On finding connected balanced partitions of trees
Maurizio Bruglieri2021, Discrete Applied Mathematics
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Abstract
Graph partitioning is a widely studied problem in the literature with several applications in real life contexts. In this paper we study the problem of partitioning a graph, with weights at its vertices, into p connected components. For each component of the partition we measure the difference between the maximum and the minimum weight of a vertex in the component. We consider two objective functions to minimize, one measuring the maximum of such differences among all the components in the partition, and the other measuring the sum of the differences between the maximum and the minimum weight of a vertex in each component. We focus our analysis on tree graphs and provide polynomial time algorithms for solving these optimization problems on such graphs. In particular, we present an O(n 2 log n) time algorithm for the min-max version of the problem on general trees and several, more efficient polynomial algorithms for some trees with a special structure, such as spiders and caterpillars. Finally, we present NP-hardness and approximation results on general graphs for both the objective functions.
Key takeaways
- In Section 5, we introduce a variant of MGGPP with an additional constraint requiring that any component of the partition has at least two vertices.
- This can be done efficiently by a binary search over all the O(n 2 ) possible values of γ, i.e., the value of all the possible differences between any two vertex weights in G. For any γ, if q(γ) > p, it is impossible to find a solution to min-max MGGPP (which requires exactly p components) having gap at most γ.
- We notice that in any partition of T v , the minimum weight of the component containing v is one of the O(n) weights of the vertices of T v .
- The minimum cost path from u 0 to u 4,3 identifies the optimal partition: v 1 and v 4 are singletons (with zero gap), v 2 and v 3 form a subgraph with gap equal to 5, the other vertices form the central subgraph, with gap equal to 10. Figure 3: Construction of the auxiliary graph to partition a spider graph into p = 4 subgraphs with v 5 and v 7 as the vertices of minimum and maximum weight in the central component A caterpillar graph is a tree which consists in a central path and, possibly, some leaves adjacent to it.
- For example, if in Figure 4 we consider the central component identified by u The min-max MGGPP2 can be solved in a similar way, testing with a binary search all possible thresholds γ, enumerating the O(n 2 ) central components that respect such a threshold, appending the white and grey vertices to build a single path, and finding whether it is possible to partition this path into p − 1 subpaths with gap ≤ γ with the algorithm described in Section 4.1.
