Parity Constrained Graph Separation M. S. Ramanujan Graph Cuts - - PowerPoint PPT Presentation

Parity Constrained Graph Separation

• M. S. Ramanujan

Graph Cuts Warsaw, 2013

April 9, 2013

(Warsaw) April 9, 2013

Outline of the Talk

Important separator framework Generalizing the framework Application: Even Multiway Cut Application: Subset Odd Cycle Transversal Conclusion

(Warsaw) April 9, 2013

Basic framework of applying Important Separators

Observe/prove that the solution contains an x-y separator of bounded size for some x and y (otherwise easily solved). Prove that the set of all x-y separators has a small dominating set. Prove that such a dominating set can be computing efficiently. Dominating set bounded by f (k) and time to compute it bounded by g(k)poly(n) = ⇒ f (k)kg(k)poly(n) algorithm.

(Warsaw) April 9, 2013

Basic framework of applying Important Separators

Observe/prove that the solution contains an x-y separator of bounded size for some x and y (otherwise easily solved). Prove that the set of all x-y separators has a small dominating set. Prove that such a dominating set can be computing efficiently. Dominating set bounded by f (k) and time to compute it bounded by g(k)poly(n) = ⇒ f (k)kg(k)poly(n) algorithm.

(Warsaw) April 9, 2013

Basic framework of applying Important Separators

Observe/prove that the solution contains an x-y separator of bounded size for some x and y (otherwise easily solved). Prove that the set of all x-y separators has a small dominating set. Prove that such a dominating set can be computing efficiently. Dominating set bounded by f (k) and time to compute it bounded by g(k)poly(n) = ⇒ f (k)kg(k)poly(n) algorithm.

(Warsaw) April 9, 2013

Basic framework of applying Important Separators

Observe/prove that the solution contains an x-y separator of bounded size for some x and y (otherwise easily solved). Prove that the set of all x-y separators has a small dominating set. Prove that such a dominating set can be computing efficiently. Dominating set bounded by f (k) and time to compute it bounded by g(k)poly(n) = ⇒ f (k)kg(k)poly(n) algorithm.

(Warsaw) April 9, 2013

Basic framework of applying Important Separators

Observe/prove that the solution contains an x-y separator of bounded size for some x and y (otherwise easily solved). Prove that the set of all x-y separators has a small dominating set. Prove that such a dominating set can be computing efficiently. Dominating set bounded by f (k) and time to compute it bounded by g(k)poly(n) = ⇒ f (k)kg(k)poly(n) algorithm.

(Warsaw) April 9, 2013

Framework of important separators: Multiway Cut

For any terminal t1 ∈ T, the solution contains a minimal t1-T \ t1 separator of size at most k (otherwise nothing to do!). (Marx, Chen et al.) There is a set of at most 4k t1-T \ t1 separators such that there is a solution containing one of them (small dominating set!). (Marx, Chen et al.) The dominating set can be computed in time 4knO(1). 4k2+kpoly(n) algorithm.

(Warsaw) April 9, 2013

Framework of important separators: Multiway Cut

For any terminal t1 ∈ T, the solution contains a minimal t1-T \ t1 separator of size at most k (otherwise nothing to do!). (Marx, Chen et al.) There is a set of at most 4k t1-T \ t1 separators such that there is a solution containing one of them (small dominating set!). (Marx, Chen et al.) The dominating set can be computed in time 4knO(1). 4k2+kpoly(n) algorithm.

(Warsaw) April 9, 2013

Framework of important separators: Multiway Cut

For any terminal t1 ∈ T, the solution contains a minimal t1-T \ t1 separator of size at most k (otherwise nothing to do!). (Marx, Chen et al.) There is a set of at most 4k t1-T \ t1 separators such that there is a solution containing one of them (small dominating set!). (Marx, Chen et al.) The dominating set can be computed in time 4knO(1). 4k2+kpoly(n) algorithm.

(Warsaw) April 9, 2013

Framework of important separators: Multiway Cut

For any terminal t1 ∈ T, the solution contains a minimal t1-T \ t1 separator of size at most k (otherwise nothing to do!). (Marx, Chen et al.) There is a set of at most 4k t1-T \ t1 separators such that there is a solution containing one of them (small dominating set!). (Marx, Chen et al.) The dominating set can be computed in time 4knO(1). 4k2+kpoly(n) algorithm.

(Warsaw) April 9, 2013

Framework of important separators: Multiway Cut

For any terminal t1 ∈ T, the solution contains a minimal t1-T \ t1 separator of size at most k (otherwise nothing to do!). (Marx, Chen et al.) There is a set of at most 4k t1-T \ t1 separators such that there is a solution containing one of them (small dominating set!). (Marx, Chen et al.) The dominating set can be computed in time 4knO(1). 4k2+kpoly(n) algorithm.

(Warsaw) April 9, 2013

Framework of important separators: Multiway Cut

X Y

S1 S2

Recall the formal definition of domination under which we have these bounds? S1 dominates S2 if the component containing X in G \ S1 is a strict superset of that containing X in G \ S2.

(Warsaw) April 9, 2013

Framework of important separators: Multiway Cut

X Y

S1 S2

Recall the formal definition of domination under which we have these bounds? S1 dominates S2 if the component containing X in G \ S1 is a strict superset of that containing X in G \ S2.

(Warsaw) April 9, 2013

Generalizing the framework

Does this definition of domination always suffice? What if we are only interested in X-Y separators where the component containing X is a tree? or a bipartite graph? or a planar graph and so on... What is the definition of domination now? Does the same one work? Clearly not!

(Warsaw) April 9, 2013

Generalizing the framework

Does this definition of domination always suffice? What if we are only interested in X-Y separators where the component containing X is a tree? or a bipartite graph? or a planar graph and so on... What is the definition of domination now? Does the same one work? Clearly not!

(Warsaw) April 9, 2013

Generalizing the framework

Does this definition of domination always suffice? What if we are only interested in X-Y separators where the component containing X is a tree? or a bipartite graph? or a planar graph and so on... What is the definition of domination now? Does the same one work? Clearly not!

(Warsaw) April 9, 2013

Generalizing the framework

Does this definition of domination always suffice? What if we are only interested in X-Y separators where the component containing X is a tree? or a bipartite graph? or a planar graph and so on... What is the definition of domination now? Does the same one work? Clearly not!

(Warsaw) April 9, 2013

Generalizing the framework

Does this definition of domination always suffice? What if we are only interested in X-Y separators where the component containing X is a tree? or a bipartite graph? or a planar graph and so on... What is the definition of domination now? Does the same one work? Clearly not!

(Warsaw) April 9, 2013

Generalizing the framework

Does this definition of domination always suffice? What if we are only interested in X-Y separators where the component containing X is a tree? or a bipartite graph? or a planar graph and so on... What is the definition of domination now? Does the same one work? Clearly not!

(Warsaw) April 9, 2013

Generalizing the framework

Does this definition of domination always suffice? What if we are only interested in X-Y separators where the component containing X is a tree? or a bipartite graph? or a planar graph and so on... What is the definition of domination now? Does the same one work? Clearly not!

(Warsaw) April 9, 2013

Generalizing the framework

We need a new definition of domination. Let us use the obvious one.

(Warsaw) April 9, 2013

Generalizing the framework

We need a new definition of domination. Let us use the obvious one.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

S1 well-dominates S2 if S1 dominates S2 in the usual sense AND the component containing X in G \ S1 has the required property. Is there a small (well-) dominating set now? Yes! Can we compute it efficiently? Yes!

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

S1 well-dominates S2 if S1 dominates S2 in the usual sense AND the component containing X in G \ S1 has the required property. Is there a small (well-) dominating set now? Yes! Can we compute it efficiently? Yes!

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

S1 well-dominates S2 if S1 dominates S2 in the usual sense AND the component containing X in G \ S1 has the required property. Is there a small (well-) dominating set now? Yes! Can we compute it efficiently? Yes!

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

S1 well-dominates S2 if S1 dominates S2 in the usual sense AND the component containing X in G \ S1 has the required property. Is there a small (well-) dominating set now? Yes! Can we compute it efficiently? Yes!

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

S1 well-dominates S2 if S1 dominates S2 in the usual sense AND the component containing X in G \ S1 has the required property. Is there a small (well-) dominating set now? Yes! Can we compute it efficiently? Yes!

(Warsaw) April 9, 2013

Generalizing the framework

Let us consider the set of X-Y separators of size at most k where the component containing X is bipartite. There is an algorithm which runs in time O∗(2O(k2)) and returns a well dominating set of size 2O(k2).

(Warsaw) April 9, 2013

Generalizing the framework

Let us consider the set of X-Y separators of size at most k where the component containing X is bipartite. There is an algorithm which runs in time O∗(2O(k2)) and returns a well dominating set of size 2O(k2).

(Warsaw) April 9, 2013

Generalizing the framework

X

S0

Compute the unique minimum X-Y separator closest to Y , S0.

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Generalizing the framework

X

S0 S1

Compute the unique minimum X-Y separator closest to S0, S1.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2

Compute the unique minimum X-Y separator closest to S1, S2.

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Generalizing the framework

X

S0 S1 S2 S3

Compute the unique minimum X-Y separator closest to S2, S3. Suppose the size of a minimum X-S3 separator is strictly greater than size of S3

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

Observation: Connectivity inside each strip is high. There is no Si+1-Si separator having the same size as the selected separators.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

Observation: Connectivity inside each strip is high. There is no Si+1-Si separator having the same size as the selected separators.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

Observation: Connectivity inside each strip is high. There is no Si+1-Si separator having the same size as the selected separators.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

Observation: Connectivity inside each strip is high. There is no Si+1-Si separator having the same size as the selected separators.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

Observation: Connectivity inside each strip is high. There is no Si+1-Si separator having the same size as the selected separators.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

Observation: Connectivity inside each strip is high. There is no Si+1-Si separator having the same size as the selected separators.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

Observation: Connectivity inside each strip is high. There is no Si+1-Si separator having the same size as the selected separators.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

We now test if these selected minimum separators satisfy the required property. In this case, testing if a subgraph is bipartite, poly time.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

We now test if these selected minimum separators satisfy the required property. In this case, testing if a subgraph is bipartite, poly time.

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Generalizing the framework

X

S0 S1 S2 S3

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

Testing S3.

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Generalizing the framework

X

S0 S1 S2 S3

S3 is found to be good, i.e. the component containing X in G \ S3 is bipartite.

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Generalizing the framework

X

S0 S1 S2 S3

Testing S2

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Generalizing the framework

X

S0 S1 S2 S3

S2 found to be good, i.e. the component containing X in G \ S3 is bipartite.

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Generalizing the framework

X

S0 S1 S2 S3

Testing S1

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Generalizing the framework

X

S0 S1 S2 S3

S1 found to be bad, i.e. the component containing X in G \ S3 is non-bipartite. What can we say about S0 then?

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

S1 found to be bad, i.e. the component containing X in G \ S3 is non-bipartite. What can we say about S0 then?

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Generalizing the framework

X

S0 S1 S2 S3

S0 is also bad.

(Warsaw) April 9, 2013

Generalizing the framework

X

S0 S1 S2 S3

We focus on the last good separator and the first bad separator. If all are good, then we set Y as the first bad separator and if all are bad, we set X as the last good separator.

(Warsaw) April 9, 2013

Generalizing the framework

X

S1 S2

Let us consider how the target separator J can interact with these separators.

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Generalizing the framework

X Y

S1 S2

Case 1: The target is dominated by S2. Then it is also well-dominated by S2. We take S2 into our well-dominating set.

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Generalizing the framework

X Y

S1 S2

Case 1: The target is dominated by S2. Then it is also well-dominated by S2. We take S2 into our well-dominating set.

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Generalizing the framework

X Y

S1 S2

Case 2: The target is dominated by S1 but itself dominates S2.

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Generalizing the framework

X Y

S1 S2

Recurse on the middle strip, i.e. make all other vertices undeletable. Progress? Connectivity increases.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

Recurse on the middle strip, i.e. make all other vertices undeletable. Progress? Connectivity increases.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

Recurse on the middle strip, i.e. make all other vertices undeletable. Progress? Connectivity increases.

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Generalizing the framework

X Y

S1 S2

J1 J2

Case 3: The target J is incomparable with one of them, say S2. Let the first piece be J1 and the second J2. Objective: Find a piece that ”locally” well-dominates J1 and a piece that ”locally” well-dominates J2 and put them together to get a separator well-dominating J.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

J1 J2

Case 3: The target J is incomparable with one of them, say S2. Let the first piece be J1 and the second J2. Objective: Find a piece that ”locally” well-dominates J1 and a piece that ”locally” well-dominates J2 and put them together to get a separator well-dominating J.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S1 S2

J1 J2

Case 3: The target J is incomparable with one of them, say S2. Let the first piece be J1 and the second J2. Objective: Find a piece that ”locally” well-dominates J1 and a piece that ”locally” well-dominates J2 and put them together to get a separator well-dominating J.

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Generalizing the framework

X Y

S2

J1 J2

Guess the vertices of S2 not reachable from X in G \ J. They are the red vertices.

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Generalizing the framework

X Y

S2

J1 J2

Guess the vertices of S2 not reachable from X in G \ J. They are the red vertices.

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Generalizing the framework

X Y

S2

J1

J1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J1 is bipartite. We recurse on this sub-instance with J1 as the new target. Progress? Size of the new target is sufficiently smaller.

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Generalizing the framework

X Y

S2

J1

J1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J1 is bipartite. We recurse on this sub-instance with J1 as the new target. Progress? Size of the new target is sufficiently smaller.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S2

J1

J1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J1 is bipartite. We recurse on this sub-instance with J1 as the new target. Progress? Size of the new target is sufficiently smaller.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S2

J1

J1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J1 is bipartite. We recurse on this sub-instance with J1 as the new target. Progress? Size of the new target is sufficiently smaller.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S2

J1

J1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J1 is bipartite. We recurse on this sub-instance with J1 as the new target. Progress? Size of the new target is sufficiently smaller.

(Warsaw) April 9, 2013

Generalizing the framework X Y

S2

J1

But, before recursing, we need to know the interaction of this subgraph with the rest of the graph through the green vertices. Since green vertices are bounded, we can guess this interaction and continue. In this case, it suffices to guess a bipartition of the green vertices and add (subdivided) edges between vertices of (same) different partitions.

(Warsaw) April 9, 2013

Generalizing the framework X Y

S2

J1

But, before recursing, we need to know the interaction of this subgraph with the rest of the graph through the green vertices. Since green vertices are bounded, we can guess this interaction and continue. In this case, it suffices to guess a bipartition of the green vertices and add (subdivided) edges between vertices of (same) different partitions.

(Warsaw) April 9, 2013

Generalizing the framework X Y

S2

J1

But, before recursing, we need to know the interaction of this subgraph with the rest of the graph through the green vertices. Since green vertices are bounded, we can guess this interaction and continue. In this case, it suffices to guess a bipartition of the green vertices and add (subdivided) edges between vertices of (same) different partitions.

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Generalizing the framework X Y

S2

J1 Q1

Suppose we find a set Q1 which well-dominates J1 in the (correct) sub-instance.

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Generalizing the framework

X Y

S2

J1 J2 Q1

We can use Q1 in the original instance to patch up J2. That is, we claim that J2 ∪ Q1 well-dominates J. Why?

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Generalizing the framework

X Y

S2

J1 J2 Q1

We can use Q1 in the original instance to patch up J2. That is, we claim that J2 ∪ Q1 well-dominates J. Why?

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S2

J1 J2 Q1

We can use Q1 in the original instance to patch up J2. That is, we claim that J2 ∪ Q1 well-dominates J. Why?

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S2

Suppose there is an odd cycle in the component containing X after deleting Q1 ∪ J2. Such an odd cycle must intersect the green vertices and contain subpaths which lie on the right side of S2.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S2

Suppose there is an odd cycle in the component containing X after deleting Q1 ∪ J2. Such an odd cycle must intersect the green vertices and contain subpaths which lie on the right side of S2.

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Generalizing the framework

X Y

S2

These subpaths can be replaced with an edge or subdivided edge which we have guessed while constructing our sub instance.

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Generalizing the framework

X Y

S2

Corresponds to an odd cycle in the sub instance disjoint from Q1 – not possible.

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Generalizing the framework

X Y

S2 J2

Q1

Now, we have a new X-Y separator which well-dominates J and we also know part of this separator.

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Generalizing the framework

X Y

S2 J2

Simply delete this part (Q1) and recurse on the resulting instance with J2 as the target. Any Q2 that well-dominates J2 in G \ Q1 can be patched up with Q1 to well-dominate J. Progress? Size of the new target is sufficiently smaller.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S2 J2

Simply delete this part (Q1) and recurse on the resulting instance with J2 as the target. Any Q2 that well-dominates J2 in G \ Q1 can be patched up with Q1 to well-dominate J. Progress? Size of the new target is sufficiently smaller.

(Warsaw) April 9, 2013

Generalizing the framework

X Y

S2 J2

Simply delete this part (Q1) and recurse on the resulting instance with J2 as the target. Any Q2 that well-dominates J2 in G \ Q1 can be patched up with Q1 to well-dominate J. Progress? Size of the new target is sufficiently smaller.

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Summary of the framework

X

S1 S2

Find the last good separator and the first bad separator. Add the first good separator to the set. In one branch, recurse on the middle strip.

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Summary of the framework

X

S1 S2

Find the last good separator and the first bad separator. Add the first good separator to the set. In one branch, recurse on the middle strip.

(Warsaw) April 9, 2013

Summary of the framework

X

S1 S2

Find the last good separator and the first bad separator. Add the first good separator to the set. In one branch, recurse on the middle strip.

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Summary of the framework

X

S1 S2

In the next set of branches, build sub-instance(s) on the graph lying to the ”left” of the good separator and recurse. For each set computed in this branch, delete it and recurse on the resulting instance. Do the same with the first bad separator as well. In each branch, either budget drops or connectivity increases. In all, 2O(k) branches = ⇒ 2O(k2) bound.

(Warsaw) April 9, 2013

Summary of the framework

X

S1 S2

In the next set of branches, build sub-instance(s) on the graph lying to the ”left” of the good separator and recurse. For each set computed in this branch, delete it and recurse on the resulting instance. Do the same with the first bad separator as well. In each branch, either budget drops or connectivity increases. In all, 2O(k) branches = ⇒ 2O(k2) bound.

(Warsaw) April 9, 2013

Summary of the framework

X

S1 S2

In the next set of branches, build sub-instance(s) on the graph lying to the ”left” of the good separator and recurse. For each set computed in this branch, delete it and recurse on the resulting instance. Do the same with the first bad separator as well. In each branch, either budget drops or connectivity increases. In all, 2O(k) branches = ⇒ 2O(k2) bound.

(Warsaw) April 9, 2013

Summary of the framework

X

S1 S2

In the next set of branches, build sub-instance(s) on the graph lying to the ”left” of the good separator and recurse. For each set computed in this branch, delete it and recurse on the resulting instance. Do the same with the first bad separator as well. In each branch, either budget drops or connectivity increases. In all, 2O(k) branches = ⇒ 2O(k2) bound.

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Parity Multiway Cut

Parity Multiway Cut

Input: A Graph G = (V , E), set Te of even terminals, set To of

• dd terminals, a positive integer k.

Parameter: k. Question: Does G have a vertex set S of size at most k intersecting every odd (even) path from t ∈ To (t ∈ Te) to T \ t? Even Multiway Cut if To = ∅, and Odd Multiway Cut if Te = ∅ and Multiway Cut if Te = To.

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Parity Multiway Cut

Special case Odd/Even Multiway Cut NP complete even for 2 terminals.

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Even Multiway Cut

Let us apply the framework to solve the Even Multiway Cut problem. We may assume that number of terminals is at most 6k.

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Even Multiway Cut

Let us apply the framework to solve the Even Multiway Cut problem. We may assume that number of terminals is at most 6k.

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Even Multiway Cut

S

Let S be a hypothetical solution. Observation: A component of G \ S contains at most 2 terminals.

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Even Multiway Cut

S

Let S be a hypothetical solution. Observation: A component of G \ S contains at most 2 terminals.

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Even Multiway Cut

S

Guess the partition of the terminals.

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Even Multiway Cut

S

Guess the partition of the terminals. Fix one set in the partition (the red terminals).

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Even Multiway Cut

S

S contains a minimal red-blue separator of size at most k.

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Even Multiway Cut

S

Clearly, the old important separator framework doesn’t work. There could be even paths between red terminals disjoint from the separator.

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Even Multiway Cut

S

Clearly, the old important separator framework doesn’t work. There could be even paths between red terminals disjoint from the separator.

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Even Multiway Cut

S

Solution has some vertices whose sole purpose is to hit all these red even paths

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Even Multiway Cut

S

Our Interest: red-blue separators of size at most k such that the component containing red vertices has an even mwc of size at most some l. Straightforward to see that it is sufficient to look at a well-dominating set of such separators.

(Warsaw) April 9, 2013

Even Multiway Cut

S

Our Interest: red-blue separators of size at most k such that the component containing red vertices has an even mwc of size at most some l. Straightforward to see that it is sufficient to look at a well-dominating set of such separators.

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Even Multiway Cut

S

We can apply the framework.

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Even Multiway Cut

S0 S1 t1 t2 S2 S3

We compute the sequence of minimum important separators as before. Find the good/bad separators. How? Need to test if there is a small even mwc for two terminals. This can be done by a separate FPT algorithm.

(Warsaw) April 9, 2013

Even Multiway Cut

S0 S1 t1 t2 S2 S3

We compute the sequence of minimum important separators as before. Find the good/bad separators. How? Need to test if there is a small even mwc for two terminals. This can be done by a separate FPT algorithm.

(Warsaw) April 9, 2013

Even Multiway Cut

S0 S1 t1 t2 S2 S3

We compute the sequence of minimum important separators as before. Find the good/bad separators. How? Need to test if there is a small even mwc for two terminals. This can be done by a separate FPT algorithm.

(Warsaw) April 9, 2013

Even Multiway Cut

S0 S1 t1 t2 S2 S3

We compute the sequence of minimum important separators as before. Find the good/bad separators. How? Need to test if there is a small even mwc for two terminals. This can be done by a separate FPT algorithm.

(Warsaw) April 9, 2013

Even Multiway Cut

S0 S1 t1 t2 S2 S3

We compute the sequence of minimum important separators as before. Find the good/bad separators. How? Need to test if there is a small even mwc for two terminals. This can be done by a separate FPT algorithm.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Again, pick the last good separator and the first bad separator. Add the good separator to our set, recursion on the middle strip is clear. Let us look at the last case.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Again, pick the last good separator and the first bad separator. Add the good separator to our set, recursion on the middle strip is clear. Let us look at the last case.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Again, pick the last good separator and the first bad separator. Add the good separator to our set, recursion on the middle strip is clear. Let us look at the last case.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Again, let J1 be the part of the target separator which lies in the green part. Guess the vertices of S2 separated from t1, t2 by J1 (red vertices).

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Build the sub instance by guessing the interaction of the green vertices with the rest of the graph. Again, it is sufficient to guess a bipartition on the green vertices.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2

Recursively, compute a set Q1 well dominating J1 in this sub instance.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2

Recursively, compute a set Q1 well dominating J1 in this sub instance.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Prove that Q1 can replace J1 in J, that is, Q1 ∪ J2 well-dominates J1 ∪ J2.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Prove that Q1 can replace J1 in J, that is, Q1 ∪ J2 well-dominates J1 ∪ J2.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Delete Q1 and recurse on the resulting instance.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Delete Q1 and recurse on the resulting instance.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Again, 2O(k)-way branching with the same progress in the measure – 2O(k2) bound. Therefore, 2O(k3) algorithm for Even Multiway Cut.

(Warsaw) April 9, 2013

Even Multiway Cut

t1 t2 S2 S1

Again, 2O(k)-way branching with the same progress in the measure – 2O(k2) bound. Therefore, 2O(k3) algorithm for Even Multiway Cut.

(Warsaw) April 9, 2013

Subset OCT

Subset OCT

Input: A Graph G = (V , E), set T of terminals, a positive integer k. Parameter: k. Question: Does G have a vertex set S of size at most k hitting all odd cycles with a vertex in T?

(Warsaw) April 9, 2013

Subset OCT

S

Assume that we are given an approximate solution (blue vertices) and we want a solution disjoint from it. Fix such a hypothetical solution S.

(Warsaw) April 9, 2013

Subset OCT

S

Guess a partition of the blue vertices into blocks. If all the blue vertices go to the same block, then it can be solved by a separate algorithm. Pick a lowest block (red vertices).

(Warsaw) April 9, 2013

Subset OCT

S

The solution (plus one other vertex) contains a red-blue separator of size at most k + 1

(Warsaw) April 9, 2013

Subset OCT

S

The component containing the red vertices after removing the separator might still contain a subset odd cycle.

(Warsaw) April 9, 2013

Subset OCT

S

There are some vertices in the solution whose sole job is to cover these subset odd cycles. Our Interest: red-blue separators of size at most k + 1 such that the component containing the red vertices has a subset OCT of size at most some l.

(Warsaw) April 9, 2013

Subset OCT

S

There are some vertices in the solution whose sole job is to cover these subset odd cycles. Our Interest: red-blue separators of size at most k + 1 such that the component containing the red vertices has a subset OCT of size at most some l.

(Warsaw) April 9, 2013

Subset OCT

S

Once again the framework can be applied. We get a worse bound here for the dominating set – 2O(k3 log k).

(Warsaw) April 9, 2013

Notes on Subset OCT

While applying the framework for Subset OCT, testing of the minimum separators reduces to solving a special case of the problem. But even this special case is not special enough. So we apply the same framework again. Repeated applications get us to a sufficiently special case which can be solved by an independent algorithm.

(Warsaw) April 9, 2013

Conclusion

A way to overload important separators with other properties and still have small dominating sets. The 2 main components – testing if the minimum separators are good/bad and the existence of a small torso. The general problem can be reduced to solving very special cases, eg. 2-terminal even mwc, Subset OCT where all the blue vertices go to the same block after deleting the solution. Can be used for more general shadow removal procedures, eg. for Subset OCT.

(Warsaw) April 9, 2013

Conclusion

A way to overload important separators with other properties and still have small dominating sets. The 2 main components – testing if the minimum separators are good/bad and the existence of a small torso. The general problem can be reduced to solving very special cases, eg. 2-terminal even mwc, Subset OCT where all the blue vertices go to the same block after deleting the solution. Can be used for more general shadow removal procedures, eg. for Subset OCT.

(Warsaw) April 9, 2013

Conclusion

A way to overload important separators with other properties and still have small dominating sets. The 2 main components – testing if the minimum separators are good/bad and the existence of a small torso. The general problem can be reduced to solving very special cases, eg. 2-terminal even mwc, Subset OCT where all the blue vertices go to the same block after deleting the solution. Can be used for more general shadow removal procedures, eg. for Subset OCT.

(Warsaw) April 9, 2013

Conclusion

A way to overload important separators with other properties and still have small dominating sets. The 2 main components – testing if the minimum separators are good/bad and the existence of a small torso. The general problem can be reduced to solving very special cases, eg. 2-terminal even mwc, Subset OCT where all the blue vertices go to the same block after deleting the solution. Can be used for more general shadow removal procedures, eg. for Subset OCT.

(Warsaw) April 9, 2013

Conclusion

A way to overload important separators with other properties and still have small dominating sets. The 2 main components – testing if the minimum separators are good/bad and the existence of a small torso. The general problem can be reduced to solving very special cases, eg. 2-terminal even mwc, Subset OCT where all the blue vertices go to the same block after deleting the solution. Can be used for more general shadow removal procedures, eg. for Subset OCT.

(Warsaw) April 9, 2013

Final Slide

Thank You!

(Warsaw) April 9, 2013