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The Minimum Vertex Cover P roblem is the optimization problem of finding a vertex cover V c of minimal cardinality in a given graph. It is a classic NP-hard problem, and various algorithms have been suggested for it. In this paper, we start with a basic algorithm for solving the problem. Using a probabilistic idea, we use it to develop an improved algorithm. The algorithm is greedy; at each step it adds to the cover a vertex such that the expected cover size, if we continue randomly after this step, is minimal. We study the new algorithm theoretically and empirically, and run simulations to compare its performance to that of some algorithms of a similar nature. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of partitioning a convex polygon P of n vertices into m polygons of equal area and perimeter. We give an algorithm for m = 2 that runs in O ( n ) time, and an algorithm for m = 2 k , where k is an integer, that runs in O ( ( 2 n ) k ) time. These are the first algorithms to solve this problem. [ABSTRACT FROM AUTHOR]

Given a graph G = ( V , E ) with a positive weight function w on the vertices of G , a global powerful alliance of G is a subset S of V such that for every vertex v at least half of the total weight in the closed neighborhood of v is contributed by the vertices of S . A global offensive alliance is a subset S of V such that the above condition holds for every vertex v not in S . Finding the smallest such set in general graphs is NP-complete for both of these problems, even when the weights are all the same. In this paper, we give linear time algorithms that find the smallest global offensive and global powerful alliance of any weighted tree T = ( V , E ) . [ABSTRACT FROM AUTHOR]