in expectation.
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).
