Approximative Kernelization: On the Trade-off between Fidelity and

Kernel Size

joint with

Michael Fellows and Frances RosamondCharles Darwin University

Hadas ShachnaiTechnion

Workshop on Kernelization, Nov 2010

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.

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Can we define a tradeoff between fidelity and kernel size?

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

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The special case where α = 1 is classic kernelization.

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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• 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 CoverInput: 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).

Application: Vertex Cover

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1. Initially C=Ф 2. Reduction step: Apply reduction rules to (G, k/α). The

resulting instance is (Ĝ, ), where = k/α –h, and h=|C|.3. If ‹ 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’= - | S1| /2.

k̂k̂ k̂

k̂

Let G=(V,E), k ≥ 1 and α [1,2].

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Algorithm : Shrinking step

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l = 2(1 – 1/ α)k =4

S1={v1,v2,v12,v14}

Ĝ = ( {v1,…,v20}, Ế)

= k/α –3=6

k̂

k=10, α=10/9

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Algorithm : Shrinking step

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G’ = ( V’, E’)V’= {v3,v4,

…,v20}

k’= -2=4

k̂

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Algorithm : example

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G=(V,E) , k=8

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Algorithm : example

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Reduction step:Omit the crownH={b,c}I={u,v,w}

α =2

l = 2(1 – 1/ α)k =8

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Algorithm : example

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Reduction step:Omit the crownH={b,c}I={u,v,w}

α =2

l = 2(1 – 1/ α)k =8

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Algorithm : example

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Reduction step:Omit the crownH={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

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G=(V,E) , k=8

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Algorithm : example

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Reduction step:Omit the crownH={b,c}I={u,v,w}

α =1

l = 2(1 – 1/ α)k =0

Analysis: α-fidelity We show that the algorithm satisfies the properties of α-

fidelity kernelization.

1. The transformation from G to G’ is polynomial.

2. If (G, k/α) L then (G’, k’) LWe note that if there is a vertex cover of size k/α for G, there

is a cover of size = k/α -h for Ĝ, and there is a cover for G’ of size k’= - | 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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k̂k̂

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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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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Analysis: Kernel Size

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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 α-fidelity, for any α ≥ 1.