4 research outputs found
Bounds for the number of meeting edges in graph partitioning
summary:Let be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that admits a bipartition such that each vertex class meets edges of total weight at least , where is the total weight of edges of size and is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph (i.e., multi-hypergraph), we show that there exists a bipartition of such that each vertex class meets edges of total weight at least , where is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with edges, except for and , admits a tripartition such that each vertex class meets at least edges, which establishes a special case of a more general conjecture of Bollobás and Scott
Balanced Judicious Bipartition is Fixed-Parameter Tractable
The family of judicious partitioning problems, introduced by Bollob\u27as and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollob\u27as and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT)