On the Containment Problem for Linear Sets

It is well known that the containment problem (as well as the equivalence problem) for semilinear sets is log-complete at the second level of the polynomial hierarchy (where hardness even holds in dimension 1). It had been shown quite recently that already the containment problem for multi-dimensional linear sets is log-complete at the same level of the hierarchy (where hardness even holds when numbers are encoded in unary). In this paper, we show that already the containment problem for 1-dimensional linear sets (with binary encoding of the numerical input parameters) is log-hard (and therefore also log-complete) at this level. However, combining both restrictions (dimension 1 and unary encoding), the problem becomes solvable in polynomial time

Decomposing 1-Sperner hypergraphs

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of $1$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of $1$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that $1$-Sperner hypergraphs are both threshold and equilizable. The study of $1$-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper

Broadcasting Automata and Patterns on Z^2

The Broadcasting Automata model draws inspiration from a variety of sources such as Ad-Hoc radio networks, cellular automata, neighbourhood se- quences and nature, employing many of the same pattern forming methods that can be seen in the superposition of waves and resonance. Algorithms for broad- casting automata model are in the same vain as those encountered in distributed algorithms using a simple notion of waves, messages passed from automata to au- tomata throughout the topology, to construct computations. The waves generated by activating processes in a digital environment can be used for designing a vari- ety of wave algorithms. In this chapter we aim to study the geometrical shapes of informational waves on integer grid generated in broadcasting automata model as well as their potential use for metric approximation in a discrete space. An explo- ration of the ability to vary the broadcasting radius of each node leads to results of categorisations of digital discs, their form, composition, encodings and gener- ation. Results pertaining to the nodal patterns generated by arbitrary transmission radii on the plane are explored with a connection to broadcasting sequences and ap- proximation of discrete metrics of which results are given for the approximation of astroids, a previously unachievable concave metric, through a novel application of the aggregation of waves via a number of explored functions

On unique recovery of finite‑valued integer signals and admissible lattices of sparse hypercubes

The paper considers the problem of unique recovery of sparse finite-valued integer signals using a single linear integer measurement. For l-sparse signals in ℤn, 2l < n, with absolute entries bounded by r, we construct an 1 × n measurement matrix with maximum absolute entry Δ = O(r^(2l−1)). Here the implicit constant depends on l and n and the exponent 2l − 1 is optimal. Additionally, we show that, in the above setting, a single measurement can be replaced by several measurements with absolute entries sub-linear in Δ. The proofs make use of results on admissible (n − 1)-dimensional integer lattices for m-sparse n-cubes that are of independent interest

Generation and analysis of networks with a prescribed degree sequence and subgraph family: higher-order structure matters

Designing algorithms that generate networks with a given degree sequence while varying both subgraph composition and distribution of subgraphs around nodes is an important but challenging research problem. Current algorithms lack control of key network parameters, the ability to specify to what subgraphs a node belongs to, come at a considerable complexity cost or, critically and sample from a limited ensemble of networks. To enable controlled investigations of the impact and role of subgraphs, especially for epidemics, neuronal activity or complex contagion, it is essential that the generation process be versatile and the generated networks as diverse as possible. In this article, we present two new network generation algorithms that use subgraphs as building blocks to construct networks preserving a given degree sequence. Additionally, these algorithms provide control over clustering both at node and global level. In both cases, we show that, despite being constrained by a degree sequence and global clustering, generated networks have markedly different topologies as evidenced by both subgraph prevalence and distribution around nodes, and large-scale network structure metrics such as path length and betweenness measures. Simulations of standard epidemic and complex contagion models on those networks reveal that degree distribution and global clustering do not always accurately predict the outcome of dynamical processes taking place on them. We conclude by discussing the benefits and limitations of both methods