Realizing symmetric set functions as hypergraph cut capacity

Realizing symmetric set functions as hypergraph cut capacity

A set function is a function defined on a set family. It is said to be symmetric if the value for each set coincides with that for its complement. A cut capacity function of an undirected graph or hypergraph is a fundamental example of symmetric set functions, and is also submodular if the capacity on each edge or hyperedge is nonnegative. Fujishige and Patkar (2001) provided necessary and suff...

Minimizing Symmetric Set Functions Faster

We describe a combinatorial algorithm which, given a monotone and consistent symmetric set function d on a finite set V in the sense of Rizzi [Riz00], constructs a non trivial set S minimizing d(S,V \ S). This includes the possibility for the minimization of symmetric submodular functions. The presented algorithm requires at most as much time as the one in [Riz00], but depending on the function...

A New Approach to Realizing Partially Symmetric Functions

Hypergraph k-Cut in Randomized Polynomial Time

In the hypergraph k-cut problem, the input is a hypergraph, and the goal is to find a smallest subset of hyperedges whose removal ensures that the remaining hypergraph has at least k connected components. This problem is known to be at least as hard as the densest k-subgraph problem when k is part of the input (Chekuri-Li, 2015). We present a randomized polynomial time algorithm to solve the hy...

The quasi-randomness of hypergraph cut properties

Let α1, . . . , αk satisfy ∑ i αi = 1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1, . . . , Ak of sizes α1n, . . . , αkn, the number of edges intersecting A1, . . . , Ak is (asymptotically) the number one would expect to find in a random k-uniform hypergraph. Can we then infer that H is quasi-random? We show t...