Efficient Algorithms for a Graph Partitioning Problem

• Published in: Frontiers in Algorithmics Vol. 10823; pp. 29 - 42
• Main Authors: Vaishali, S., Atulya, M. S., Purohit, Nidhi
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2018; Springer International Publishing
• Series: Lecture Notes in Computer Science
• ISBN: 3319784544; 9783319784540
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-78455-7_3

Motivated by an expensive computation performed by a computational topology software RIVET [9], Madkour et al. [1] introduced and studied the following graph partitioning problem. Given an edge weighted graph and an integer k, partition the vertex set of the graph into k connected components such that the weight of the heaviest component is as small as possible, where the weight of each component is the weight of a minimum spanning tree of the graph induced by the vertices in that component. They showed that the problem is NP-hard even for $$k=2$$ and provided some heuristic algorithms. They asked whether the problem is polynomial time solvable when the input is a tree. Our first result is an affirmative answer to their question. We give a polynomial time algorithm to provide such a partition in a tree. We also give an exact exponential algorithm taking $$O^*(2^n)$$ time on general graphs (improving on the naive $$O^*(k^n)$$ algorithm) ( $$O^*$$ notation ignores polynomial factors). We also prove that the problem remains NP-complete even when the weights on all the edges are the same and give a linear time algorithm for this version of the problem when the graph is a tree.