Transformations of Steiner tree problem variants have been frequently discussed in the literature. Besides allowing to easily transfer complexity results, they constitute a central pillar of exact state-of-the-art sol...Transformations of Steiner tree problem variants have been frequently discussed in the literature. Besides allowing to easily transfer complexity results, they constitute a central pillar of exact state-of-the-art solvers for well-known variants such as the Steiner tree problem in graphs. In this article transformations for both the prize-collecting Steiner tree problem and the maximum-weight connected subgraph problem to the Steiner arborescence problem are introduced for the first time. Furthermore, the considerable implications for practical solving approaches will be demonstrated, including the computation of strong upper and lower bounds.展开更多
最优Steiner树问题(Steiner tree problem,STP)是一个经典的组合优化问题,许多工程问题都可以归结为最优Steiner树问题。STP被广泛应用于通信网络、电路设计、VLSI设计等领域。然而,STP是典型的NP难问题,还没有多项式时间的精确算法求...最优Steiner树问题(Steiner tree problem,STP)是一个经典的组合优化问题,许多工程问题都可以归结为最优Steiner树问题。STP被广泛应用于通信网络、电路设计、VLSI设计等领域。然而,STP是典型的NP难问题,还没有多项式时间的精确算法求解该问题。目前,求解该问题的算法主要集中在基于启发式的近似算法、智能优化算法、信息传播算法等,并取得了很好的效果。在不同规模的网络中,基于传统遗传算法给出一种叶交叉机制(leaf crossover,LC),使用该机制的算法性能表现更好。通过对这些算法的原理、性能、精度等方面进行梳理,归纳出算法的优缺点,并指出STP的研究方向和算法设计路径,对于相关问题的研究有指导意义。展开更多
In this paper,we study the prize-collecting k-Steiner tree(PCkST) problem.We are given a graph G=(V,E) and an integer k.The graph is connected and undirected.A vertex r ∈ V called root and a subset R?V called termina...In this paper,we study the prize-collecting k-Steiner tree(PCkST) problem.We are given a graph G=(V,E) and an integer k.The graph is connected and undirected.A vertex r ∈ V called root and a subset R?V called terminals are also given.A feasible solution for the PCkST is a tree F rooted at r and connecting at least k vertices in R.Excluding a vertex from the tree incurs a penalty cost,and including an edge in the tree incurs an edge cost.We wish to find a feasible solution with minimum total cost.The total cost of a tree F is the sum of the edge costs of the edges in F and the penalty costs of the vertices not in F.We present a simple approximation algorithm with the ratio of 5.9672 for the PCkST.This algorithm uses the approximation algorithms for the prize-collecting Steiner tree(PCST) problem and the k-Steiner tree(kST) problem as subroutines.Then we propose a primal-dual based approximation algorithm and improve the approximation ratio to 5.展开更多
Spanning tree problems with specialized constraints can be difficult to solve in real-world scenarios,often requiring intricate algorithmic design and exponential time.Recently,there has been growing interest in end-t...Spanning tree problems with specialized constraints can be difficult to solve in real-world scenarios,often requiring intricate algorithmic design and exponential time.Recently,there has been growing interest in end-to-end deep neural networks for solving routing problems.However,such methods typically produce sequences of vertices,which make it difficult to apply them to general combinatorial optimization problems where the solution set consists of edges,as in various spanning tree problems.In this paper,we propose NeuroPrim,a novel framework for solving various spanning tree problems by defining a Markov decision process for general combinatorial optimization problems on graphs.Our approach reduces the action and state space using Prim's algorithm and trains the resulting model using REINFORCE.We apply our framework to three difficult problems on the Euclidean space:the degree-constrained minimum spanning tree problem,the minimum routing cost spanning tree problem and the Steiner tree problem in graphs.Experimental results on literature instances demonstrate that our model outperforms strong heuristics and achieves small optimality gaps of up to 250 vertices.Additionally,we find that our model has strong generalization ability with no significant degradation observed on problem instances as large as 1,000.Our results suggest that our framework can be effective for solving a wide range of combinatorial optimization problems beyond spanning tree problems.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
文摘Transformations of Steiner tree problem variants have been frequently discussed in the literature. Besides allowing to easily transfer complexity results, they constitute a central pillar of exact state-of-the-art solvers for well-known variants such as the Steiner tree problem in graphs. In this article transformations for both the prize-collecting Steiner tree problem and the maximum-weight connected subgraph problem to the Steiner arborescence problem are introduced for the first time. Furthermore, the considerable implications for practical solving approaches will be demonstrated, including the computation of strong upper and lower bounds.
文摘最优Steiner树问题(Steiner tree problem,STP)是一个经典的组合优化问题,许多工程问题都可以归结为最优Steiner树问题。STP被广泛应用于通信网络、电路设计、VLSI设计等领域。然而,STP是典型的NP难问题,还没有多项式时间的精确算法求解该问题。目前,求解该问题的算法主要集中在基于启发式的近似算法、智能优化算法、信息传播算法等,并取得了很好的效果。在不同规模的网络中,基于传统遗传算法给出一种叶交叉机制(leaf crossover,LC),使用该机制的算法性能表现更好。通过对这些算法的原理、性能、精度等方面进行梳理,归纳出算法的优缺点,并指出STP的研究方向和算法设计路径,对于相关问题的研究有指导意义。
基金supported by the National Natural Science Foundation of China (Nos. 12001523,11971046,12131003,and 11871081)the Scientific Research Project of Beijing Municipal Education Commission (No. KM201910005012)Beijing Natural Science Foundation Project (No. Z200002)。
文摘In this paper,we study the prize-collecting k-Steiner tree(PCkST) problem.We are given a graph G=(V,E) and an integer k.The graph is connected and undirected.A vertex r ∈ V called root and a subset R?V called terminals are also given.A feasible solution for the PCkST is a tree F rooted at r and connecting at least k vertices in R.Excluding a vertex from the tree incurs a penalty cost,and including an edge in the tree incurs an edge cost.We wish to find a feasible solution with minimum total cost.The total cost of a tree F is the sum of the edge costs of the edges in F and the penalty costs of the vertices not in F.We present a simple approximation algorithm with the ratio of 5.9672 for the PCkST.This algorithm uses the approximation algorithms for the prize-collecting Steiner tree(PCST) problem and the k-Steiner tree(kST) problem as subroutines.Then we propose a primal-dual based approximation algorithm and improve the approximation ratio to 5.
基金supported by National Key R&D Program of China(Grant No.2021YFA1000403)National Natural Science Foundation of China(Grant No.11991022)+1 种基金the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA27000000)the Fundamental Research Funds for the Central Universities。
文摘Spanning tree problems with specialized constraints can be difficult to solve in real-world scenarios,often requiring intricate algorithmic design and exponential time.Recently,there has been growing interest in end-to-end deep neural networks for solving routing problems.However,such methods typically produce sequences of vertices,which make it difficult to apply them to general combinatorial optimization problems where the solution set consists of edges,as in various spanning tree problems.In this paper,we propose NeuroPrim,a novel framework for solving various spanning tree problems by defining a Markov decision process for general combinatorial optimization problems on graphs.Our approach reduces the action and state space using Prim's algorithm and trains the resulting model using REINFORCE.We apply our framework to three difficult problems on the Euclidean space:the degree-constrained minimum spanning tree problem,the minimum routing cost spanning tree problem and the Steiner tree problem in graphs.Experimental results on literature instances demonstrate that our model outperforms strong heuristics and achieves small optimality gaps of up to 250 vertices.Additionally,we find that our model has strong generalization ability with no significant degradation observed on problem instances as large as 1,000.Our results suggest that our framework can be effective for solving a wide range of combinatorial optimization problems beyond spanning tree problems.
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.
