Packing Topological Minors Half-Integrally

The packing problem and the covering problem are two of the most general questions in graph theory. The Erd\H{o}s-P\'{o}sa property characterizes the cases when the optimal solutions of these two problems are bounded by functions of each other. Robertson and Seymour proved that when packing and covering $H$-minors for any fixed graph $H$, the planarity of $H$ is equivalent with the Erd\H{o}s-P\'{o}sa property. Thomas conjectured that the planarity is no longer required if the solution of the packing problem is allowed to be half-integral. In this paper, we prove that this half-integral version of Erd\H{o}s-P\'{o}sa property holds with respect to the topological minor containment, which easily implies Thomas' conjecture. Indeed, we prove an even stronger statement in which those subdivisions are rooted at any choice of prescribed subsets of vertices. Precisely, we prove that for every graph $H$, there exists a function $f$ such that for every graph $G$, every sequence $(R_v: v \in V(H))$ of subsets of $V(G)$ and every integer $k$, either there exist $k$ subgraphs $G_1,G_2,...,G_k$ of $G$ such that every vertex of $G$ belongs to at most two of $G_1,...,G_k$ and each $G_i$ is isomorphic to a subdivision of $H$ whose branch vertex corresponding to $v$ belongs to $R_v$ for each $v \in V(H)$, or there exists a set $Z \subseteq V(G)$ with size at most $f(k)$ intersecting all subgraphs of $G$ isomorphic to a subdivision of $H$ whose branch vertex corresponding to $v$ belongs to $R_v$ for each $v \in V(H)$. Applications of this theorem include generalizations of algorithmic meta-theorems and structure theorems for $H$-topological minor free (or $H$-minor free) graphs to graphs that do not half-integrally pack many $H$-topological minors (or $H$-minors)

Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any hyperplane of the arrangement. We show that f(M) is a piecewise polynomial function of positive integers M, composed of terms that appear gradually as M increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo M; the number of proper modular colorations is a different piecewise polynomial that for large M becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.Comment: 13 p

Coloring non-crossing strings

For a family of geometric objects in the plane $\mathcal{F}=\{S_1,\ldots,S_n\}$, define $\chi(\mathcal{F})$ as the least integer $\ell$ such that the elements of $\mathcal{F}$ can be colored with $\ell$ colors, in such a way that any two intersecting objects have distinct colors. When $\mathcal{F}$ is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most $k$ pseudo-disks, it can be proven that $\chi(\mathcal{F})\le 3k/2 + o(k)$ since the problem is equivalent to cyclic coloring of plane graphs. In this paper, we study the same problem when pseudo-disks are replaced by a family $\mathcal{F}$ of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of $\mathcal{F}$ are only allowed to "touch" each other. Such a family is said to be $k$-touching if no point of the plane is contained in more than $k$ elements of $\mathcal{F}$. We give bounds on $\chi(\mathcal{F})$ as a function of $k$, and in particular we show that $k$-touching segments can be colored with $k+5$ colors. This partially answers a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems of strings.Comment: 19 pages. A preliminary version of this work appeared in the proceedings of EuroComb'09 under the title "Coloring a set of touching strings

Lossy Kernelization

In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions combine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size $\alpha$-approximate kernel. Loosely speaking, a polynomial size $\alpha$-approximate kernel is a polynomial time pre-processing algorithm that takes as input an instance $(I,k)$ to a parameterized problem, and outputs another instance $(I',k')$ to the same problem, such that $|I'|+k' \leq k^{O(1)}$. Additionally, for every $c \geq 1$, a $c$-approximate solution $s'$ to the pre-processed instance $(I',k')$ can be turned in polynomial time into a $(c \cdot \alpha)$-approximate solution $s$ to the original instance $(I,k)$. Our main technical contribution are $\alpha$-approximate kernels of polynomial size for three problems, namely Connected Vertex Cover, Disjoint Cycle Packing and Disjoint Factors. These problems are known not to admit any polynomial size kernels unless $NP \subseteq coNP/poly$. Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems. On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an $\alpha$-approximate kernel of polynomial size, for any $\alpha \geq 1$, unless $NP \subseteq coNP/poly$. In order to prove this lower bound we need to combine in a non-trivial way the techniques used for showing kernelization lower bounds with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and approximate kernel lower bounds for Set Cover and Hitting Set parameterized by universe siz

Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs