Round- and Message-Optimal Distributed Graph Algorithms

Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication. In this paper we provide algorithms that are simultaneously round- and message-optimal for a number of well-studied distributed optimization problems. Our main result is such a distributed algorithm for the fundamental primitive of computing simple functions over each part of a graph partition. From this algorithm we derive round- and message-optimal algorithms for multiple problems, including MST, Approximate Min-Cut and Approximate Single Source Shortest Paths, among others. On general graphs all of our algorithms achieve worst-case optimal $\tilde{O}(D+\sqrt n)$ round complexity and $\tilde{O}(m)$ message complexity. Furthermore, our algorithms require an optimal $\tilde{O}(D)$ rounds and $\tilde{O}(n)$ messages on planar, genus-bounded, treewidth-bounded and pathwidth-bounded graphs.Comment: To appear in PODC 201

First Order Logic and Twin-Width in Tournaments

We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments T, first-order model checking either is fixed parameter tractable, or is AW[*]-hard. This dichotomy coincides with the fact that T has either bounded or unbounded twin-width, and that the growth of T is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions: T has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs. The key for these results is a polynomial time algorithm which takes as input a tournament T and computes a linear order < on V(T) such that the twin-width of the birelation (T, <) is at most some function of the twin-width of T. Since approximating twin-width can be done in FPT time for an ordered structure (T, <), this provides a FPT approximation of twin-width for tournaments

Linear Programming Bounds for Randomly Sampling Colorings

Here we study the problem of sampling random proper colorings of a bounded degree graph. Let $k$ be the number of colors and let $d$ be the maximum degree. In 1999, Vigoda showed that the Glauber dynamics is rapidly mixing for any $k > \frac{11}{6} d$. It turns out that there is a natural barrier at $\frac{11}{6}$, below which there is no one-step coupling that is contractive, even for the flip dynamics. We use linear programming and duality arguments to guide our construction of a better coupling. We fully characterize the obstructions to going beyond $\frac{11}{6}$. These examples turn out to be quite brittle, and even starting from one, they are likely to break apart before the flip dynamics changes the distance between two neighboring colorings. We use this intuition to design a variable length coupling that shows that the Glauber dynamics is rapidly mixing for any $k\ge \left(\frac{11}{6} - \epsilon_0\right)d$ where $\epsilon_0 \geq 9.4 \cdot 10^{-5}$. This is the first improvement to Vigoda's analysis that holds for general graphs.Comment: 30 pages, 3 figures; fixed some typo

On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic

Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomial-sized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter. The second part of our work deals with providing a lower bound to Courcelle's famous theorem, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Using our results from the first part of our work we establish a strong lower bound for tractability of MSO on classes of colored graphs

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum