About equivalent interval colorings of weighted graphs

Abstract

Given a graph G = (V, E) with strictly positive integer weights ω i on the vertices i ∈ V , a k-interval coloring of G is a function I that assigns an interval I(i) ⊆ {1, • • • , k} of ω i consecutive integers (called colors) to each vertex i ∈ V. If two adjacent vertices x and y have common colors, i.e. I(i) ∩ I(j) = ∅ for an edge [i, j] in G, then the edge [i, j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ int (G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω i = 1 for all vertices i ∈ V). Two k-interval colorings I 1 and I 2 are said equivalent if there is a permutation π of the integers 1, • • • , k such that ℓ ∈ I 1 (i) if and only if π(ℓ) ∈ I 2 (i) for all vertices i ∈ V. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.

Memory Analysis and Optimized Allocation of Dataflow Applications on Shared-Memory MPSoCs

“…Unfortunately, as presented in [4], this problem is known to be NP-Hard, therefore it would be prohibitively long to solve for applications with hundreds or thousands of memory objects. Moreover, a sub-optimal solution to the Interval Coloring problem corresponds to an allocation that uses more memory than the minimum possible: more memory than the greatest lower bound.…”

“…Finding this optimal allocation based on a MEG can be achieved by solving the equivalent Interval Coloring Problem [4,12].…”

“…Each vertex of the graph is associated with a set of colors such that two connected vertices have no color in common. The purpose of graph coloring technique is to minimize the total number of colors used in the graph [4]. -Using constraint programming [31] where memory constraints can be specified together with resource usage and execution time constraints.…”

Memory bounds for the distributed execution of a hierarchical Synchronous Data-Flow graph

“…Unfortunately, as presented in [15], this problem is known to be NP-Hard, therefore it would be prohibitively long to solve in the rapid prototyping context which involves applications with hundreds or thousands of buffers. Moreover, a suboptimal solution to the Interval Coloring problem corresponds to an allocation that uses more memory than the minimum possible: more memory than the greatest lower bound.…”

“…Finding this optimal allocation based on an exclusion graph can be done by solving the equivalent Interval Coloring Problem [15], [3].…”