Notes On Inverse Interval Graph Coloring Problems (original) (raw)

Exact Algorithms for Intervalizing Coloured Graphs

Theory of Computing Systems, 2015

In the INTERVALIZING COLOURED GRAPHS problem, one must decide for a given graph G = (V , E) with a proper vertex colouring of G whether G is the subgraph of a properly coloured interval graph. For the case that the number of colors is fixed, we give an exact algorithm that uses 2 O(n/ log n) time. We also give an O * (2 n) algorithm for the case that the number of colors is not fixed.

Exact Algorithms for Intervalizing Colored Graphs

Lecture Notes in Computer Science, 2011

In the Intervalizing Colored Graphs problem, one must decide for a given graph G = (V, E) with a proper vertex coloring of G whether G is the subgraph of a properly colored interval graph. For the case that the number of colors k is xed, we give an exact algorithm that uses O * (2 n/log 1− (n)) time for all > 0. We also give an O * (2 n) algorithm for the case that the number of colors k is not xed.

A Note on First-Fit Coloring of Interval Graphs

A Journal on The Theory of Ordered Sets and Its Applications, 2008

We apply the Column Construction Method (Varadarajan et al., Proceedings of the Fifteenth Annual ACM-SIAM Symposium On Discrete Algorithms, pp. 562–571, 2004) to a minimal clique cover of an interval graph to obtain a new proof that First-Fit is 8-competitive for online coloring interval graphs. This proof also yields a new discovery that in each minimal clique cover of an interval graph G, there is a clique of size \(\frac{\omega(G)}{8}\) .

SUB-COLORING AND HYPO-COLORING INTERVAL GRAPHS

Discrete Mathematics, Algorithms and Applications, 2010

In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as "subroutines" for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a "forbidden subgraph" characterization of graphs with sub-chromatic number k and use this to derive a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete, and then via reduction to the max-coloring problem, show how to obtain an O(log n)-approximation algorithm for it.

On the k-coloring of intervals

Discrete Applied Mathematics, 1987

The problem of coloring a set of n intervals (from the real line) with a set of k colors is studied. In such a coloring, two intersecting intervals must receive distinct colors. Our main result is an O(k + n) algorithm for k-coloring a maximum cardinality subset of the intervals, assuming that the endpoints of the intervals are presorted. Previous methods are linear only in n, and assume that k is a fixed constant. In addition to the main result, we provide an O(kS(n)) algorithm for k-coloring a set of weighted intervals of maximum total weight. Here, S(n) is the running time of any algorithm for finding shortest paths in graphs with O(n) edges. The best previous algorithm for this problem required time O(nS(n)). Since in most applications, k is substantially smaller than n, the saving is significant.

On a reduction of the interval coloring problem to a series of bandwidth coloring problems

Journal of Scheduling, 2010

Given a graph G = (V, E) with strictly positive integer weights ω i on the vertices i ∈ V , an interval coloring of G is a function I that assigns an interval I(i) of ω i consecutive integers (called colors) to each vertex i ∈ V so that I(i) ∩ I(j) = ∅ for all edges {i, j} ∈ E. The interval coloring problem is to determine an interval coloring that uses as few colors as possible. Assuming that a strictly positive integer weight δ ij is associated with each edge {i, j} ∈ E, a bandwidth coloring of G is a function c that assigns an integer (called a color) to each vertex i ∈ V so that |c(i) − c(j)| ≥ δ ij for all edges {i, j} ∈ E. The bandwidth coloring problem is to determine a bandwidth coloring with minimum difference between the largest and the smallest colors used. We prove that an optimal solution of the interval coloring problem can be obtained by solving a series of bandwidth coloring problems. Computational experiments demonstrate that such a reduction can help to solve larger instances or to obtain better upper bounds on the optimal solution value of the interval coloring problem.

A Technique for Exact Computation of Precoloring Extension on Interval Graphs

International Journal of Foundations of Computer Science, 2013

Inspired by a real-life application, we investigate the computationally hard problem of extending a precoloring of an interval graph to a proper coloring under some bound on the number of available colors. We are interested in quickly determining whether or not such an extension exists on instances occurring in practice in connection with campsite bookings on a campground. A naive exhaustive search does not terminate in reasonable time. We have formulated a new approach which moves the computation time within the usable range on all the data samples available to us.

Online Interval Coloring and Variants

Lecture Notes in Computer Science, 2005

We study interval coloring problems and present new upper and lower bounds for several variants. We are interested in four problems, online coloring of intervals with and without bandwidth and a new problem called lazy online coloring again with and without bandwidth. We consider both general interval graphs and unit interval graphs. Specifically, we establish the difference between the two main problems which are interval coloring with and without bandwidth. We present the first nontrivial lower bound of 3.2609 for the problem with bandwidth. This improves the lower bound of 3 that follows from the tight results for interval coloring without bandwidth presented in [9].

The harmonious coloring problem is NP-complete for interval and permutation graphs

2007

In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.