Abstract
Graph Theory Dynamic Storage Allocation is a problem concerned with storing items that each have weight and time restrictions. Approximate algorithms have been constructed through online coloring of interval graphs. We present a generalization that uses online coloring of tolerance graphs. We utilize online-with-representation algorithms on tolerance graphs, which are online algorithms in which the corresponding tolerance representation of a vertex is also presented. We find linear bounds for the online-with-representation chromatic number of various classes of tolerance graphs and apply these results to a generalization of Dynamic Storage Allocation, giving us a polynomial time approximation algorithm with linear performance ratio.
Highlights
In this paper we study a generalization of Dynamic Storage Allocation (DSA) through the use of tolerance graphs and online coloring with representation
The Dynamic Storage Allocation (DSA) problem is defined as follows: Given a maximum storage size D and a set A of items such that each item a ∈ A has integral size s(a) and storage interval Ia = [L(a), R(a)], is it possible to store A in linear space D, i.e., does there exists a function f : A → {1, . . . , D} so that f (a)+s(a)−1 ≤ D for every a ∈ A, and if I(a)∩I(b) = ∅ either f (a)+s(a) ≤ f (b) or f (b) + s(b) ≤ f (a) for all a, b ∈ A? It is known that DSA is NP-complete
The following result was stated in [1] and we find a similar result for totally bounded tolerance graphs
Summary
In this paper we study a generalization of Dynamic Storage Allocation (DSA) through the use of tolerance graphs and online coloring with representation. For a graph G, denote its clique number by ω(G), its maximum degree by ∆(G) and its chromatic number by χ(G). Let ∆+(D) denote the maximum out-degree of a digraph D, and δ+(x) the out-degree of a vertex x ∈ V (D). For a weighted graph G∗ = (V, E, w) with weight function w : V → N, the weight of a set of vertices U is defined to be w(U ) := v∈U w(v) and ω∗(G∗) := max{w(U ) : U is a clique}. An interval t-coloring is a function c : V → {1, . The weighted chromatic number, χ∗(G∗), is defined to be the least t such that G∗ has a interval t-coloring. For a weighted digraph D∗ = (V, E, w), define the weighted out-degree of a digraph as ∆+w(D) = {w(v) : v ∈ S, S is maximum out-star}. 1365–8050 c 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
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