On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values

Colorful Strips

Given a planar point set and an integer $k$, we wish to color the points with $k$ colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of $k$. We show that if the strip size is at least $2k{-}1$, such a coloring can always be found. We prove that the size of the strip is also bounded in any fixed number of dimensions. In contrast to the planar case, we show that deciding whether a 3D point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. We also consider the problem of coloring a given set of axis-aligned strips, so that any sufficiently covered point in the plane is covered by $k$ colors. We show that in $d$ dimensions the required coverage is at most $d(k{-}1)+1$. Lower bounds are given for the two problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. Finally, we study a variant where strips are replaced by wedges

Conflict-Free Coloring Made Stronger

In FOCS 2002, Even et al. showed that any set of $n$ discs in the plane can be Conflict-Free colored with a total of at most $O(\log n)$ colors. That is, it can be colored with $O(\log n)$ colors such that for any (covered) point $p$ there is some disc whose color is distinct from all other colors of discs containing $p$. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: \begin{enumerate} \item [(i)] Any set of $n$ discs in the plane can be colored with a total of at most $O(k \log n)$ colors such that (a) for any point $p$ that is covered by at least $k$ discs, there are at least $k$ distinct discs each of which is colored by a color distinct from all other discs containing $p$ and (b) for any point $p$ covered by at most $k$ discs, all discs covering $p$ are colored distinctively. We call such a coloring a {\em $k$-Strong Conflict-Free} coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. \item [(ii)] More generally, for families of $n$ simple closed Jordan regions with union-complexity bounded by $O(n^{1+\alpha})$, we prove that there exists a $k$-Strong Conflict-Free coloring with at most $O(k n^\alpha)$ colors. \item [(iii)] We prove that any set of $n$ axis-parallel rectangles can be $k$-Strong Conflict-Free colored with at most $O(k \log^2 n)$ colors. \item [(iv)] We provide a general framework for $k$-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of $k$-Strong Conflict-Free coloring and the recently studied notion of $k$-colorful coloring. \end{enumerate} All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings

Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces

Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This can be interpreted as coloring point sets in R^2 with k colors such that any bottomless rectangle containing at least 3k-2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function $p(k)$ exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the European Workshop on Computational Geometry, held in Assisi (Italy) on March 19-21, 201

On the Drach superintegrable systems

Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated St\"ackel problems with quadratic integrals of motion. For the superintegrable St\"ackel systems the cubic invariant is shown to admit new algebro-geometric representation that is far more elementary than the all the known representations in physical variables. A complete list of all known systems on the plane which admit a cubic invariant is discussed.Comment: 16 pages, Latex2e+Amssym

Combinatorial Bounds for Conflict-free Coloring on Open Neighborhoods

In an undirected graph $G$, a conflict-free coloring with respect to open neighborhoods (denoted by CFON coloring) is an assignment of colors to the vertices such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for a CFON coloring of $G$ is the CFON chromatic number of $G$, denoted by $\chi_{ON}(G)$. The decision problem that asks whether $\chi_{ON}(G) \leq k$ is NP-complete. We obtain the following results: * Bodlaender, Kolay and Pieterse [WADS 2019] showed the upper bound $\chi_{ON}(G)\leq {\sf fvs}(G)+3$, where ${\sf fvs}(G)$ denotes the size of a minimum feedback vertex set of $G$. We show the improved bound of $\chi_{ON}(G)\leq {\sf fvs}(G)+2$, which is tight, thereby answering an open question in the above paper. * We study the relation between $\chi_{ON}(G)$ and the pathwidth of the graph $G$, denoted ${\sf pw}(G)$. The above paper from WADS 2019 showed the upper bound $\chi_{ON}(G) \leq 2{\sf tw}(G)+1$ where ${\sf tw}(G)$ stands for the treewidth of $G$. This implies an upper bound of $\chi_{ON}(G) \leq 2{\sf pw}(G)+1$. We show an improved bound of $\chi_{ON}(G) \leq \lfloor \frac{5}{3}({\sf pw}(G)+1) \rfloor$. * We prove new bounds for $\chi_{ON}(G)$ with respect to the structural parameters neighborhood diversity and distance to cluster, improving existing results. * We also study the partial coloring variant of the CFON coloring problem, which allows vertices to be left uncolored. Let $\chi^*_{ON}(G)$ denote the minimum number of colors required to color $G$ as per this variant. Abel et. al. [SIDMA 2018] showed that $\chi^*_{ON}(G) \leq 8$ when $G$ is planar. They asked if fewer colors would suffice for planar graphs. We answer this question by showing that $\chi^*_{ON}(G) \leq 5$ for all planar $G$. All our bounds are a result of constructive algorithmic procedures.Comment: 30 page

On the exchange of intersection and supremum of sigma-fields in filtering theory

We construct a stationary Markov process with trivial tail sigma-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page