arXiv subjects

Silvia Di Gregorio

Publications and source records attributed to Silvia Di Gregorio.

6 recordsLinked to original sources

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↗

arXiv2025

Partial Optimality in Cubic Correlation Clustering for General Graphs

The higher-order correlation clustering problem for a graph $G$ and costs associated with cliques of $G$ consists in finding a clustering of $G$ so as to minimize the sum of the costs of those cliques whose nodes all belong to the same cluster. To tackle this NP-hard problem in practice, local search heuristics have been proposed and studied in the context of applications. Here, we establish partial optimality conditions for cubic correlation clustering, i.e., for the special case of at most 3-cliques. We define and implement algorithms for deciding these conditions and examine their effectiveness numerically, on two data sets.

cs.DM↗

arXiv2024

A Knowledge Compilation Take on Binary Polynomial Optimization

The Binary Polynomial Optimization (BPO) problem is defined as the problem of maximizing a given polynomial function over all binary points. The main contribution of this paper is to draw a novel connection between BPO and the field of Knowledge Compilation. This connection allows us to unify and significantly extend the state-of-the-art for BPO, both in terms of tractable classes, and in terms of existence of extended formulations. In particular, for instances of BPO with hypergraphs that are either $β$-acyclic or with bounded incidence treewidth, we obtain strongly polynomial algorithms for BPO, and extended formulations of polynomial size for the corresponding multilinear polytopes. The generality of our technique allows us to obtain the same type of results for extensions of BPO, where we enforce extended cardinality constraints on the set of binary points, and where variables are replaced by literals. We also obtain strongly polynomial algorithms for the variant of the above problems where we seek $k$ best feasible solutions, instead of only one optimal solution. Computational results show that the resulting algorithms can be significantly faster than current state-of-the-art.

math.OC↗

arXiv2023

Partial Optimality in Cubic Correlation Clustering

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

cs.DM↗

arXiv2022

On the complexity of binary polynomial optimization over acyclic hypergraphs

In this work we advance the understanding of the fundamental limits of computation for Binary Polynomial Optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we provide a novel class of BPO that can be solved efficiently both from a theoretical and computational perspective. In fact, we give a strongly polynomial-time algorithm for instances whose corresponding hypergraph is beta-acyclic. We note that the beta-acyclicity assumption is natural in several applications including relational database schemes and the lifted multicut problem on trees. Due to the novelty of our proving technique, we obtain an algorithm which is interesting also from a practical viewpoint. This is because our algorithm is very simple to implement and the running time is a polynomial of very low degree in the number of nodes and edges of the hypergraph. Our result completely settles the computational complexity of BPO over acyclic hypergraphs, since the problem is NP-hard on alpha-acyclic instances. Our algorithm can also be applied to any general BPO problem that contains beta-cycles. For these problems, the algorithm returns a smaller instance together with a rule to extend any optimal solution of the smaller instance to an optimal solution of the original instance.

cs.DM↗

arXiv2022

A Polyhedral Study of Lifted Multicuts

Fundamental to many applications in data analysis are the decompositions of a graph, i.e. partitions of the node set into component-inducing subsets. One way of encoding decompositions is by multicuts, the subsets of those edges that straddle distinct components. Recently, a lifting of multicuts from a graph $G = (V, E)$ to an augmented graph $\hat G = (V, E \cup F)$ has been proposed in the field of image analysis, with the goal of obtaining a more expressive characterization of graph decompositions in which it is made explicit also for pairs $F \subseteq \tbinom{V}{2} \setminus E$ of non-neighboring nodes whether these are in the same or distinct components. In this work, we study in detail the polytope in $\mathbb{R}^{E \cup F}$ whose vertices are precisely the characteristic vectors of multicuts of $\hat G$ lifted from $G$, connecting it, in particular, to the rich body of prior work on the clique partitioning and multilinear polytope.

cs.DM↗