Approximation algorithm for the balanced 2-connected k-partition problem - Wu, Di; Zhang, Zhao; Wu, Weili (Theoretical computer science, 2016)
For two positive integers m,k and a connected graph G=(V,E) with a nonnegative vertex weight function w, the balanced m-connected k-partition problem, denoted as BCmPk, is to find a partition of V into k disjoint nonempty vertex subsets (V1,V2,…,Vk) such that each G[Vi] (the subgraph of G induced by Vi) is m-connected, and min1≤i≤k⁡{w(Vi)} is maximized. The optimal value of BCmPk on graph G is denoted as βm⁎(G,k), that is, βm⁎(G,k)=max⁡min1≤i≤k⁡{w(Vi)}, where the maximum is taken over all m-connected k-partition of G. In this paper, we study the BC2Pk problem on interval graphs, and obtain the following results. (1) For k=2, a 4/3-approximation algorithm is given for BC2P2 on 4-connected interval graphs. (2) In the case that there exists a vertex v with weight at least W/k, where W is the total weight of the graph, we prove that the BC2Pk problem on a 2k-connected interval graph G can be reduced to the BC2Pk−1 problem on the (2k−1)-connected interval graph G−v. In the case that every vertex has weight at most W/k, we prove a lower bound β2⁎(G,k)≥W/(2k−1) for 2k-connected interval graph G. (3) Assuming that weight w is integral, a pseudo-polynomial time algorithm is obtained. Combining this pseudo-polynomial time algorithm with the above lower bound, a fully polynomial time approximation scheme (FPTAS) is obtained for the BC2Pk problem on 2k-connected interval graphs.