Exact Algorithms for L(2,1)-Labeling of Graphs
The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G =( V , E ) into an interval of integers {0,…, k } is an L (2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integer...
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| Published in: | Algorithmica Vol. 59; no. 2; pp. 169 - 194 |
|---|---|
| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer-Verlag
01.02.2011
Springer Springer Verlag |
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online Access: | Get full text |
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| Summary: | The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph
G
=(
V
,
E
) into an interval of integers {0,…,
k
} is an
L
(2,1)-labeling of
G
of span
k
if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed
k
≥4, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive
O
*
((
k
+1)
n
) algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of
k
=4, where the running time of our algorithm is
O
(1.3006
n
). Furthermore we show that dynamic programming can be used to establish an
O
(3.8730
n
) algorithm to compute an optimal
L
(2,1)-labeling. |
|---|---|
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-009-9302-7 |