Given a population with dynamic pairwise connections, we ask if the entire population could be (indirectly) infected by a small group of k initially infected individuals. We formalise this problem as the Temporal Reachability Dominating Set (TaRDiS) problem on temporal graphs. We provide positive and negative parameterized complexity results in four different parameters: the number k of initially infected, the lifetime τ of the graph, the number of locally earliest edges in the graph, and the treewidth of the footprint graph G↓. We additionally introduce and study the MaxMinTaRDiS problem, where the aim is to schedule connections between individuals so that at least k individuals must be infected for the entire population to become fully infected. We classify three variants of the problem: Strict, Nonstrict, and Happy. We show these to be coNP-complete, NP-hard, and ΣP2-complete, respectively. Interestingly, we obtain hardness of the Nonstrict variant by showing that a natural restriction is exactly the well-studied Distance-3 Independent Set problem on static graphs.

A simple greedy algorithm to find a maximal independent set (MIS) in a graph starts with the empty set and visits every vertex, adding it to the set if and only if none of its neighbours are already in the set. In this paper, we consider (the complexity of decision problems related to) the generalisation of this MIS algorithm wherein any starting set is allowed. Two main approaches are leveraged. Firstly, we view the MIS algorithm as a sequential update of a Boolean network according to a permutation of the vertex set. Secondly, we introduce the concept of a constituency of a graph: a set of vertices that is dominated by an independent set. Recognizing a constituency is NP -complete, a fact we leverage repeatedly in our investigation. Our contributions are multiple: we establish that deciding whether all maximal independent sets can be reached from some configuration is coNP -complete; that fixing words (which reach a MIS from any starting configuration) and fixing permutations (briefly, permises) are coNP -complete to recognize; and that permissible graphs (graphs with a permis) are coNP -hard to recognize. We also exhibit large classes of permissible and non-permissible graphs, notably near-comparability graphs which may be of independent interest. Lastly, we extend our study to digraphs, where we search for kernels. Since the natural generalisation of our approach may not necessarily find a kernel, we introduce two further Boolean networks for digraphs: one always finds an independent set, and the other always finds a dominating set.

The network-based study of financial systems has received considerable attention in recent years but has seldom explicitly incorporated the dynamic aspects of such systems. We consider this problem setting from the temporal point of view and introduce the Interval Debt Model (IDM) and some scheduling problems based on it, namely: Bankruptcy Minimization/Maximization, in which the aim is to produce a payment schedule with at most/at least a given number of bankruptcies; Perfect Scheduling, the special case of the minimization variant where the aim is to produce a schedule with no bankruptcies (that is, a perfect schedule); and Bailout Minimization, in which a financial authority must allocate a smallest possible bailout package to enable a perfect schedule. We show that each of these problems is NP-complete, in many cases even on very restricted input instances. On the positive side, we provide for Perfect Scheduling a polynomial-time algorithm on (rooted) out-trees although in contrast we prove NP-completeness on directed acyclic graphs, as well as on instances with a constant number of nodes (and hence also constant treewidth). When we allow non-integer payments, we show by a linear programming argument that the problem Bailout Minimization can be solved in polynomial time.

A hybrid network is a static (electronic) network that is augmented with optical switches. The Reconfigurable Routing Problem (RRP) in hybrid networks is the problem of finding settings for the optical switches augmenting a static network so as to achieve optimal delivery of some given workload. The problem has previously been studied in various scenarios with both tractability and NP-hardness results obtained. However, the data center and interconnection networks to which the problem is most relevant are almost always such that the static network is highly structured (and often node-symmetric) whereas all previous results assume that the static network can be arbitrary (which makes existing computational hardness results less technologically relevant and also easier to obtain). In this paper, and for the first time, we prove various intractability results for RRP where the underlying static network is highly structured, for example consisting of a hypercube, and also extend some existing tractability results.