Algorithms for Frequency Assignment Problems
Feodor F. Dragan, Yang Xiang, Chenyu Yan and Udaykiran V. Viyyure
Algorithmics lab, Spring 2006, Kent State University
FAP (Frequency Assignment Problem)
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The Frequency Assignment Problem (FAP) in multihop radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to
acceptable levels. FAP can be viewed as a variant of the graph coloring problem.
FAP is usually modeled as Distance-k-Coloring or L(δ1, δ2 ,δ3 ,…,δk)-Coloring of a graph.
L(δ1 , δ2 ,δ3 ,…,δk)-coloring of a graph G=(V, E), where δis are positive integers, is an assignment function Ф: V  N∪{0} such that |Ф(u) - Ф(v)|δi when the distance between u and v in G is equal to
i (i∈{1,2,…,k}). The aim is to minimize λ such that G admits a L(δ1 ,δ2 ,δ3 ,…,δk)- coloring with frequencies between 0 and λ.
Distance-k-Coloring is defined as coloring Gk, the kth power of G, with minimum number of colors. Two vertices v and u are adjacent in Gk if and only if their distance in G is at most k.
FAP evolution
Frequency Assignment in Cellular
networks
(reuse distance 2, or L(1) –coloring)
Map Coloring
Map coloring: Adjacent faces
have different colors
Map coloring: The color contrast
between two adjacent faces
should be large.
Channel Assignment: The same
channel cannot be assigned to
close cells in cellular networks
L(δ1, δ2 ,δ3 ,…,δk)-coloring problem
Hexagonal system
Its dual graph
Color k-powers of graphs
Frequency Assignment in Cellular networks
reuse distance 3, or L(1,1)-coloring
Color generalized powers of
graphs
Our Research Contributions
Frequency Assignment in Cellular networks
L(2,1,1)-coloring (only the dual graph is shown)
Generalized powers
• We define r-coloring of G as an assignment Ф: V{0,1,2,…}
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Frequency Assignment in Cellular networks
L(3,1,1)-coloring. Optimal solution uses 14 colors.
of colors to vertices such that Ф(u) = Ф(v) implies
dG(u,v)>r(v)+r(u), and r+-coloring of G as an assignment
Ф: V{0,1,2,…} of colors to vertices such that Ф(u) = Ф(v)
implies dG(u,v)>r(v)+r(u)+1.
This is our new formulation which generalizes the Distancek-Coloring, approximates L(δ1, δ2 ,δ3 ,…,δk)-coloring, and is
suitable for heterogeneous multihop radio networks.
(Andreas Brandstädt, Feodor F. Dragan, Yang Xiang, Chenyu Yan, “Ceneralized Powers of Graphs
and Their Algorithmic Use”, Accepted by SWAT06.)
Conclusion:
Ongoing research: Frequency Assignment in
trigraphs, modeling irregular cellular networks.
• L(δ , δ ,δ ,…,δ )-coloring is NP-complete for arbitrary
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Graph G with r-values
L graph
Γ graph
• L graph: vertices u,v∈V form an edge if and only if dG(u,v)
r(v)+r(u)
• Γ graph: vertices u,v∈V form an edge if and only if dG(u,v) 
r(v)+r(u)+1
Complexity results for the r-Coloring and
r +Coloring problems on several graph families
k
graphs. We can L(3,1,1)-color any cellular network using the
optimal number of colors (14) in linear time.
r-Coloring is NP-complete in general. But, as we show, for
many graph families, the problem can be solved in
polynomial time, by applying known coloring algorithms to
L graphs or Γ graphs. This gives also approximation
algorithms for the L(δ1 , δ2 ,δ3 ,…,δk)-coloring problem on
those families of graphs.
Results will be partially presented at SWAT’ 2006 Conference, July 6-8, 2006 Riga, Latvia