This document summarizes Venkatesh Raman's presentation on randomized techniques in fixed-parameter tractable algorithms. It discusses randomized algorithms for the undirected feedback vertex set problem and the k-path problem. For the k-path problem, it describes the color coding technique of assigning random colors to graph vertices in order to find colorful subgraphs. It analyzes the running time and success probability of algorithms for these problems.

1. Color Coding and Chromatic Coding
Venkatesh Raman
The Institute of Mathematical Sciences
3 March 2014
Venkatesh Raman (IMSc)
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2. Randomized Algorithms
Let Π ⊆ Σ∗ be a problem.
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3. Randomized Algorithms
Let Π ⊆ Σ∗ be a problem.
Algorithm
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4. Randomized Algorithms
Let Π ⊆ Σ∗ be a problem.
x ∈ Σ∗
Algorithm
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5. Randomized Algorithms
Let Π ⊆ Σ∗ be a problem.
x ∈ Σ∗
Algorithm
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6. Randomized Algorithms
Let Π ⊆ Σ∗ be a problem.
x ∈ Σ∗
Algorithm
x ∈ Π?
(Output correct answer with good probability)
Venkatesh Raman (IMSc)
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7. Randomized Algorithms
Let Π ⊆ Σ∗ be a problem.
x ∈ Σ∗
Algorithm
x ∈ Π?
(Output correct answer with good probability)
Here we are interested in randomized algorithms for parameterized
problems taking running time f (k)|x|O(1) with constant success
probability.
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8. Randomized Algorithm for UFVS
Definition (Feedback Vertex Set)
Let G = (V , E ) be an undirected graph. S ⊆ V is called feedback vertex
set if G S is a forest.
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9. Randomized Algorithm for UFVS
Definition (Feedback Vertex Set)
Let G = (V , E ) be an undirected graph. S ⊆ V is called feedback vertex
set if G S is a forest.
Undirected Feedback Vertex Set (UFVS)
Input:
Parameter:
Question:
An undirected graph G = (V , E ) and a positive integer k
k
Does there exists a feedback vertex set of size at most k
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10. Randomized Algorithm for UFVS
Do the following preprocessing rules
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11. Randomized Algorithm for UFVS
Do the following preprocessing rules
Delete degree ≤ 1 vertices.
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12. Randomized Algorithm for UFVS
Do the following preprocessing rules
Delete degree ≤ 1 vertices.
Short circuit degree 2 vertices.
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13. Randomized Algorithm for UFVS
Do the following preprocessing rules
Delete degree ≤ 1 vertices.
Short circuit degree 2 vertices.
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14. Randomized Algorithm for UFVS
Do the following preprocessing rules
Delete degree ≤ 1 vertices.
Short circuit degree 2 vertices.
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15. Randomized Algorithm for UFVS
Do the following preprocessing rules
Delete degree ≤ 1 vertices.
Short circuit degree 2 vertices.
Add vertex x to FVS, if x has self loop.
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16. Randomized Algorithm for UFVS
Do the following preprocessing rules
Delete degree ≤ 1 vertices.
Short circuit degree 2 vertices.
Add vertex x to FVS, if x has self loop.
Now we can assume every vertex in G has degree ≥ 3.
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17. Randomized Algorithm for UFVS
Claim : If G is graph with minimum degree ≥ 3, then number of edges
incident to any FVS F is ≥ E (G ) .
2
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18. Randomized Algorithm for UFVS
Claim : If G is graph with minimum degree ≥ 3, then number of edges
incident to any FVS F is ≥ E (G ) .
2
Proof:
F
H=G - F
;
Venkatesh Raman (IMSc)
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19. Randomized Algorithm for UFVS
Claim : If G is graph with minimum degree ≥ 3, then number of edges
incident to any FVS F is ≥ E (G ) .
2
Proof:
Let
V≤1 = set of degree ≤ 1 vertices in H,
V2 = set of degree 2 vertices in H, and
F
V≥3 = set of degree ≥ 3 vertices in H.
H=G - F
;
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20. Randomized Algorithm for UFVS
Claim : If G is graph with minimum degree ≥ 3, then number of edges
incident to any FVS F is ≥ E (G ) .
2
Proof:
Let
V≤1 = set of degree ≤ 1 vertices in H,
V2 = set of degree 2 vertices in H, and
F
V≥3 = set of degree ≥ 3 vertices in H.
Number of edges between H and F ,
E (H, F ) ≥ 2V≤1 + V2
H=G - F
;
Venkatesh Raman (IMSc)
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21. Randomized Algorithm for UFVS
Claim : If G is graph with minimum degree ≥ 3, then number of edges
incident to any FVS F is ≥ E (G ) .
2
Proof:
Let
V≤1 = set of degree ≤ 1 vertices in H,
V2 = set of degree 2 vertices in H, and
F
V≥3 = set of degree ≥ 3 vertices in H.
Number of edges between H and F ,
E (H, F ) ≥ 2V≤1 + V2
> V≤1 + V2 + V≥3 ( V≤1 > V≥3 )
H=G - F
;
Venkatesh Raman (IMSc)
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22. Randomized Algorithm for UFVS
Claim : If G is graph with minimum degree ≥ 3, then number of edges
incident to any FVS F is ≥ E (G ) .
2
Proof:
Let
V≤1 = set of degree ≤ 1 vertices in H,
V2 = set of degree 2 vertices in H, and
F
V≥3 = set of degree ≥ 3 vertices in H.
Number of edges between H and F ,
E (H, F ) ≥ 2V≤1 + V2
> V≤1 + V2 + V≥3 ( V≤1 > V≥3 )
= V (H)> E (H).
H=G - F
;
Venkatesh Raman (IMSc)
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23. Randomized Algorithm for UFVS
Algorithm
Initially we set S ← ∅. Now run the following steps k times.
1: Apply preprocessing rules and create an equivalent instance (G , k )
2: Pick an edge e, u.a.r (i.e with probability
1
E (G ) )
3: Choose an end point of e, u.a.r (i.e with probability 1 ) into S and
2
delete it from the graph.
Now if S is a FVS, output S, otherwise output No.
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24. Analysis
If the input instance is No instance Pr[Algorithm output No] = 1.
Let the input instance is Yes instance and F is a FVS od size ≤ k.
Pr[Choosing an edge incident to F in step 2] ≥
Pr[Chosen vertex is in F in step 3] ≥
Pr[S=F] ≥
1
2
1 1
·
2 2
1
4k
Now we repeat the algorithm 4k times.
1
Pr[Algorithm fails in all repetitions] ≤ 1 − 4k
1
Pr[Algorithm succeed at least once] ≥ 1 − e ≥
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4k
1
2
1
≤ e.
3 March 2014
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25. Color Coding
This technique is introduced by Alon et al. (1995)
This technique is used to solve constant treewidth k-sized subgraph
isomorphism problem.
Venkatesh Raman (IMSc)
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26. Color Coding
This technique is introduced by Alon et al. (1995)
This technique is used to solve constant treewidth k-sized subgraph
isomorphism problem.
We illustrate this technique by applying to k-path
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27. k-path
k-path
Input:
Parameter:
Question:
An undirected graph G and a positive integer k
k
Does there exists a path on k vertices in G
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28. Colorful path
v1
v5
v7
v2
v6
v3
v4
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29. Colorful path
v1
v5
v7
v2
v6
v3
v4
colorful path on 5 vertices in the graph G ?
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30. Algorithm for k-path
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31. Algorithm for k-path
Color each vertex of the input graph G , u.a.r using from the set of k
colors, [k]
Check whether there exists a colorful k-path in the colored graph and
output Yes/No accordingly
Venkatesh Raman (IMSc)
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32. Algorithm for k-path
Color each vertex of the input graph G , u.a.r using from the set of k
colors, [k]
Check whether there exists a colorful k-path in the colored graph and
output Yes/No accordingly
Running Time : Time to check colorful k-path × nO(1)
Venkatesh Raman (IMSc)
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33. Algorithm for k-path
Color each vertex of the input graph G , u.a.r using from the set of k
colors, [k]
Check whether there exists a colorful k-path in the colored graph and
output Yes/No accordingly
Running Time : Time to check colorful k-path × nO(1)
Probability of success
If (G , k) is a No instance, the probability of success is 1.
If (G , k) is an Yes instance, the probability of success is at least
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k!
.
kk
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34. DP for Checking colorful k-path
We introduce 2k · |V (G )| Boolean variables:
x(v , C ) = TRUE for some v ∈ V (G ) and C ⊆ [k]
There is path ends at v where each color in C appears
exactly once and no other color appears.
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35. DP for Checking colorful k-path
Clearly, x(v , {color (v )}) = TRUE. Recurrence for vertex v with color r :
x(u, C {r })
x(v , C ) =
uv ∈E (G )
If we know every x(v , C ) with |C | = i, then we can determine every
x(v , C ) with |C | = i + 1. All the values can be determined in time
O(2k · |E (G )|).
There is a colorful path ends at t ⇐⇒ x(t, [k]) = TRUE for some t.
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36. Analysis for k-path
The algorithm for k-path has
Running time 2k nO(1)
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37. Analysis for k-path
The algorithm for k-path has
Running time 2k nO(1)
Success probability
k!
at least k k if (G , k) is an Yes instance
1 if (G , k) is an No instance
Venkatesh Raman (IMSc)
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38. Analysis for k-path
The algorithm for k-path has
Running time 2k nO(1)
Success probability
k!
at least k k if (G , k) is an Yes instance
1 if (G , k) is an No instance
k
Now run the algorithm k (≤ e k ) times and output Yes if at least once we
k!
get an Yes answer, otherwise output No.
Venkatesh Raman (IMSc)
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39. Analysis for k-path
The algorithm for k-path has
Running time 2k nO(1)
Success probability
k!
at least k k if (G , k) is an Yes instance
1 if (G , k) is an No instance
k
Now run the algorithm k (≤ e k ) times and output Yes if at least once we
k!
get an Yes answer, otherwise output No.
Probability of failure ≤ 1 −
Venkatesh Raman (IMSc)
k
k! k /k!
kk
≤ e −1
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40. Analysis for k-path
The algorithm for k-path has
Running time 2k nO(1)
Success probability
k!
at least k k if (G , k) is an Yes instance
1 if (G , k) is an No instance
k
Now run the algorithm k (≤ e k ) times and output Yes if at least once we
k!
get an Yes answer, otherwise output No.
Probability of failure ≤ 1 −
k
k! k /k!
kk
≤ e −1
k-path can be solved in randomized (2e)k nO(1) time, with constant success probability.
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41. Derandomization
Suppose we have a list of colorings col1 , col2 , . . . , colm such that for
any S ⊆ V with |S| = k there exists i, coli is one-to-one on S, then
instead of random coloring we can use these list of colorings to get a
deterministic algorithm.
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42. Derandomization
Suppose we have a list of colorings col1 , col2 , . . . , colm such that for
any S ⊆ V with |S| = k there exists i, coli is one-to-one on S, then
instead of random coloring we can use these list of colorings to get a
deterministic algorithm.
Such a list of colorings is called an (n, k)-family of perfect hash
functions
Venkatesh Raman (IMSc)
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43. Derandomization
Suppose we have a list of colorings col1 , col2 , . . . , colm such that for
any S ⊆ V with |S| = k there exists i, coli is one-to-one on S, then
instead of random coloring we can use these list of colorings to get a
deterministic algorithm.
Such a list of colorings is called an (n, k)-family of perfect hash
functions
There exists an (n, k)-family of perfect hash functions of size
e k k O(log k) log2 n and can be constructed in time linear in the output
size
Venkatesh Raman (IMSc)
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44. Derandomization
Suppose we have a list of colorings col1 , col2 , . . . , colm such that for
any S ⊆ V with |S| = k there exists i, coli is one-to-one on S, then
instead of random coloring we can use these list of colorings to get a
deterministic algorithm.
Such a list of colorings is called an (n, k)-family of perfect hash
functions
There exists an (n, k)-family of perfect hash functions of size
e k k O(log k) log2 n and can be constructed in time linear in the output
size
k-path can be solved deterministically in (2e)k k O(log k) nO(1) time.
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45. Chromatic Coding
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46. Thank You
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