More Like this

1. On Classes of Bounded Tree Rank, Their Interpretations, and Efficient Sparsification

   https://doi.org/10.4230/LIPIcs.ICALP.2024.137  

   Gajarský, Jakub; McCarty, Rose (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded treedepth, and they are a special case of graph classes of bounded expansion. We introduce a notion of decomposition for these classes and show that these decompositions can be efficiently computed. Also, a natural extension of our decomposition leads to a new characterization and decomposition for graph classes of bounded expansion (and an efficient algorithm computing this decomposition). We then focus on interpretations of graph classes of bounded tree rank. We give a characterization of graph classes interpretable in graph classes of tree rank 2. Importantly, our characterization leads to an efficient sparsification procedure: For any graph class 𝒞 interpretable in a graph class of tree rank at most 2, there is a polynomial time algorithm that to any G ∈ 𝒞 computes a (sparse) graph H from a fixed graph class of tree rank at most 2 such that G = I(H) for a fixed interpretation I. To the best of our knowledge, this is the first efficient "interpretation reversal" result that generalizes the result of Gajarský et al. [LICS 2016], who showed an analogous result for graph classes interpretable in classes of graphs of bounded degree. 

   more » « less
2. Subdivided Claws and the Clique-Stable Set Separation Property

   https://doi.org/10.1007/978-3-030-62497-2  

   Chudnovsky, Maria and (February 2021, MATRIX book series)

   null (Ed.)

   Let C be a class of graphs closed under taking induced subgraphs. We say that C has the clique-stable set separation property if there exists c ∈ N such that for every graph G ∈ C there is a collection P of partitions (X, Y ) of the vertex set of G with |P| ≤ |V (G)| c and with the following property: if K is a clique of G, and S is a stable set of G, and K ∩ S = ∅, then there is (X, Y ) ∈ P with K ⊆ X and S ⊆ Y . In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by M. G¨o¨os in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families S, K of graphs and show that for every S ∈ S and K ∈ K, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property. 

   more » « less
3. Subdivided Claws and the Clique-Stable Set Separation Property

   https://doi.org/10.1007/978-3-030-62497-2_29  

   Chudnovsky, Maria and (February 2021, MATRIX book series)

   Let C be a class of graphs closed under taking induced subgraphs. We say that C has the clique-stable set separation property if there exists c∈N such that for every graph G∈C there is a collection P of partitions (X,Y) of the vertex set of G with |P|≤|V(G)|c and with the following property: if K is a clique of G, and S is a stable set of G, and K∩S=∅, then there is (X,Y)∈P with K⊆X and S⊆Y. In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by Göös in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families S,K of graphs and show that for every S∈S and K∈K, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property. 

   more » « less
4. Indiscernibles in monadically NIP theories

   https://doi.org/10.1112/blms.70155  

   Braunfeld, Samuel; Laskowski, Michael C (July 2025, Bulletin of the London Mathematical Society)

   Abstract We prove various results around indiscernibles in monadically NIP theories. First, we provide several characterizations of monadic NIP in terms of indiscernibles, mirroring previous characterizations in terms of the behavior of finite satisfiability. Second, we study (monadic) distality in hereditary classes and complete theories. Here, via finite combinatorics, we prove a result implying that every planar graph admits a distal expansion. Finally, we prove a result implying that no monadically NIP theory interprets an infinite group, and note an example of a (monadically) stable theory with no distal expansion that does not interpret an infinite group. 

   more » « less
5. Distributed Graph Diameter Approximation

   https://doi.org/10.3390/a13090216  

   Ceccarello, Matteo; Pietracaprina, Andrea; Pucci, Geppino; Upfal, Eli (September 2020, Algorithms)

   null (Ed.)

   We present an algorithm for approximating the diameter of massive weighted undirected graphs on distributed platforms supporting a MapReduce-like abstraction. In order to be efficient in terms of both time and space, our algorithm is based on a decomposition strategy which partitions the graph into disjoint clusters of bounded radius. Theoretically, our algorithm uses linear space and yields a polylogarithmic approximation guarantee; most importantly, for a large family of graphs, it features a round complexity asymptotically smaller than the one exhibited by a natural approximation algorithm based on the state-of-the-art Δ-stepping SSSP algorithm, which is its only practical, linear-space competitor in the distributed setting. We complement our theoretical findings with a proof-of-concept experimental analysis on large benchmark graphs, which suggests that our algorithm may attain substantial improvements in terms of running time compared to the aforementioned competitor, while featuring, in practice, a similar approximation ratio. 

   more » « less