Coloring and Recognizing Directed Interval Graphs

A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph $G$, an interval $u$ receives a lower (different) color than an interval $v$ if $G$ contains arc $(u,v)$ (edge $\{u,v\}$). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a $\min \{\omega(G), \lambda(G)+1 \}$-approximation algorithm, where $\omega(G)$ is the size of a largest clique and $\lambda(G)$ is the length of a longest directed path in $G$. For the subclass of \emph{bidirectional interval graphs} (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call \emph{containment interval graphs}. In such a graph, there is an arc $(u,v)$ if interval $u$ contains interval $v$, and there is an edge $\{u,v\}$ if $u$ and $v$ overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.Comment: To appear in Proc. ISAAC 202

On List Coloring and List Homomorphism of Permutation and Interval Graphs

List coloring is an NP-complete decision problem even if the total number of colors is three. It is hard even on planar bipartite graphs. We give a polynomial-time algorithm for solving list coloring of permutation graphs with a bounded total number of colors. More generally, we give a polynomial-time algorithm that solves the list-homomorphism problem to any fixed target graph for a large class of input graphs, including all permutation and interval graphs.&nbsp

Recoloring Interval Graphs with Limited Recourse Budget

We consider the problem of coloring an interval graph dynamically. Intervals arrive one after the other and have to be colored immediately such that no two intervals of the same color overlap. In each step only a limited number of intervals may be recolored to maintain a proper coloring (thus interpolating between the well-studied online and offline settings). The number of allowed recolorings per step is the so-called recourse budget. Our main aim is to prove both upper and lower bounds on the required recourse budget for interval graphs, given a bound on the allowed number of colors. For general interval graphs with n vertices and chromatic number k it is known that some recoloring is needed even if we have 2k colors available. We give an algorithm that maintains a 2k-coloring with an amortized recourse budget of $\u1d4aa(log n)$. For maintaining a k-coloring with k ≤ n, we give an amortized upper bound of \u1d4aa(k⋅ k! ⋅ √n), and a lower bound of $Ω(k) for k ∈ \u1d4aa(√n)$, which can be as large as $Ω(√n$). For unit interval graphs it is known that some recoloring is needed even if we have k+1 colors available. We give an algorithm that maintains a (k+1)-coloring with at most $\u1d4aa(k²)$ recolorings per step in the worst case. We also give a lower bound of $Ω(log n)$ on the amortized recourse budget needed to maintain a k-coloring. Additionally, for general interval graphs we show that if one does not insist on maintaining an explicit coloring, one can have a k-coloring algorithm which does not incur a factor of $\u1d4aa(k ⋅ k! ⋅ √n)$ in the running time. For this we provide a data structure, which allows for adding intervals in $\u1d4aa(k² log³ n)$ amortized time per update and querying for the color of a particular interval in $\u1d4aa(log n) time$. Between any two updates, the data structure answers consistently with some optimal coloring. The data structure maintains the coloring implicitly, so the notion of recourse budget does not apply to it

OBDD-Based Representation of Interval Graphs

A graph $G = (V,E)$ can be described by the characteristic function of the edge set $\chi_E$ which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store $\chi_E$ can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e.g. quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is $O(\ | V \ | /\log \ | V \ |)$ and the OBDD size of interval graphs is $O(\ | V \ | \log \ | V \ |)$which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is$\Omega(\ | V \ | \log \ | V \ |)$. We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using$O(\log \ | V \ |)$operations and a coloring algorithm for unit and general intervals graphs using$O(\log^2 \ | V \ |)$ operations and evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic Concepts 201