Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.13244

Equitable Subpartitionability Does Not Characterize Box-TDI Polyhedra

Emiliano Lancini·Francesco Pisanu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Equitable subpartitionability (ESP) is a sufficient condition for an integral system to be box-totally dual integral. We show that the converse fails even when alternative descriptions of the same polyhedron are allowed. We provide a geometrical characterization of polyhedra that can be described by ESP systems in terms of normal cones. We construct a four-dimensional integer polytope described by a totally equimodular matrix that admits no integral ESP description. This answers negatively a previously posed question by the same authors.

Explore related subjects

cs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Emiliano Lancini, Francesco Pisanu. 2026-10-09. Equitable Subpartitionability Does Not Characterize Box-TDI Polyhedra. https://arxiv.org/abs/2610.13244

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Differentiable Repairability & Mutations for Scalable Combinatorial Optimization

Combinatorial optimization problems such as Maximum Independent Set (MIS) and Maximum Cut (MaxCut) arise across a wide range of applications, yet solving large-scale instances remains computationally challenging. While discrete local-search procedures can produce high-quality solutions, their inherently sequential and iterative nature can become increasingly expensive as problem size and density grow. In this paper, we present a differentiable perspective on combinatorial optimization that leverages continuous relaxation with parallelizable gradient-based updates to improve scalability while reducing dependence on discrete local search. More specifically, for MIS, we show that solutions that remain repairable through a discrete (1,2)-swap can nevertheless be fixed points of projected gradient ascent, revealing a fundamental limitation of purely local gradient dynamics. For MaxCut, we develop a new quadratic formulation whose landscape provably allows projected gradient ascent to escape 1-flip-repairable solutions while ensuring binary local maximizers. Building on these insights, and inspired by discrete perturb-and-improve methods, we introduce a framework that combines projected gradient ascent with differentiable mutations (partial resets) to efficiently explore new regions of the solution space}. Experimental results on MIS and MaxCut demonstrate the potential of this approach for scalable combinatorial optimization, particularly in regimes where conventional discrete local search becomes computationally prohibitive.

cs.DM↗
arXiv2026

Detection and Construction of Small Blocking Sets via Critical Independent Sets and Matching Decomposition

The small blocking set detection and construction problem asks whether a given connected simple graph contains a nonempty independent set whose cardinality equals the size of its neighborhood and is strictly less than half the number of vertices, and, if so, how to construct such a set. Previous work characterized restless explorability in vertex-permuted temporal graphs using small blocking sets, but left the computational complexity of detecting their existence as an open problem. We give a polynomial-time algorithm for detection and construction. Specifically, we first compute a maximum-cardinality critical independent set and use its difference and cardinality to handle the cases that admit an immediate solution. In the remaining boundary case, where the critical difference is zero and the selected independent set contains exactly half the vertices, we establish a projection theorem using a perfect matching and neighborhood counting: the graph contains a small blocking set if and only if the selected independent set contains a nonempty proper tight subset. We then use matching contraction to translate the tight-subset condition into the absence of arcs leaving a vertex subset in an auxiliary digraph. This yields a strong-connectivity test for the existence of a small blocking set and, when one exists, a construction based on a sink strongly connected component. The result resolves the open complexity question for small blocking set detection and provides an explicit construction algorithm.

cs.DM↗
arXiv2026

The Gallai Vertex Problem is $Θ_2^p$-Complete

When a graph $G$ admits a vertex $v$ that is contained in all its longest paths, we call $v$ a Gallai vertex. These are named after Gallai, who in 1966 asked the question if it is true that every connected graph contains such a vertex. This was soon answered in the negative by Walther and Zamfirescu, who presented a graph in which every vertex is omitted by some longest path of the graph. In spite of its long history, the Gallai Vertex Problem, i.e. determining whether a graph has a Gallai vertex, was until now neither known to be NP- nor co-NP-hard. In this work, we show something much stronger, as we completely settle the computational complexity of determining whether a graph has a Gallai vertex: we show that it is complete for the complexity class $Θ_2^p = \text{P}^{\text{NP}[\log n]}$. This class, also known as parallel access to NP, is a complexity class larger than NP situated just below the class $Σ^p_2$ in Stockmeyer's polynomial hierarchy. In more generality, the longest path transversal number of a connected graph is the minimum size of a set of vertices that intersects all its longest paths. I.e. if the graph has a Gallai vertex, its longest path transversal number is $1$. Thus, as a consequence of our theorem, the longest path transversal number of a graph cannot be approximated in polynomial time by a factor better than 2, unless $\text{P} = \text{NP}$. In fact, using related techniques, we show a strengthening of this result: For any constant $C$, if there is a graph with longest path transversal number $C$, then there is no polynomial time algorithm for approximating the longest path transversal number by a factor better than $C$, unless $\text{P} = \text{NP}$. In particular, this excludes approximation by a factor below $3$. Similar results hold for the longest cycle transversal.

cs.DM↗