Dynamic Kernels for Hitting Sets and Set Packing
Computing small kernels for the hitting set problem is a well-studied computational problem where we are given a hypergraph with n vertices and m hyperedges, each of size d for some small constant d , and a parameter k . The task is to compute a new hypergraph, called a kernel , whose size is p...
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| Vydáno v: | Algorithmica Ročník 84; číslo 11; s. 3459 - 3488 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.11.2022
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Computing small kernels for the hitting set problem is a well-studied computational problem where we are given a hypergraph with
n
vertices and
m
hyperedges, each of size
d
for some small constant
d
, and a parameter
k
. The task is to compute a new hypergraph, called a
kernel
, whose size is polynomial with respect to the parameter
k
and which has a size-
k
hitting set if, and only if, the original hypergraph has one. State-of-the-art algorithms compute kernels of size
k
d
(which is a polynomial as
d
is a constant), and they do so in time
m
·
2
d
poly
(
d
)
for a small polynomial
poly
(
d
)
(which is linear in the hypergraph size for
d
fixed). We generalize this task to the
dynamic
setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a size-
k
d
kernel. This paper presents a
deterministic
solution with
worst-case
time
3
d
poly
(
d
)
for updating the kernel upon inserts and time
5
d
poly
(
d
)
for updates upon deletions. These bounds nearly match the time
2
d
poly
(
d
)
needed by the best static algorithm per hyperedge. Let us stress that for constant
d
our algorithm maintains a hitting set kernel with
constant, deterministic, worst-case
update time that is independent of
n
,
m
, and the parameter
k
. As a consequence, we also get a deterministic dynamic algorithm for keeping track of size-
k
hitting sets in
d
-hypergraphs with update times
O
(1) and query times
O
(
c
k
)
where
c
=
d
-
1
+
O
(
1
/
d
)
equals the best base known for the static setting. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-022-00986-0 |