共查询到20条相似文献,搜索用时 187 毫秒
1.
假设G=(V,E,F)是一个平面图。如果e1和e2是G中两条相邻边且在关联的面的边界上连续出现,那么称e1和e2面相邻。图G的一个弱完备k-染色是指存在一个从V ∪ E ∪ F到k色集合{1, …, K}的映射,使得任意两个相邻点,两个相邻面,两条面相邻的边,以及V ∪ E ∪ F中任意两个相关联的元素都染不同的颜色。若图G有一个弱完备k-染色,则称G是弱完备k-可染的。平面图G的弱完备色数是指G是弱完备k-可染的正整数k的最小值,记成χ vef(G)。2016年,Fabrici等人猜想:每个无环且无割边的连通平面图是弱完备7-可染的。证明外平面图满足猜想,即外平面图是弱完备7-可染的。 相似文献
2.
给定两个非负整数s和t,图G的(s,t)-松弛强k边着色可表示为映射c:E(G)→[k],这个映射满足对G中的任意一条边e,颜色c(e)在e的1-邻域中最多出现s次并且在e的2-邻域中最多出现t次。图G的(s,t)-松弛强边着色指数,记作χ'(s,t)(G),表示使得图G有(s,t)-松弛强k边着色的最小k值。在图G中,如果mad(G) < 3并且Δ≤4,那么χ'(1,0)(G)≤3Δ。并证明如果G是平面图,最大度Δ≥4并且围长最少为7,那么χ'(1,0)(G)≤3Δ-1。 相似文献
3.
G是一个k-连通图,T是G的一个k-点割,若G-T可被划分成两个子图G1,G2,且|G1|≥2,|G2|≥2,则称T是G的一个非平凡点割。假定G是一个不含非平凡(k-1)点割的(k-1)-连通图,则称G是一个拟k-连通图。证明了对任意一个k≥5且t> $ \frac{k}{2}$ 的整数,若G是一个不含(K2+tK1)的k-连通图,且G中任意两个不同点对v,w,有d(v)+d(w)≥ $\frac{{3k}}{2} $ +t,则对G中的任意一个点,存在一条与之关联的边收缩后可以得到一个拟k-连通图,且G中至少有$\frac{{\left| {V\left( G \right)} \right|}}{2} $ 条边使得收缩其中任意一条边后仍是拟k-连通的。 相似文献
4.
5.
考虑工件具有加工位置上限最小化总加权误工量的单机排序问题.在此排序问题中,每个工件Jj都具有一个加工位置上限kj.也就是说,如果工件Jj是一个可行排序中的第x个工件,那么就需要满足x ≤ kj.证明了(i)当工件具有相同工期时,该排序问题是二元NP-难的并且是拟多项式时间可解的,(ii)当工件具有单位权重时,该排序问题是一元NP-难的. 相似文献
6.
7.
8.
9.
10.
11.
The postman problem requires finding a lowest cost tour in a connected graph that traverses each edge at least once. In this paper we first give a brief survey of the literature on postman problems including, the original Chinese postman problem on undirected graphs, the windy Chinese postman problem on graphs where the cost of an arc depends on the direction the arc is transversed, the directed postman problem on graphs with directed edges, and the mixed postman problem on graphs in which there are some directed and some undirected arcs.We show how the mixed postman problem can be solved as an integer program, using the formulation of Gendreau, Laporte and Zhao, by a new row addition branch and bound algorithm, which is a modification of the column subtraction algorithm for set partitioning problems of Harche and Thompson. Computational experience shows that a slack variable heuristic is very effective in finding good solutions that are frequently optimal for these problems. 相似文献
12.
Gündüz Ulusoy 《European Journal of Operational Research》1985,22(3):329-337
There have been several attempts to solve the capacitated arc routing problem with m vehicles starting their tours from a central node. The objective has been to minimize the total distance travelled. In the problem treated here we also have the fixed costs of the vehicles included in the objective function. A set of vehicle capacities with their respective costs are used. Thus the objective function becomes a combination of fixed and variable costs. The solution procedure consists of four phases. In the first phase, a Chinese or rural postman problem is solved depending on whether all or some of the arcs in the network demand service with the objective of minimizing the total distance travelled. It results in a tour called the giant tour. In the second phase, the giant tour is partitioned into single vehicle subtours feasible with respect to the constraints. A new network is constructed with the node set corresponding to the arcs of the giant tour and with the arc set consisting of the subtours of the giant tour. The arc costs include both the fixed and variable costs of the subtours. The third phase consists of solving the shortest path problem on this new network to result in the least cost set of subtours represented on the new network. In the last phase a postprocessor is applied to the solution to improve it. The procedure is repeated for different giant tours to improve the final solution. The problem is extended to the case where there can be upper bounds on the number of vehicles with given capacities using a branch and bound method. Extension to directed networks is given. Some computational results are reported. 相似文献
13.
Gerold Jäger Changxing Dong Boris Goldengorin Paul Molitor Dirk Richter 《Journal of Heuristics》2014,20(1):107-124
We introduce a reduction technique for large instances of the traveling salesman problem (TSP). This approach is based on the observation that tours with good quality are likely to share many edges. We exploit this observation by neglecting the less important tour space defined by the shared edges, and searching the important tour subspace in more depth. More precisely, by using a basic TSP heuristic, we obtain a set of starting tours. We call the set of edges which are contained in each of these starting tours as pseudo-backbone edges. Then we compute the maximal paths consisting only of pseudo-backbone edges, and transform the TSP instance to another one with smaller size by contracting each such path to a single edge. This reduced TSP instance can be investigated more intensively, and each tour of the reduced instance can be expanded to a tour of the original instance. Combining our reduction technique with the currently leading TSP heuristic of Helsgaun, we experimentally investigate 32 difficult VLSI instances from the well-known TSP homepage. In our experimental results we set world records for seven VLSI instances, i.e., find better tours than the best tours known so far (two of these world records have since been improved upon by Keld Helsgaun and Yuichi Nagata, respectively). For the remaining instances we find tours that are equally good or only slightly worse than the world record tours. 相似文献
14.
On the Windy Postman Problem on eulerian graphs 总被引:1,自引:0,他引:1
Zaw Win 《Mathematical Programming》1989,44(1-3):97-112
15.
Hybrid metaheuristics for the profitable arc tour problem 总被引:1,自引:0,他引:1
The profitable arc tour problem is a variant in the vehicle routing problems. It is included in the family of the vehicle routing with profit problems in which a set of vehicle tours are constructed. The objective is to find a set of cycles in the vehicle tours that maximize the collection of profits minus travel costs, subject to constraints limiting the length of cycles that profit is available on arcs. To solve this variant we adopted two metaheuristics based on adaptive memory. We show that our algorithms provide good results in terms of solution quality and running times. 相似文献
16.
Carmine Cerrone Benjamin Dussault Xingyin Wang Bruce Golden Edward Wasil 《European Journal of Operational Research》2019,272(2):754-765
In this paper, we consider the Directed Rural Postman Problem with Turn Penalties (DRPP-TP). A solution is a tour that traverses all required arcs of the graph. The total cost of the tour is the sum of the lengths of the traversed arcs plus the penalties associated with the turns. One solution approach involves transforming the arc routing problem into an equivalent node routing problem. An alternative direct approach (without graph transformation) that involves two stages has been proposed in the literature. In the first part of this paper, we investigate the applicability of the direct approach. We identify several characteristics of the input instance that make this approach effective and present several limitations of this approach. In the second part of this paper, we describe an integer linear program that is combined with a local search algorithm. This combination produces high-quality solutions to the DRPP-TP in a reasonable amount of computing time. 相似文献
17.
Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0. 相似文献
18.
Given an edge-weighted tree T and an integer p1, the minmax p-traveling salesmen problem on a tree T asks to find p tours such that the union of the p tours covers all the vertices. The objective is to minimize the maximum of length of the p tours. It is known that the problem is NP-hard and has a (2−2/(p+1))-approximation algorithm which runs in O(pp−1np−1) time for a tree with n vertices. In this paper, we consider an extension of the problem in which the set of vertices to be covered now can be chosen as a subset S of vertices and weights to process vertices in S are also introduced in the tour length. For the problem, we give an approximation algorithm that has the same performance guarantee, but runs in O((p−1)!·n) time. 相似文献
19.
Let M=(V,E,A) be a mixed graph with vertex set V, edge set E and arc set A. A cycle cover of M is a family C={C1,…,Ck} of cycles of M such that each edge/arc of M belongs to at least one cycle in C. The weight of C is . The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small as possible. The Chinese postman problem is: given a strongly connected mixed graph M, find a minimum length closed walk using all edges and arcs of M. These problems are NP-hard. We show that they can be solved in polynomial time if M has bounded tree-width. 相似文献