Friedrich, Tobias; Göbel, Andreas; Katzmann, Maximilian; Schiller, Leon Cliques in High-Dimensional Geometric Inhomogeneous Random GraphsInternational Colloquium on Automata, Languages and Programming (ICALP) 2023: 62:1–62:13
A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.
Fomin, Fedor; Golovach, Petr; Sagunov, Danil; Simonov, Kirill Approximating Long Cycle Above Dirac’s GuaranteeInternational Colloquium on Automata, Languages and Programming (ICALP) 2023: 60:1–60:18
Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit “natural” guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m/2 + k clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n/2 + k for some \(k \geq 1\) and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree \(\delta(G) \leq n/2\), the length of a longest cycle L is at least \(2\delta(G)\). Thus the “essential” part in finding the longest cycle is in approximating the “offset” \(k = L − 2\delta(G)\). The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length \(2\delta(G) + \Omega(f(k))\).
Bilò, Davide; Choudhary, Keerti; Cohen, Sarel; Friedrich, Tobias; Krogmann, Simon; Schirneck, Martin Fault-Tolerant ST-Diameter OraclesInternational Colloquium on Automata, Languages and Programming (ICALP) 2023: 24:1–24:20
We study the problem of estimating the \(ST\)-diameter of a graph that is subject to a bounded number of edge failures. An \(f\)-edge fault-tolerant \(ST\)-diameter oracle (\(f\)-FDO-\(ST\)) is a data structure that preprocesses a given graph \(G\), two sets of vertices \(S,T\), and positive integer \(f\). When queried with a set \(F\) of at most \(f\) edges, the oracle returns an estimate \(\widehat{D}\) of the \(ST\)-diameter \(\mathrm{diam}(G-F,S,T)\), the maximum distance between vertices in \(S\) and \(T\) in \(G-F\). The oracle has stretch \(\sigma \geq 1\) if \(\mathrm{diam}(G-F,S,T) leq \widehat{D} leq sigma \mathrm{diam}(G-F,S,T)\). If \(S\) and \(T\) both contain all vertices, the data structure is called an \(f\)-edge fault-tolerant diameter oracle (\(f\)-FDO). An \(f\)-edge fault-tolerant distance sensitivity oracles (\(f\)-DSO) estimates the pairwise graph distances under up to \(f\) failures. We design new \(f\)-FDOs and \(f\)-FDO-\(ST\)s by reducing their construction to that of all-pairs and single-source \(f\)-DSOs. We obtain several new tradeoffs between the size of the data structure, stretch guarantee, query and preprocessing times for diameter oracles by combining our black-box reductions with known results from the literature. We also provide an information-theoretic lower bound on the space requirement of approximate \(f\)-FDOs. We show that there exists a family of graphs for which any \(f\)-FDO with sensitivity \(f \ge 2\) and stretch less than \(5/3\) requires \(\Omega(n^{3/2})\) bits of space, regardless of the query time.