Please join us for our inaugural DMG Mathematics Seminar.
*Title: Soft Happy Colouring and Community
A talk by Iman Shekkariz
Abstract: For a coloured graph G and 0 ≤ ρ ≤ 1, a vertex v is ρhappy if at least ρ deg(v) of its neighbours share its colour. The soft happy colouring problem seeks a colouring σ that extends a given precolouring and maximises the number of ρ-happy vertices [3]. This NP-hard problem is closely linked to community detection in graphs. For example, for a graph in the stochastic block model (SBM) and for suitable ρ, with high probability, complete soft happy colourings can be achieved by the planted community structure [1]. Moreover, for 0 ≤ ρ1 < ρ2 ≤ 1, complete ρ2 -happy colourings achieve higher detection accuracy than complete ρ1 -happy colourings, and when ρ surpasses a critical threshold, it is unlikely to find a complete ρ-happy colouring with near-equal class sizes [2]. Finally, we survey existing algorithms and propose novel heuristic, local search, evolutionary, metaheuristic, and matheuristic approaches that enhance solution quality for soft happy colouring.
References [1] Mohammad H. Shekarriz, Dhananjay Thiruvady, Asef Nazari, and Rhyd Lewis. Soft happy colourings and community structure of networks. Computers & Operations Research, 174:106893, 2025. [2] Mohammad H. Shekarriz, Dhananjay Thiruvady, Asef Nazari, and Wilfried Imrich. Local search improvements for soft happy colouring. Preprint available on ArXiv: 2506.19284, 2025 1 [3] Peng Zhang and Angsheng Li. Algorithmic aspects of homophyly of networks. Theoretical Computer Science, 593:117–131, 2015.
*Title: Faces of convex sets and its relation with the generalised interior
A talk by Reinier Diaz Millan
Abstract: This presentation explores the relationship between the facial structure of convex sets and various notions of generalised interiors, particularly in general vector spaces. We review the fundamental concepts of faces and minimal faces, with a central focus on the intrinsic core (icr), a generalisation of the relative interior that is characterised by its connection to minimal faces. A key result is the decomposition of any convex set into a disjoint union of the intrinsic cores of its faces. Building on this, we introduce the face relative interior (fri), a new concept of interiority for convex sets in topological vector spaces. We establish its fundamental properties and its place within the hierarchy of interiority notions. The talk also covers conditions that guarantee the face relative interior is nonempty. Ultimately, this work provides a deeper understanding of the geometric structure of convex sets in infinite-dimensional settings by linking their facial geometry to these generalised interior points.