Monomial cut-dials

In combinatorial optimization the MaxCut problem is a well-studied problem appearing in particular as one of the 21 NP-complete problems of Karp . Sturmfels and Sullivant started an interesting connection to algebraic geometry and commutative algebra by introducing toric-cut ideals and -algebras, which have been intensively studied in the last decade. A different point of studying the problem is through the monomial cut ideals I(G) of a graph G. They are monomial ideals generated by square-free monomials associated to cut vectors. The main goal of this project is to study their algebraic properties extending the work of J. Herzog, M. Rahimbeigi and T. Romer.

Some important questions:

Extending the already carried-out work by J. Herzog, M. Rahimbeigi and T. Romer, explore the case of decomposition of a given graph with subgraphs having edge-sharing. Possible point to focus, what happened to the degree of generators, the behavior of their betti numbers and other related invariants.

Translating results known for toric-ideals for monomial cut-ideals by Tim Romer and Mitra Koley.

Finding algebraic retracts for monomial cut-ideals following the exposition by Tim Romer and Sara S. Madani.

Exploring the existence and non-existence cases of algebra retracts for a given graph through neighborhood minors. As Tim Romer and Sara S. Madani explored it for toric cases.

Interesting to search will be exploring graphs through monomial cut-ideals with Cohen-Macaulay property and bounding regularity or projective dimension.

Moreover, Matroid version of the study is done for toric-cut ideals and would be interesting for monomial cut-ideals.