Boolean-width of graphs
We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums of neighborhoods–across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an n -vertex graph given with...
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| Vydáno v: | Theoretical computer science Ročník 412; číslo 39; s. 5187 - 5204 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Oxford
Elsevier B.V
09.09.2011
Elsevier |
| Témata: | |
| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums of neighborhoods–across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an
n
-vertex graph given with a decomposition tree of boolean-width
k
, we solve Maximum Weight Independent Set in time
O
(
n
2
k
2
2
k
)
and Minimum Weight Dominating Set in time
O
(
n
2
+
n
k
2
3
k
)
. With an additional
n
2
factor in the runtime, we can also count all independent sets and dominating sets of each cardinality.
Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of
G
F
(
2
)
-sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on
Θ
(
n
2
)
vertices has boolean-width
Θ
(
log
n
)
and rank-width, clique-width, tree-width, and branch-width
Θ
(
n
)
. |
|---|---|
| AbstractList | We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums of neighborhoods–across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an
n
-vertex graph given with a decomposition tree of boolean-width
k
, we solve Maximum Weight Independent Set in time
O
(
n
2
k
2
2
k
)
and Minimum Weight Dominating Set in time
O
(
n
2
+
n
k
2
3
k
)
. With an additional
n
2
factor in the runtime, we can also count all independent sets and dominating sets of each cardinality.
Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of
G
F
(
2
)
-sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on
Θ
(
n
2
)
vertices has boolean-width
Θ
(
log
n
)
and rank-width, clique-width, tree-width, and branch-width
Θ
(
n
)
. We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods-Boolean sums of neighborhoods-across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an n -vertex graph given with a decomposition tree of boolean-width k , we solve Maximum Weight Independent Set in time O ( n 2 k 2 2 k ) and Minimum Weight Dominating Set in time O ( n 2 + n k 2 3 k ) . With an additional n 2 factor in the runtime, we can also count all independent sets and dominating sets of each cardinality. Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of G F ( 2 ) -sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Theta ( n 2 ) vertices has boolean-width Theta ( log n ) and rank-width, clique-width, tree-width, and branch-width Theta ( n ) . |
| Author | Vatshelle, Martin Bui-Xuan, Binh-Minh Telle, Jan Arne |
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| Issue | 39 |
| Keywords | Graph decomposition FPT algorithm Boolean algebra Width parameter Bottleneck Clique width Grid pattern Dominating set Computer theory Graph clique Vertex(graph) Rank Algorithm Minimum time Tree width Graph cut Independent set |
| Language | English |
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| Snippet | We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums of neighborhoods–across a cut of a... We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods-Boolean sums of neighborhoods-across a cut of a... |
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| SubjectTerms | Algorithmics. Computability. Computer arithmetics Applied sciences Boolean algebra Combinatorics. Ordered structures Computer Science Computer science; control theory; systems Counting Data Structures and Algorithms Decomposition Exact sciences and technology FPT algorithm Graph decomposition Graphs Information retrieval. Graph Mathematics Minimum weight Miscellaneous Order, lattices, ordered algebraic structures Sciences and techniques of general use Sums Theoretical computing Trees Unions Width parameter |
| Title | Boolean-width of graphs |
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