An Efficient Algorithm for Judicious Partition of Hypergraphs

Data Distribution for Phylogenetic Inference with Site Repeats via Judicious Hypergraph Partitioning

“…The problem is known to be NP-hard [8] and has mainly been studied in the context of extremal combinatorics [9]- [11]. To the best of our knowledge, Tan et al [3] were the first to investigate algorithmic aspects of the problem.…”

“…Furthermore, this behavior can deteriorate solution quality, since more iterations will be required for merging two blocks and the solution cost (i.e., ∆ + d) increases with each iteration. By employing the greedy set cover algorithm, the judicious partitioning algorithm of Tan et al [3] chooses an arbitrary filler element out of all combinations of ∆ + d nets that constitute a superset of u. Since the chosen filler set S j will be included in S * and thus becomes a new element of U in the next iteration, some choices of S j may be more difficult to cover using ∆ + d + 1 nets than others.…”

“…: Both, the elements of U, and the sets of S comprise elements that are represented by hyperedges of the hypergraph. While Tan et al [3] do not explicitly discuss how set elements are represented, their example (an enhanced version of which is shown in Figure 3) already hints at a potential implementation: each element can be represented as a m-bit bitvector, where a set bit at the ith position denotes that hyperedge i is part of the set element. Thus, set unions and intersections become bit-wise and and or operations, respectively.…”

“…We initially transform the problem into a hypergraph partitioning problem and show that it corresponds to a particular instance of the so-called judicious hypergraph partitioning problem [3] (henceforth, denoted as judicious partitioning). As no efficient implementation of a judicious partitioning algorithm was available, we develop, parallelize, and present HyperPhylo 1 an efficient open-source implementation of the specific judicious partitioning flavor where all vertices have the same degree.…”

“…Modified example from Tan et al[3]: A hypergraph with 6 vertices and 4 nets is partitioned into k = 2 blocks using their algorithm. The final partition has a cost of ∆ + d = 3.…”