Approximative Kernelization:
On the Trade-off between Fidelity and
Kernel Size
Hadas Shachnai
Technion
Workshop on Kernelization, Nov 2010
joint with
Michael Fellows and Frances Rosamond
Charles Darwin University
Kernelization – Fidelity vs. Kernel Size
• Traditionally: used as a preprocessing tool in FPT
algorithms, which does not harm the classification of
the instance (as a ‘yes’ or ‘no’ w.r.t. the parameterized
problem).
• Many FPT algorithms for NP-hard problems use
kernels whose sizes are lower bounded by a function
f(k) = Ω(poly(k)), where k is the parameter.
• Suppose that in solving an FPT problem Π, we want to
obtain a kernel of smaller size (=better running time),
with some compromise on its fidelity when lifting a
solution for the kernelized instance back to a solution
for the original instance.
Can we define a tradeoff between fidelity and
kernel size?
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Approximative Kernelization
Let L be a parameterized problem, i.e., L consists of input
pairs (x, k), where x is a problem instance, and k is the
parameter. Given α ≥ 1, an α-fidelity kernelization of the
problem
(i) Transforms in polynomial time the input (x, k) to
‘reduced’ input (x’, k’), such that k’ ≤ k and |x’| ≤ g(k, α),
and
(ii) If (x, k/α)  L then (x’, k’)  L
(iii) If (x’, k’)  L then (x, k)  L
The special case where α = 1 is classic kernelization.
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Approximative Kernelization
•
Combine approximation with kernelization: While lifting
up to a solution for the original problem, we may get the
value k, whereas there exists a solution of value k/α.
• The definition refers to Minimization problems (similar
for maximization problems with k/α replaced by kα).
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Application: Vertex Cover
Input: An undirected graph G=(V,E), an integer k ≥ 1.
Output: A subset of vertices C  V, |C| ≤ k such that each
edge in E has at least one endpoint in C (if one exists).
• Many 2- approximation polynomial-time algorithms
• A 3/2- approximation known for maximum degree four
[Hochbaum 1983].
• Unless Unique Game Conjecture fails: No factor-(2- ε)approximation polynomial time algorithm exists [Khot, Regev
2008].
• Vertex Cover is in FPT for general graphs: can be solved
in time O*(1.28k).
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Application: Vertex Cover
Let G=(V,E), k ≥ 1 and α  [1,2].
1. Initially C=Ф
2. Reduction step: Apply reduction rules to (G, k/α). The
resulting instance is (Ĝ, k̂ ), where k̂ = k/α –h, and h=|C|.
3. If k̂ ‹ 0 return failure, else
(a) Let l= 2(1 – 1/ α)k. Find a maximum matching M in Ĝ.
(b) Partition the edges in M to m ≥ 1 sets, each (except
maybe the last) contains l vertices. Denote the vertex
sets by {S1,…, Sm}.
(c) Shrinking step: C= C U S1. Omit from Ĝ the vertices in
S1 and all neighboring edges.
4. Omit from the resulting graph, G’, isolated vertices. Return
G’ with parameter k’= k̂ - | S1| /2.
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Algorithm : Shrinking step
v1
v2
v12
v3
v13
v4
v14
v5
v15
v6
v16
v7
v17
v8
v18
v9
v19
v10
v20
k=10, α=10/9
Ĝ = ( {v1,…,v20}, Ế)
k̂ = k/α –3=6
l = 2(1 – 1/ α)k =4
S1={v1,v2,v12,v14}
v11
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Algorithm : Shrinking step
G’ = ( V’, E’)
V’= {v3,v4,…,v20}
v3
v13
v4
v5
v15
v6
v16
v7
v17
v8
v18
v9
v19
v10
v20
k’= k̂ -2=4
v11
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Algorithm : example
r
s
y
G=(V,E) , k=8
z
t
u
v
w
a
b
c
x
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Algorithm : example
r
s
y
z
t
u
v
w
Reduction step:
Omit the crown
H={b,c}
I={u,v,w}
a
b
c
α =2
l = 2(1 – 1/ α)k =8
x
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Algorithm : example
r
s
y
z
t
Reduction step:
Omit the crown
H={b,c}
I={u,v,w}
a
α =2
l = 2(1 – 1/ α)k =8
x
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Algorithm : example
s
z
t
Reduction step:
Omit the crown
H={b,c}
I={u,v,w}
α =2
l = 2(1 – 1/ α)k =8
|M| ‹ l/2 : G’ is a 2-fidelity kernel of size 0!
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Algorithm : example
r
s
y
G=(V,E) , k=8
z
t
u
v
w
a
b
c
x
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Algorithm : example
r
s
y
z
t
u
v
w
Reduction step:
Omit the crown
H={b,c}
I={u,v,w}
a
b
c
α =1
l = 2(1 – 1/ α)k =0
x
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Analysis: α-fidelity
We show that the algorithm satisfies the properties of αfidelity kernelization.
1. The transformation from G to G’ is polynomial.
2. If (G, k/α)  L then (G’, k’)  L
We note that if there is a vertex cover of size k/α for G, there
is a cover of size k̂ = k/α -h for Ĝ, and there is a cover for
G’ of size k’= k̂ - | S1| /2.
3. If (G’, k’)  L then (G, k)  L
Assume that there is a vertex cover C(G’) of size k’ for G’.
Consider the cover C*= C(G’) U S1 U C, where C is the cover
found in the Reduction step.
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Analysis : α-fidelity (Cont’d)
Then,
|C*| = |C(G’) U S1 U C |
= k’ + | S1| + |C|
= k/α – h - |S1| /2 + | S1| + h
= k/α + |S1| /2
≤ k/α + (1 – 1/α)k = k
Last inequality follows from the definition of l.
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Analysis: Kernel Size
Suppose there is a cover of size k/α for G, then the number
of vertices in Ĝ is at most 2k/α (using, e.g., crown rules).
Distinguish between two cases:
(i) If |M| ≥ l /2 = (1-1/ α)k then the number of vertices in G’
is at most
2k/ α - l = 2k/ α – 2k(1 – 1/ α)= 2k(2- α)/ α.
(ii) If |M| ‹ l /2, then S1 contains all the matched vertices in
M, therefore G’ is empty.
It follows that the kernel size is at most 2k(2- α)/ α.
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Related Work
FPT approximation
• Obtain a solution of value g(k) for a problem parameterized
by k (e.g., Downey, F, McCartin and R, 2008; Many more..)
Parameterized approximations for NP-hard problems by
moderately exponential time algorithms
• Improve best known approximation ratios for subgraph
maximization, minimum covering (Bourgeois, Escoffier and
Paschos, 2009)
• β-approximation algorithms for vertex cover, β(1,2), through
accelerated branching (Fernau, Brankovic and Cakic, 2009)
Links between approximation and kernelization
• Exploit polynomial time approximation results in kernelization
(Bevern, Moser and Niedermeier, 2010)
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Future work
Explore further approximative kernelization:

Better tradeoff for vertex cover? (Current algorithm
does not optimize on kernel size.)

Define tradeoffs for other FPT problems

A general framework for combining exact reduction
rules with approximation algorithms to guarantee αfidelity, for any α ≥ 1.
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