Constraint Minimum Vertex Cover in K Partite Graph, Approximation Algorithm and Complexity Analysis
Kamanashis Biswas, S. A. M. Harun
备注：
5 Pages IEEE format, International Journal of Computer Science and Information Security, IJCSIS 2009, ISSN 1947 5500, Impact factor 0.423, http://sites.google.com/site/ijcsis/
发表信息：
International Journal of Computer Science and Information Security, IJCSIS, Vol. 4, No. 1 & 2, August 2009, USA
Generally, a graph G, an independent set is a subset S of vertices in G such that no two vertices in S are adjacent (connected by an edge) and a vertex cover is a subset S of vertices such that each edge of G has at least one of its endpoints in S. Again, the minimum vertex cover problem is to find a vertex cover with the smallest number of vertices. This study shows that the constrained minimum vertex cover problem in k-partite graph (MIN CVCK) is NP-Complete which is an important property of k partite graph. Many combinatorial problems on general graphs are NP-complete, but when restricted to k partite graph with at most k vertices then many of these problems can be solved in polynomial time. This paper also illustrates an approximation algorithm for MIN CVCK and analyzes its complexity. In future work section, we specified a number of dimensions which may be interesting for the researchers such as developing algorithm for maximum matching and polynomial algorithm for constructing k-partite graph from general graph.
一个伪森林是一个图，它的连通分支最多有一个圈。设X是图G的伪森林调制器，i. e. G的一个顶点子集，使得G-X是一个伪森林。我们表明，顶点覆盖承认一个多项式核参数化的大小的pseudoforest调制器。换句话说，我们提供了一个多项式时间算法，对于输入图G和整数k，输出图G'和整数k'，使得G'具有O（|X| 12）个顶点，并且G具有大小为k的顶点覆盖当且仅当G'具有大小为k '的顶点覆盖。我们补充我们的研究结果证明，有没有多项式内核的顶点覆盖参数化的大小调制器到一个模拟森林（图中没有周期共享一个顶点），除非NP是一个子集的coNP/聚。特别是，这也排除了多项式内核时，参数化的调制器的大小外平面和仙人掌图。
An $O(1)$-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity
本研究考虑动态环境中的（软）容量限制顶点覆盖问题。该问题推广了近年来得到广泛研究的顶点覆盖问题的动态模型。给定一个动态变化的顶点加权图$G=(V,E)$，它允许边插入和边删除，目标是设计一个数据结构，保持一个近似的最小顶点覆盖，同时满足每个顶点的容量约束。也就是说，当拾取覆盖中的顶点$v$的副本时，由副本覆盖的$v$的关联边的数量达到给定容量的$v$。我们扩展Bhattacharya等人。的工作[SODA'15和ICALP'15]，以获得一个确定性的原始-对偶算法，用于保持一个常数因子近似最小容量顶点覆盖与$O(\log n / ε)$摊销更新时间，其中$n$是在图中的顶点数。该算法可以扩展到（1）一个更一般的模型，其中每个边与一个非均匀和不可分裂的需求，（2）更一般的容量限制集覆盖问题。
Graph Pricing Problem on Bounded Treewidth, Bounded Genus and k-partite graphs
Consider the following problem. A seller has infinite copies of $n$ products represented by nodes in a graph. There are $m$ consumers, each has a budget and wants to buy two products. Consumers are represented by weighted edges. Given the prices of products, each consumer will buy both products she wants, at the given price, if she can afford to. Our objective is to help the seller price the products to maximize her profit. This problem is called { \em graph vertex pricing} ({ \sf GVP}) problem and has resisted several recent attempts despite its current simple solution. This motivates the stu…
