Cutting planes for families implying Frankl’s conjecture

Abstract

We find previously unknown families of sets which ensure Frankl's conjecture holds for all families that contain them using an algorithmic framework. The conjecture states that for any nonempty union-closed (UC) family there exists an element of the ground set in at least half the sets of the considered UC family. Poonen's Theorem characterizes the existence of weights which determine whether a given UC family implies the conjecture for all UC families which contain it. We design a cutting-plane method that computes the explicit weights which imply the existence conditions of Poonen's Theorem. This method enables us to answer several open questions regarding structural properties of UC families, including the construction of a counterexample to a conjecture of Morris from 2006.

Fully Automatic, Verified Classification of all Frankl-Complete (FC(6)) Set Families

“…An alternative to using a specialized, verified procedure ssn F c w is to encode the existence of a union-closed extension F with a negative share as a linear integer programming problem and to employ an existing solver to do the search [22]. In our case, we need to formally prove (within the Isabelle/HOL) that our characteristic FC-families are indeed FC, so the SMT solver Z3 integrated within Isabelle/HOL can be used [3].…”

“…It is very desired to have a procedure that can obtain them fully automatically. Pulaj suggested the first algorithm capable of checking the FC-status of an arbitrary family based on the cutting planes method and linear (integer) programming implemented in SCIP [22], and it can easily be modified to provide required witnesses (both for the FC and the nonFC case). Note that such procedure need not be implemented within Isabelle/HOL -its purpose is to determine the status and give witnesses that can be used for Theorem 1 or 2, which are formally checked within Isabelle/HOL.…”

“…It has recently been shown that this process can be fully automated 1 . Namely, Pulaj recently proposed a fully automated method for determining the FC-status of an arbitrary given family [22]. The method is based on linear integer programming, and, although not integrated within a proof-assistant, it is very reliable, as it uses exact arithmetic.…”

A computational status update for exact rational mixed integer programming

“…This holds in particular when the solution of mixed integer programs is used as a tool in mathematics itself. Examples of recent work that employs MIP to investigate open mathematical questions include [11,12,19,29,30,33]. Some of these approaches are forced to rely on floating-point solvers because the availability, the flexibility, and most importantly the computational performance of MIP solvers with numerically rigorous guarantees is currently limited.…”

A Computational Status Update for Exact Rational Mixed Integer Programming

“…This holds in particular when the solution of mixed integer programs is used as a tool in mathematics itself. Examples of recent work that employs MIP to investigate open mathematical questions include [11,12,18,28,29,32]. Some of these approaches are forced to rely on floating-point solvers because the availability, the flexibility, and most importantly the computational performance of MIP…”