Balanced Interval Coloring

We consider the discrepancy problem of coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time $O(n \log n + kn \log k)$ for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where $n$ tasks with given start- and endtimes have to be distributed among $k$ servers. Our results imply that this can be done ideally balanced. When generalizing to $d$-dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any $d \ge 2$ and any $k \ge 2$ it is NP-complete to decide if such a solution exists, which implies also NP-hardness of the respective minimization problem. In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.Comment: Accepted at STACS 201

Balanced Independent and Dominating Sets on Colored Interval Graphs

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely \emph{$f$-Balanced Independent Set} ($f$-BIS) and \emph{$f$-Balanced Dominating Set} ($f$-BDS). Let $G=(V,E)$ be a vertex-colored interval graph with a $k$-coloring $\gamma \colon V \rightarrow \{1,\ldots,k\}$ for some $k \in \mathbb N$. A subset of vertices $S\subseteq V$ is called \emph{$f$-balanced} if $S$ contains $f$ vertices from each color class. In the $f$-BIS and $f$-BDS problems, the objective is to compute an independent set or a dominating set that is $f$-balanced. We show that both problems are \NP-complete even on proper interval graphs. For the BIS problem on interval graphs, we design two \FPT\ algorithms, one parameterized by $(f,k)$ and the other by the vertex cover number of $G$. Moreover, we present a 2-approximation algorithm for a slight variation of BIS on proper interval graphs

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