arXiv subjects

Lucas Fabian Naumann

Publications and source records attributed to Lucas Fabian Naumann.

5 recordsLinked to original sources

arXiv2026

On Partial Optimality in QUBO via Quadratic Posiform Autarkies

Partially optimal assignments (also called persistencies) allow combinatorial optimization problems to be reduced by fixing a subset of variables to values consistent with at least one optimal solution. For the quadratic unconstrained binary optimization (QUBO) problem, such assignments can be computed efficiently via autarkies of quadratic posiforms. However, the largest autarky domain and thus the number of fixed variables can depend on the specific posiform representation. We show that every quadratic posiform with the greatest possible constant term (the roof-dual bound) yields the largest autarky domain obtainable from any quadratic posiform. By computing such a roof-dual-optimal quadratic posiform and a corresponding autarky with largest domain in polynomial time, one can therefore attain the maximum partial optimality obtainable by quadratic posiform autarkies in polynomial time. This justifies the computation of roof-dual-optimal quadratic posiforms and corresponding autarkies in popular polynomial-time algorithms and establishes that finding additional partial optimality in QUBO requires going beyond autarkies of quadratic posiforms.

math.OC↗

arXiv2026

The canonical facets of multi-separator polytopes

We initiate a polyhedral study of the graph multi-separator problem proposed by Irmai et al. (2024) as an alternative to the lifted multicut problem for application to the task of image segmentation. Starting with an integer linear program (ILP) formulation and the multi-separator polytope spanned by its feasible solutions, we characterize in terms of efficiently-decidable, graph-theoretic conditions all facets induced by inequalities of the ILP. We proceed by strengthening these inequalities and describing additional facets of some multi-separator polytopes induced by the stronger inequalities. Specifically, we obtain a totally dual integral description of the multi-separator polytope for paths in the case where separation is considered for all vertex pairs. Finally, we relate the multi-separator polytope to the boolean quadric polytope, showing that facets induced by odd-cycle inequalities do not transfer generally, and to the lifted multicut polytope, showing that either polytope is a projection of a face of the other.

cs.DM↗

arXiv2026

Graph Neural Networks with Triangle-Based Messages for the Multicut Problem

The multicut problem is an NP-hard combinatorial optimization problem with diverse applications in fields such as bioinformatics, data mining and computer vision. Graph neural networks have been defined for the multicut problem but can be adapted further to its specific objective function and constraints. In this article, we introduce such an adapted graph neural network architecture in which features are assigned only to edges, and the computation of messages is based on triangles in the underlying graph. Experiments with synthetic and real-world instances with up to 200 nodes show that our method outperforms state-of-the-art heuristic solvers in terms of solution quality while maintaining feasible runtimes. For some instances, our method finds optimal solutions in seconds whereas exact solvers need hours to find and certify optimal solutions.

cs.LG↗

arXiv2025

Chorded cycle facets of the clique partitioning polytope

The $q$-chorded $k$-cycle inequalities are a class of valid inequalities for the clique partitioning polytope. It is known that for $q \in \{2, \tfrac{k-1}{2}\}$, these inequalities induce facets of the clique partitioning polytope if and only if $k$ is odd. Here, we characterize such facets for arbitrary $k$ and $q$. More specifically, we prove that the $q$-chorded $k$-cycle inequalities induce facets of the clique partitioning polytope if and only if two conditions are satisfied: $k = 1$ mod $q$, and if $k=3q+1$ then $q=3$ or $q$ is even. This establishes the existence of many facets induced by $q$-chorded $k$-cycle inequalities beyond those previously known.

cs.DM↗

arXiv2024

Box Facets and Cut Facets of Lifted Multicut Polytopes

The lifted multicut problem is a combinatorial optimization problem whose feasible solutions relate one-to-one to the decompositions of a graph $G = (V, E)$. Given an augmentation $\widehat{G} = (V, E \cup F)$ of $G$ and given costs $c \in \mathbb{R}^{E \cup F}$, the objective is to minimize the sum of those $c_{uw}$ with $uw \in E \cup F$ for which $u$ and $w$ are in distinct components. For $F = \emptyset$, the problem specializes to the multicut problem, and for $E = \tbinom{V}{2}$ to the clique partitioning problem. We study a binary linear program formulation of the lifted multicut problem. More specifically, we contribute to the analysis of the associated lifted multicut polytopes: Firstly, we establish a necessary, sufficient and efficiently decidable condition for a lower box inequality to define a facet. Secondly, we show that deciding whether a cut inequality of the binary linear program defines a facet is NP-hard.

cs.DM↗