Given a connected graph G = (V, E), a Safe Set S is a subset of the vertex set V such that the cardinality of each connected component in the subgraph induced by V \ S does not exceed the cardinality of any neighbor connected component in the subgraph induced by S. When the vertices … Read more
In this article we address the problem of finding lower bounds for small Ramsey numbers $R(m,n)$ using circulant graphs. Our constructive approach is based on finding feasible colorings of circulant graphs using Integer Programming (IP) techniques. First we show how to model the problem as a Stackelberg game and, using the tools of bilevel optimization, … Read more
The s-clubs model cohesive social subgroups as vertex subsets that induce subgraphs of diameter at most s. In defender-attacker settings, for low values of s, they can represent tightly-knit communities whose operation is undesirable for the defender. For instance, in online social networks, large communities of malicious accounts can effectively propagate undesirable rumors. In this … Read more
The Graphical Traveling Salesperson Problem (GTSP) is the problem of assigning, for a given weighted graph, a nonnegative number x_e to each edge e such that the induced multi-subgraph is of minimum weight among those that are spanning, connected and Eulerian. Naturally, known mixed-integer programming formulations use integer variables x_e in addition to others. Denis … Read more
We introduce a new graph-theoretic paradigm called a graph signature that describes persistent patterns in a sequence of graphs. This framework is motivated by the need to detect subgraphs of significance in temporal networks, e.g., social and biological networks that evolve over time. Because the subgraphs of interest may not all “look alike” in the … Read more
The clique problem with multiple-choice constraints (CPMC), i.e. the problem of finding a k-clique in a k-partite graph with known partition, occurs as a substructure in many real-world applications, in particular scheduling and railway timetabling. Although CPMC is NP-complete in general, it is known to be solvable in polynomial time when the so-called dependency graph … Read more
This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound for both the $k$-multichromatic number of a graph and the maximum $k$-colorable subgraph problem. We present various properties of $\vartheta_k(G)$, such as that the series $(\vartheta_k(G))_k$ is increasing and bounded above by the order … Read more
For the problem to find an m-clique in an m-partite graph, staircase compatibility has recently been introduced as a polynomial-time solvable special case. It is a property of a graph together with an m-partition of the vertex set and total orders on each subset of the partition. In optimization problems involving m-cliques in m-partite graphs … Read more
All feasible flows in potential-driven networks induce an orientation on the undirected graph underlying the network. Clearly, these orientations must satisfy two conditions: they are acyclic and there are no “dead ends” in the network, i.e. each source requires outgoing flows, each sink requires incoming flows, and each transhipment vertex requires both an incoming and … Read more
Timetabling of railway traffic and other modes of transport is among the most prominent applications of discrete optimization in practice. However, it has only been recently that the connection between timetabling and energy consumption has been studied more extensively. In our joint project VAG Verkehrs-Aktiengesellschaft, the transit authority and operator of underground transport in the … Read more