New Tools and Connections for Exponential-Time Approximation
In this paper, we develop new tools and connections for exponential time approximation . In this setting, we are given a problem instance and an integer r > 1 , and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establis...
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| Published in: | Algorithmica Vol. 81; no. 10; pp. 3993 - 4009 |
|---|---|
| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.10.2019
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541, 1432-0541 |
| Online Access: | Get full text |
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| Summary: | In this paper, we develop new tools and connections for
exponential time approximation
. In this setting, we are given a problem instance and an integer
r
>
1
, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of
r
for maximum independent set in
O
∗
(
exp
(
O
~
(
n
/
r
log
2
r
+
r
log
2
r
)
)
)
time,
r
for chromatic number in
O
∗
(
exp
(
O
~
(
n
/
r
log
r
+
r
log
2
r
)
)
)
time,
(
2
-
1
/
r
)
for minimum vertex cover in
O
∗
(
exp
(
n
/
r
Ω
(
r
)
)
)
time, and
(
k
-
1
/
r
)
for minimum
k
-hypergraph vertex cover in
O
∗
(
exp
(
n
/
(
k
r
)
Ω
(
k
r
)
)
)
time.
(Throughout,
O
~
and
O
∗
omit
polyloglog
(
r
)
and factors polynomial in the input size, respectively.) The best known time bounds for all problems were
O
∗
(
2
n
/
r
)
(Bourgeois et al. in Discret Appl Math 159(17):1954–1970,
2011
; Cygan et al. in Exponential-time approximation of hard problems,
2008
). For maximum independent set and chromatic number, these bounds were complemented by
exp
(
n
1
-
o
(
1
)
/
r
1
+
o
(
1
)
)
lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379,
2013
; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis,
2014
). Our results show that the naturally-looking
O
∗
(
2
n
/
r
)
bounds are not tight for all these problems. The key to these results is a
sparsification
procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new
randomized branching rule
. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan’s PCP (Chan in J. ACM 63(3):27:1–27:32,
2016
). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128,
2016
; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs,
2016
). |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 |
| ISSN: | 0178-4617 1432-0541 1432-0541 |
| DOI: | 10.1007/s00453-018-0512-8 |