Connected k-partition of k-connected graphs and c-claw-free graphs
share
A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. We consider Balanced Connected Partitions (BCP), where the two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on c-claw-free graphs, the class of graphs that do not have K_1,c as an induced subgraph, and present efficient (c-1)-approximation algorithms for both objectives. In particular, due to the (3-)claw-freeness of line graphs, this also implies a 2-approximations for the edge-partition version of BCP in general graphs. In the 1970s Győri and Lovász showed for natural numbers w_1,…,w_k where ∑_i w_i is the vertex size, that if G is k-connected, then there exist a connected k-partition with part sizes w_1,…,w_k. However, to this day no polynomial algorithm to compute such partitions exists for k>4. Towards finding such a partition T_1,…, T_k, we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each w_i is greater than the weight of the heaviest vertex. In particular, we give a 3-approximation for both the lower and the upper bounded version i.e. we guarantee that each T_i has weight at least w_i/3 or that each T_i has weight most 3w_i, respectively. Also, we present a both-side bounded version that produces a connected partition where each T_i has size at least w_i/3 and at most max({r,3}) w_i, where r ≥ 1 is the ratio between the largest and smallest value in w_1, …, w_k. In particular for the balanced version, i.e. w_1=w_2=, …,=w_k, this gives a partition with 1/3w_i ≤ w(T_i) ≤ 3w_i.
READ FULL TEXT