Coloring triangle-free rectangle overlap graphs with O(log⁡log⁡n)O(\log\log n) colors

Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta(\log\log n)$. Essentially the same construction produces $\Theta(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number $O(\log\log n)$, improving on the previous bound of $O(\log n)$. To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with $O(\log\log n)$ colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).Comment: Minor revisio

Restricted frame graphs and a conjecture of Scott

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once. It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once. We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.Comment: 21 pages, 8 figures - Revised version (note that we moved some of our results to an appendix

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding $\Sigma_2$-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018

Space-Efficient Algorithms and Verification Schemes for Graph Streams

Structured data-sets are often easy to represent using graphs. The prevalence of massive data-sets in the modern world gives rise to big graphs such as web graphs, social networks, biological networks, and citation graphs. Most of these graphs keep growing continuously and pose two major challenges in their processing: (a) it is infeasible to store them entirely in the memory of a regular server, and (b) even if stored entirely, it is incredibly inefficient to reread the whole graph every time a new query appears. Thus, a natural approach for efficiently processing and analyzing such graphs is reading them as a stream of edge insertions and deletions and maintaining a summary that can be (a) stored in affordable memory (significantly smaller than the input size) and (b) used to detect properties of the original graph. In this thesis, we explore the strengths and limitations of such graph streaming algorithms under three main paradigms: classical or standard streaming, adversarially robust streaming, and streaming verification. In the classical streaming model, an algorithm needs to process an adversarially chosen input stream using space sublinear in the input size and return a desired output at the end of the stream. Here, we study a collection of fundamental directed graph problems like reachability, acyclicity testing, and topological sorting. Our investigation reveals that while most problems are provably hard for general digraphs, they admit efficient algorithms for the special and widely-studied subclass of tournament graphs. Further, we exhibit certain problems that become drastically easier when the stream elements arrive in random order rather than adversarial order, as well as problems that do not get much easier even under this relaxation. Furthermore, we study the graph coloring problem in this model and design color-efficient algorithms using novel parameterizations and establish complexity separations between different versions of the problem. The classical streaming setting assumes that the entire input stream is fixed by an adversary before the algorithm reads it. Many randomized algorithms in this setting, however, fail when the stream is extended by an adaptive adversary based on past outputs received. This is the so-called adversarially robust streaming model. We show that graph coloring is significantly harder in the robust setting than in the classical setting, thus establishing the first such separation for a ``natural\u27\u27 problem. We also design a class of efficient robust coloring algorithms using novel techniques. In classical streaming, many important problems turn out to be ``intractable\u27\u27, i.e., provably impossible to solve in sublinear space. It is then natural to consider an enhanced streaming setting where a space-bounded client outsources the computation to a space-unbounded but untrusted cloud service, who replies with the solution and a supporting ``proof\u27\u27 that the client needs to verify. This is called streaming verification or the annotated streaming model. It allows algorithms or verification schemes for the otherwise intractable problems using both space and proof length sublinear in the input size. We devise efficient schemes that improve upon the state of the art for a variety of fundamental graph problems including triangle counting, maximum matching, topological sorting, maximal independent set, graph connectivity, and shortest paths, as well as for computing frequency-based functions such as distinct items and maximum frequency, which have broad applications in graph streaming. Some of our schemes were conjectured to be impossible, while some others attain smooth and optimal tradeoffs between space and communication costs

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version