Parameterized Complexity VS

Approximation Algorithms

Boaz Ophir

Winter 2012-13

Intro • NP-hard optimization problems - there is no

polynomial-time algorithm that finds the exact value of the optimum.

• Our options (until now):

1. Exact Algorithms :

• Run time is not polynomial.

• Parameterized Complexity .

2. Approximation Algorithms :

• Polynomial time.

• Solution is not optimal but is worse-case bound.

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Motivation • By combining ideas from:

1. Parameterized Complexity

AND

2. Approximation Theory

we may be able to tackle problems that are intractable in each theory by itself!

• Our goal – FPT Approximation Algorithms

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Talk Outline 1. Motivation.

2. Concepts in Approximation Theory.

3. Approximation with Instance Parameters.

1. Example - Vertex Cover.

2. Example - Vertex Coloring.

4. Other forms of Parameterization.

1. Parameterization by Cost.

2. Performance Functions.

3. Parameterization by Quality of Approximation.

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Approximation Algorithms 101 • Definition: an NP optimization problem is a 4-tuple

𝐼, 𝑠𝑜𝑙, 𝑐𝑜𝑠𝑡, 𝑔𝑜𝑎𝑙 :

o 𝐼 – the set of all instances.

o For an instance x ∈ 𝐼, 𝑠𝑜𝑙(𝑥) is the set of feasible solutions of 𝑥.

• The length of each solution 𝑦 ∈ 𝑠𝑜𝑙(𝑥) is polynomial in the size of the input, |𝑥|.

• It can be decided in poly time whether 𝑦 ∈ 𝑠𝑜𝑙(𝑥) holds for a given 𝑦, 𝑥.

o Given 𝑥 and 𝑦 ∈ 𝑠𝑜𝑙(𝑥), cost 𝑥, 𝑦 is a poly-time computable positive integer.

o 𝑔𝑜𝑎𝑙 ∈ 𝑚𝑖𝑛,𝑚𝑎𝑥 .

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Approximation Algorithms 101(2)

• Goal:

o Find a feasible solution 𝑦 that achieves the best objective: 𝑐𝑜𝑠𝑡 𝑥, 𝑦 = 𝑔𝑜𝑎𝑙{𝑐𝑜𝑠𝑡 𝑥, 𝑦′ ∶ 𝑦′ ∈ 𝑠𝑜𝑙 𝑥 }

o 𝑜𝑝𝑡 𝑥 - the cost of the optimum solution.

o The performance ratio 𝑐(𝑦) is defined as:

𝑐 𝑦 =

𝑐𝑜𝑠𝑡(𝑥, 𝑦)

𝑜𝑝𝑡(𝑥) 𝑖𝑓 𝑔𝑜𝑎𝑙 𝑖𝑠 𝑚𝑖𝑛

𝑜𝑝𝑡(𝑥)

𝑐𝑜𝑠𝑡(𝑥, 𝑦) 𝑖𝑓 𝑔𝑜𝑎𝑙 𝑖𝑠 𝑚𝑎𝑥

o A c-approximation algorithm produces a solution with performance ration at most 𝑐.

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Approximation Algorithms 101(3)

• Approximation algorithm race:

o A constant ratio approx. alg. is published

o An improved alg. with better ratio is published.

o Repeat … (sometimes endlessly)

• A problem admits a Poly-time Approximation Scheme (PTAS) if for every 𝜀 > 0 there is a poly-time (1 + ε)-approximation algorithm.

o PTAS – run-time is of the form |𝑥|𝑓(1/𝜀)

o Efficient PTAS(EPTAS)-run-time of the form 𝑓(1/𝜀)|𝑥|𝑂(1)

o Fully PTAS(FPTAS)- run-time of the form (1/𝜀)𝑂(1)|𝑥|𝑂(1)

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Approximation with Instance

Parameters • Definition: An FPT c-Approximation Algorithm with

parameter κ : o Input: 𝑥 ∈ 𝐼

o Output: a c-approximate solution

o Run-time: 𝑓 𝜅 𝑥 |𝑥|𝑂(1)

• Definition: An FPT Approximation Scheme (FPT-AS) with parameter κ : o Input: 𝑥 ∈ 𝐼 and 𝜀 > 0

o Output: an (1 + ε)-approximate solution

o Run-time: 𝑓 𝜀, 𝜅 𝑥 |𝑥|𝑂(1)

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Approximation with Instance

Parameters (2)

Categories of parameterizations (what is κ ?):

1. Measure Parameters – an obvious measure of the problem instance.

2. Structural Parameters – a structural property of the input that describes how complicated the input is. o Maximum degree

o Diameter

o Tree width

o Genus

o Distance from a class

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Example 1 – Vertex Cover • Vertex Cover: Cover all edges with as few vertices as

possible. o VC is FPT.

o VC has a trivial 2-approximation algorithm.

o VC does not admit PTAS.

• Partial VC: Cover as many edges as possible with k vertices. o W[1]-hard with parameter k.

o Simple greedy algorithm gives 1.582-approximation.

o Does not admit PTAS.

o Does admit FPT-AS with parameter k !!!

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Example 1 – Vertex Cover (2)

• Proof:

o Denote 𝐷 ≔ 2 𝑘2

/𝜀.

o Mark vertices as 𝑣1 …𝑣𝑛 ordered by non-increasing degree 𝑑 𝑣1 ≥ 𝑑 𝑣2 ≥ ⋯ ≥ 𝑑(𝑣𝑛).

• Case 1: 𝑑(𝑣1) ≥ 𝐷 o Output 𝑣1, . . 𝑣𝑘 .

o These 𝑘 vertices cover at least 𝑑 𝑣𝑖 − 𝑘2

𝑘𝑖=1 edges.

o The optimum is at most 𝑑 𝑣𝑖𝑘𝑖=1 .

𝑑 𝑣𝑖 − 𝑘2

𝑘𝑖=1

𝑑 𝑣𝑖𝑘𝑖=1

≥ 1 −

𝑘2

𝑑 𝑣1≥ 1 −

𝑘2

𝐷= 1 −

𝜀

2≥

1

1 + 𝜀

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Example 1 – Vertex Cover (3)

• Case 2: 𝑑 𝑣1 < 𝐷 o The optimum is at most 𝑘𝐷. o Determine the exact cover in FPT-time: For 𝑙 = 1. . 𝑘𝐷 check whether it is possible to cover 𝑙 edges

with 𝑘 vertices. Method used – Color Coding.

• Color the edges randomly with 𝑙 colors. • Try to find solution where the 𝑙 edges covered have distinct

colors. • For a particular coloring consider every partition 𝑃 = 𝑃1, . . 𝑃𝑘

of the 𝑙 colors into k classes (there are at most 𝑘𝑙 partitions). • Find vertices 𝑣1 …𝑣𝑘 so that for each color 𝑐 in partition 𝑃𝑖,

vertex 𝑣𝑖 covers at least one edge with color 𝑐. This is done in poly-time.

• Repeat for different colorings or de-randomize. □

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Example 2– Vertex Coloring • Problem: given a graph 𝐺, find a coloring using the minimal

number of colors so that adjacent vertices have different colors.

• Denote 𝜒(𝐺) the chromatic number of 𝐺.

• Hard even for planar graphs ! o Easy to check if graph is 2-colorable.

o NP-complete to decide whether graph is 3-colorable.

o 4-Color theorem – 4 colors are sufficient and coloring can be found in poly-time.

=> A 4

3-approximate coloring can be found in poly-time.

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Example 2– Vertex Coloring (2)

• What about graphs that are almost planar?

o Bounded genus graphs.

o Planar+𝑘𝑣 graphs.

• Theorem: Vertex Coloring has an FPT 2-approx. if the parameter is the genus of the graph.

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Example 2– Vertex Coloring (3)

• Theorem: Vertex Coloring has an FPT 7

3-approx. if the graph

is planar+𝑘𝑣.

• Proof

o 𝑋 = {𝑣1, … 𝑣𝑘} (can be found in 𝑂(𝑓 𝑘 𝑛2)).

o Look at 𝐺 𝑋 - use brute force - try all 𝑘𝑘 colorings to find 𝜒(𝐺 𝑋 ).

o Look at 𝐺\X - Find 4

3-approximation of 𝜒 𝐺\X .

o Combine to color the whole graph:

𝑐 = 𝜒 𝐺 𝑋 + 4

3∙ 𝜒 𝐺\X ≤

7

3∙ 𝜒 𝐺

(the worst case is 𝜒 𝐺 = 𝜒 𝐺\X = 𝜒 𝐺 𝑋 = 3)

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Parameterization by Cost • The most obvious parameter for optimization:

The Optimum Cost • But – this is tricky

o The optimum cost is hard to determine o Instead we assume the input contains a parameter 𝑘 – the cost that

should be reached.

• Definition: FPT-approximation alg. with performance ratio 𝑐 for optimization problem 𝑋 = (𝐼, 𝑠𝑜𝑙, 𝑐𝑜𝑠𝑡, 𝑔𝑜𝑎𝑙).

Given input (𝑥, 𝑘) satisfying 𝑜𝑝𝑡 𝑥 ≤ 𝑘, 𝑔𝑜𝑎𝑙 = 𝑚𝑖𝑛

𝑜𝑝𝑡 𝑥 ≥ 𝑘, 𝑔𝑜𝑎𝑙 = 𝑚𝑎𝑥

the algorithm computes a solution 𝑦 ∈ 𝑠𝑜𝑙(𝑥) in time 𝑓(𝑘)|𝑥|𝑂(1) such that:

𝑐𝑜𝑠𝑡 𝑥, 𝑦 ≤ 𝑘 ∙ 𝑐, 𝑔𝑜𝑎𝑙 = 𝑚𝑖𝑛

𝑐𝑜𝑠𝑡 𝑥, 𝑦 ≥ 𝑘/𝑐, 𝑔𝑜𝑎𝑙 = 𝑚𝑎𝑥

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Performance Functions • Typically approximation algorithms output a solution

that differs from the optimum by (at most) a constant factor.

• The alg. performs equally well regardless of optimum size.

• More precise analysis is achieved by bounding by a function of the optimum.

• Let 𝑙: ℕ → ℕ be a non-decreasing function.

• We change the FPT-approximation alg. definition:

𝑐𝑜𝑠𝑡 𝑥, 𝑦 ≤ 𝑘 ∙ 𝑙(𝑘), 𝑔𝑜𝑎𝑙 = 𝑚𝑖𝑛

𝑐𝑜𝑠𝑡 𝑥, 𝑦 ≥ 𝑘/𝑙(𝑘), 𝑔𝑜𝑎𝑙 = 𝑚𝑎𝑥

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Performance Functions (cntd.)

• Positive result:

o Theorem: If maximization problem 𝑋 has an FPT-approximation alg. with performance ration function 𝑙(𝑘) then there is a poly-time 𝑙′(𝑘)-approximation for 𝑋.

o Does not hold for minimization problems.

• Negative result:

o For certain problems there is no FPT-approximation algorithm for any performance ratio function.

o Example: Minimum Independent Dominating Set

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Parameterization by Quality of

Approximation

• Efficient PTAS(EPTAS)- poly-time approximation scheme where run-time is of the form 𝑓(1/𝜀)|𝑥|𝑂(1).

• Proposition: If an optimization problem 𝑋 admits an EPTAS then the parameterization of 𝑋 is FPT, where the parameter is the cost.

• We can use this to prove that certain problems do not admit EPTAS.

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Parameterization by Quality of

Approximation (2)

• Proof: o Assume – we have an algorithm that gives an (1 + ε)-

approximate solution in time 𝑓(1/𝜀)|𝑥|𝑂(1).

o Set 𝜀 ≔ 1

2𝑘 .

o Run the EPTAS – runtime is 𝑓(2𝑘)|𝑥|𝑂(1). o If the (exact) optimum is at most 𝑘, then the cost of the

approximate solution is at most

1 + 𝜀 𝑘 = 𝑘 +1

2< 𝑘 + 1.

Cost is an integer => the cost of the approximate solution is at most 𝑘. o If the (exact) optimum is greater than 𝑘 then so is the

approximate solution. o So – if the approximate solution is at most 𝑘 so is the exact

solution.

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Conclusions • Parameterized Complexity and Approximation

Algorithms can benefit from each other.

• We saw a few examples – there are many other examples of FPT-approximability.

• There are more connections:

o Kernelization – lower bounds on the approximation ratio translate to lower bounds on kernel size.

o Iterative compression

o Counting problems

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Fin

Thank you

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