On the fine-grained complexity of rainbow coloring

The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in $k$ colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any $k\ge 2$, there is no algorithm for Rainbow k-Coloring running in time $2^{o(n^{3/2})}$, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set $S$ of pairs of vertices and we ask if there is a coloring in which all the pairs in $S$ are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by $|S|$. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer $q$ and we ask if there is a coloring in which at least $q$ anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by $q$ and has a kernel of size $O(q)$ for every $k\ge 2$ (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011]

Fine-Grained Complexity of Rainbow Coloring and its Variants

Consider a graph G and an edge-coloring c_R:E(G) rightarrow [k]. A rainbow path between u,v in V(G) is a path P from u to v such that for all e,e\u27 in E(P), where e neq e\u27 we have c_R(e) neq c_R(e\u27). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all u,v in V(G) there is a rainbow path between u and v in G. Several variants of Rainbow k-Coloring have been studied, two of which are defined as follows. The Subset Rainbow k-Coloring takes as an input a graph G and a set S subseteq V(G) times V(G), and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all (u,v) in S there is a rainbow path between u and v in G. The problem Steiner Rainbow k-Coloring takes as an input a graph G and a set S subseteq V(G), and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all u,v in S there is a rainbow path between u and v in G. In an attempt to resolve open problems posed by Kowalik et al. (ESA 2016), we obtain the following results. - For every k geq 3, Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|E(G)|)}n^{O(1)}, unless ETH fails. - For every k geq 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|S|^2)}n^{O(1)}, unless ETH fails. - Subset Rainbow k-Coloring admits an algorithm running in time 2^{OO(|S|)}n^{O(1)}. This also implies an algorithm running in time 2^{o(|S|^2)}n^{O(1)} for Steiner Rainbow k-Coloring, which matches the lower bound we obtain

Algorithms and Bounds for Very Strong Rainbow Coloring

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (\src(G)) of the graph. When proving upper bounds on \src(G), it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} (\vsrc(G)). In this paper, we give upper bounds on \vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on \src(G) for these classes, showing that the study of \vsrc(G) enables meaningful progress on bounding \src(G). Then we study the complexity of the problem to compute \vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that \vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for \src(G). We also observe that deciding whether \vsrc(G) = k is fixed-parameter tractable in $k$ and the treewidth of $G$. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor $n^{1-\varepsilon}$, unless P$=$NP

Open questions about Ramsey-type statements in reverse mathematics

Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.Comment: 15 page

Complexity of Computing the Anti-Ramsey Numbers for Paths

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erd\" os, Simonovits and S\' os. For given graphs $G$ and $H$ the \emph{anti-Ramsey number} $\textrm{ar}(G,H)$ is defined to be the maximum number $k$ such that there exists an assignment of $k$ colors to the edges of $G$ in which every copy of $H$ in $G$ has at least two edges with the same color. There are works on the computational complexity of the problem when $H$ is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number $\textrm{ar}(G,P_k)$, where $P_k$ is a path of length $k$. First, we observe that when $k = \Omega(n)$, the problem is hard; hence, the challenging part is the computational complexity of the problem when $k$ is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing $\textrm{ar}(G,P_k)$ for every integer $k>2$ is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating $\textrm{ar}(G,P_3)$ to a factor of $n^{-1/2 - \epsilon}$ is hard already in $3$-partite graphs, unless P=NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant $k$. Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. We introduce the notion of color connected coloring and employing this structural property. We obtain a linear time algorithm to compute $\textrm{ar}(G,P_k)$, for every integer $k$, when the host graph, $G$, is a tree

Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds

We study rainbow connectivity of graphs from the algorithmic and graph-theoretic points of view. The study is divided into three parts. First, we study the complexity of deciding whether a given edge-colored graph is rainbow-connected. That is, we seek to verify whether the graph has a path on which no color repeats between each pair of its vertices. We obtain a comprehensive map of the hardness landscape of the problem. While the problem is NP-complete in general, we identify several structural properties that render the problem tractable. At the same time, we strengthen the known NP-completeness results for the problem. We pinpoint various parameters for which the problem is fixed-parameter tractable, including dichotomy results for popular width parameters, such as treewidth and pathwidth. The study extends to variants of the problem that consider vertex-colored graphs and/or rainbow shortest paths. We also consider upper and lower bounds for exact parameterized algorithms. In particular, we show that when parameterized by the number of colors k, the existence of a rainbow s-t path can be decided in O∗ (2k ) time and polynomial space. For the highly related problem of finding a path on which all the k colors appear, i.e., a colorful path, we explain the modest progress over the last twenty years. Namely, we prove that the existence of an algorithm for finding a colorful path in (2 − ε)k nO(1) time for some ε > 0 disproves the so-called Set Cover Conjecture.Second, we focus on the problem of finding a rainbow coloring. The minimum number of colors for which a graph G is rainbow-connected is known as its rainbow connection number, denoted by rc(G). Likewise, the minimum number of colors required to establish a rainbow shortest path between each pair of vertices in G is known as its strong rainbow connection number, denoted by src(G). We give new hardness results for computing rc(G) and src(G), including their vertex variants. The hardness results exclude polynomial-time algorithms for restricted graph classes and also fast exact exponential-time algorithms (under reasonable complexity assumptions). For positive results, we show that rainbow coloring is tractable for e.g., graphs of bounded treewidth. In addition, we give positive parameterized results for certain variants and relaxations of the problems in which the goal is to save k colors from a trivial upper bound, or to rainbow connect only a certain number of vertex pairs.Third, we take a more graph-theoretic view on rainbow coloring. We observe upper bounds on the rainbow connection numbers in terms of other well-known graph parameters. Furthermore, despite the interest, there have been few results on the strong rainbow connection number of a graph. We give improved bounds and determine exactly the rainbow and strong rainbow connection numbers for some subclasses of chordal graphs. Finally, we pose open problems and conjectures arising from our work

TextureGAN: Controlling Deep Image Synthesis with Texture Patches

In this paper, we investigate deep image synthesis guided by sketch, color, and texture. Previous image synthesis methods can be controlled by sketch and color strokes but we are the first to examine texture control. We allow a user to place a texture patch on a sketch at arbitrary locations and scales to control the desired output texture. Our generative network learns to synthesize objects consistent with these texture suggestions. To achieve this, we develop a local texture loss in addition to adversarial and content loss to train the generative network. We conduct experiments using sketches generated from real images and textures sampled from a separate texture database and results show that our proposed algorithm is able to generate plausible images that are faithful to user controls. Ablation studies show that our proposed pipeline can generate more realistic images than adapting existing methods directly.Comment: CVPR 2018 spotligh

Manual / Issue 10 / Polychrome

Manual, a journal about art and its making. Polychrome. In art, especially, polychrome invites us to the dialogue that colors are always having amongst themselves. A history of polychrome could be a series of poems exchanged among colors. The exchange might exhibit something like perpetual newness, again and again revealing differently bent hues and movingly novel blends. It would be a short-line poetry, excruciatingly sensitive to tone. Its speakers would have no names, so it would confuse the psychology of human orientation. In this connection, a warning against rendering polychrome as a pure positive seems in order: the parties to this dialogue talk at cross-purposes, always on the brink of divorcing. Polychrome can offend and destroy. It conscripts discrete colors in order to sacrifice them. Does polychrome offend by mocking our own failure to connect? In any case, polychrome has an advanced idiom for dealing with conflict. It’s at home with uncertainty. —Darby English, from the introduction to Issue 10: Polychrome. Softcover, 80 pages. Published 2018 by the RISD Museum. Manual 10 (Polychrome) contributors include David Batchelor, Gina Borromeo, Nicole Buchanan, Catherine Cooper, Darby English, Mara L. Hermano, Elon Cook Lee, Josephine Lee, Evelyn Lincoln, Dominic Molon, Maureen C. O\u27Brien, RISD Museum 2017 Summer Teen Intensive Students, and Elizabeth A. Williams.https://digitalcommons.risd.edu/risdmuseum_journals/1036/thumbnail.jp