28.04.2023

Three Papers accepted at ICALP

For the competitive International Colloquium on Automata, Languages and Programming (ICALP) with an acceptance rate of 30% three papers of our group were accepted. The conference will take place on 10-14 July in Paderborn, Germany. The first paper Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs is based on the master thesis of Leon Schiller and investigated the influence of the dimension of the underlying metric space on Geometric Inhomogeneous Random Graphs (GIRGs). The authors prove that the model loses its geometry (i.e. converges in total variation distance to its non-geometric counterpart) in the infinite dimensional limit, and further prove tight bounds on the number of cliques and the clique number (size of the largest clique) that explicitly depend on the dimension. They find that, if the dimension scales superlogarithmically, the model essentially behaves like its non-geometric counterpart w.r.t. the clique structure.

The paper Fault-Tolerant ST-Diameter Oracles is a cooperation of a large group with authors from several universities and introduces structures that estimate the diameter of a graph under the existence of edge failures. To create these structures they use reductions from existing Fault-Tolerant Distance Sensitivity Oracles which estimate distances between pairs of nodes in a graph with edge failures. Additionally they give a lower bound on the space requirement of such diameter oracles.

Lastly, in the paper Approximating Long Cycle Above Dirac’s Guarantee, the authors show that any approximation algorithm for Longest Cycle could be effectively applied on top of Dirac's existential guarantee: given a polynomial-time algorithm constructing a cycle of length \(f(L)\) in an undirected graph \(G\) containing a cycle of length \(L\), their result yields a polynomial-time algorithm constructing a cycle of length \(2δ(G) + Ω(f(k))\), where \(δ(G)\) is the minimum degree of \(G\) and \(k\) is the "offset" \(L - 2δ(G)\).

Clean Citation Style 005

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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

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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

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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 \(2\delta(G) + \Omega(f(k))\).

@inproceedings{fomin2023approximating,
  abstract = {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 \(2\delta(G) + \Omega(f(k))\).},
  author = {Fomin, Fedor and Golovach, Petr and Sagunov, Danil and Simonov, Kirill},
  booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)},
  keywords = {sys:relevantfor:ae icalp kirillsimonov year2023},
  pages = {60:1&ndash;60:18},
  title = {Approximating Long Cycle Above Dirac’s Guarantee},
  year = 2023
}

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

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@inproceedings{bilo2023faulttolerant,
  abstract = {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.},
  author = {Bilò, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Krogmann, Simon and Schirneck, Martin},
  booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)},
  keywords = {davidebilo icalp keertichoudhary martinschirneck sarelcohen simonkrogmann tobiasfriedrich year2023},
  pages = {24:1-24:20},
  title = {Fault-Tolerant ST-Diameter Oracles},
  year = 2023
}