Matroid products via submodular coupling

The study of matroid products traces back to the 1970s, when Lovász and Mason studied the existence of various types of matroid products with different strengths. Among these, the tensor product is arguably the most important, which can be considered as an extension of the tensor product from linear algebra. However, Las Vergnas showed that…

Cyclic ordering of split matroids

There is a long list of open questions rooted in the same underlying problem: understanding the structure of bases or common bases of matroids. These conjectures suggest that matroids may possess much stronger structural properties than are currently known. One example is related to cyclic orderings of matroids. A rank-r matroid is called cyclically orderable…

Multiway cuts

In the paper Multiway Cuts with a Choice of Representatives, appearing at MFCS 2024, we study several generalizations of multiway cut where the terminals can be chosen as representatives from sets of candidates $T_1,\dots,T_q$. In this setting, one is allowed to choose these representatives so that the minimum-weight cut separating these sets via their representatives…

Matroid intersection under minimum rank oracle

In the paper paper Matroid Intersection under Minimum Rank Oracle, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements, and returns as its output the minimum of the ranks of the given set…

Spanning trees with perfect matchings

In our recent paper titled Finding Spanning Trees with Perfect Matchings, we investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight spanning tree among those containing a perfect matching. On the…