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 ∗ (2 O ( k 2 ) ) and returns a well dominating set of size 2 O ( k 2 ) . (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 ∗ (2 O ( k 2 ) ) and returns a well dominating set of size 2 O ( k 2 ) . (Warsaw) April 9, 2013
Generalizing the framework X S 0 Compute the unique minimum X - Y separator closest to Y , S 0 . (Warsaw) April 9, 2013
Generalizing the framework X S 1 S 0 Compute the unique minimum X - Y separator closest to S 0 , S 1 . (Warsaw) April 9, 2013
Generalizing the framework X S 2 S 1 S 0 Compute the unique minimum X - Y separator closest to S 1 , S 2 . (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Compute the unique minimum X - Y separator closest to S 2 , S 3 . Suppose the size of a minimum X - S 3 separator is strictly greater than size of S 3 (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Observation: Connectivity inside each strip is high. There is no S i +1 - S i separator having the same size as the selected separators. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Observation: Connectivity inside each strip is high. There is no S i +1 - S i separator having the same size as the selected separators. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Observation: Connectivity inside each strip is high. There is no S i +1 - S i separator having the same size as the selected separators. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Observation: Connectivity inside each strip is high. There is no S i +1 - S i separator having the same size as the selected separators. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Observation: Connectivity inside each strip is high. There is no S i +1 - S i separator having the same size as the selected separators. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Observation: Connectivity inside each strip is high. There is no S i +1 - S i separator having the same size as the selected separators. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Observation: Connectivity inside each strip is high. There is no S i +1 - S i separator having the same size as the selected separators. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 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 S 3 S 2 S 1 S 0 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 S 3 S 2 S 1 S 0 (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Testing S 3 . (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 S 3 is found to be good, i.e. the component containing X in G \ S 3 is bipartite. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Testing S 2 (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 S 2 found to be good, i.e. the component containing X in G \ S 3 is bipartite. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 Testing S 1 (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 0 S 1 S 1 found to be bad, i.e. the component containing X in G \ S 3 is non-bipartite. What can we say about S 0 then? (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 0 S 1 S 1 found to be bad, i.e. the component containing X in G \ S 3 is non-bipartite. What can we say about S 0 then? (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 S 0 is also bad. (Warsaw) April 9, 2013
Generalizing the framework X S 3 S 2 S 1 S 0 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 S 2 S 1 Let us consider how the target separator J can interact with these separators. (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 S 1 Case 1: The target is dominated by S 2 . Then it is also well-dominated by S 2 . We take S 2 into our well-dominating set. (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 S 1 Case 1: The target is dominated by S 2 . Then it is also well-dominated by S 2 . We take S 2 into our well-dominating set. (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 S 1 Case 2: The target is dominated by S 1 but itself dominates S 2 . (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 S 1 Recurse on the middle strip, i.e. make all other vertices undeletable. Progress? Connectivity increases. (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 S 1 Recurse on the middle strip, i.e. make all other vertices undeletable. Progress? Connectivity increases. (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 S 1 Recurse on the middle strip, i.e. make all other vertices undeletable. Progress? Connectivity increases. (Warsaw) April 9, 2013
Generalizing the framework J 1 Y X J 2 S 2 S 1 Case 3: The target J is incomparable with one of them, say S 2 . Let the first piece be J 1 and the second J 2 . Objective: Find a piece that ”locally” well-dominates J 1 and a piece that ”locally” well-dominates J 2 and put them together to get a separator well-dominating J . (Warsaw) April 9, 2013
Generalizing the framework J 1 Y X J 2 S 2 S 1 Case 3: The target J is incomparable with one of them, say S 2 . Let the first piece be J 1 and the second J 2 . Objective: Find a piece that ”locally” well-dominates J 1 and a piece that ”locally” well-dominates J 2 and put them together to get a separator well-dominating J . (Warsaw) April 9, 2013
Generalizing the framework J 1 Y X J 2 S 2 S 1 Case 3: The target J is incomparable with one of them, say S 2 . Let the first piece be J 1 and the second J 2 . Objective: Find a piece that ”locally” well-dominates J 1 and a piece that ”locally” well-dominates J 2 and put them together to get a separator well-dominating J . (Warsaw) April 9, 2013
Generalizing the framework J 1 Y X J 2 S 2 Guess the vertices of S 2 not reachable from X in G \ J . They are the red vertices. (Warsaw) April 9, 2013
Generalizing the framework J 1 Y X J 2 S 2 Guess the vertices of S 2 not reachable from X in G \ J . They are the red vertices. (Warsaw) April 9, 2013
Generalizing the framework J 1 X Y S 2 J 1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J 1 is bipartite. We recurse on this sub-instance with J 1 as the new target. Progress? Size of the new target is sufficiently smaller. (Warsaw) April 9, 2013
Generalizing the framework J 1 X Y S 2 J 1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J 1 is bipartite. We recurse on this sub-instance with J 1 as the new target. Progress? Size of the new target is sufficiently smaller. (Warsaw) April 9, 2013
Generalizing the framework J 1 X Y S 2 J 1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J 1 is bipartite. We recurse on this sub-instance with J 1 as the new target. Progress? Size of the new target is sufficiently smaller. (Warsaw) April 9, 2013
Generalizing the framework J 1 X Y S 2 J 1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J 1 is bipartite. We recurse on this sub-instance with J 1 as the new target. Progress? Size of the new target is sufficiently smaller. (Warsaw) April 9, 2013
Generalizing the framework J 1 X Y S 2 J 1 intersects all paths from X to red vertices in this subgraph. Also the component containing X after removing J 1 is bipartite. We recurse on this sub-instance with J 1 as the new target. Progress? Size of the new target is sufficiently smaller. (Warsaw) April 9, 2013
Generalizing the framework J 1 X Y S 2 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 J 1 X Y S 2 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 J 1 X Y S 2 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 J 1 Q 1 X Y S 2 Suppose we find a set Q 1 which well-dominates J 1 in the (correct) sub-instance. (Warsaw) April 9, 2013
Generalizing the framework J 1 Q 1 Y X J 2 S 2 We can use Q 1 in the original instance to patch up J 2 . That is, we claim that J 2 ∪ Q 1 well-dominates J . Why? (Warsaw) April 9, 2013
Generalizing the framework J 1 Q 1 Y X J 2 S 2 We can use Q 1 in the original instance to patch up J 2 . That is, we claim that J 2 ∪ Q 1 well-dominates J . Why? (Warsaw) April 9, 2013
Generalizing the framework J 1 Q 1 Y X J 2 S 2 We can use Q 1 in the original instance to patch up J 2 . That is, we claim that J 2 ∪ Q 1 well-dominates J . Why? (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 Suppose there is an odd cycle in the component containing X after deleting Q 1 ∪ J 2 . Such an odd cycle must intersect the green vertices and contain subpaths which lie on the right side of S 2 . (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 Suppose there is an odd cycle in the component containing X after deleting Q 1 ∪ J 2 . Such an odd cycle must intersect the green vertices and contain subpaths which lie on the right side of S 2 . (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 These subpaths can be replaced with an edge or subdivided edge which we have guessed while constructing our sub instance. (Warsaw) April 9, 2013
Generalizing the framework Y X S 2 Corresponds to an odd cycle in the sub instance disjoint from Q 1 – not possible. (Warsaw) April 9, 2013
Generalizing the framework Q 1 Y X J 2 S 2 Now, we have a new X - Y separator which well-dominates J and we also know part of this separator. (Warsaw) April 9, 2013
Generalizing the framework Y X J 2 S 2 Simply delete this part ( Q 1 ) and recurse on the resulting instance with J 2 as the target. Any Q 2 that well-dominates J 2 in G \ Q 1 can be patched up with Q 1 to well-dominate J . Progress? Size of the new target is sufficiently smaller. (Warsaw) April 9, 2013
Generalizing the framework Y X J 2 S 2 Simply delete this part ( Q 1 ) and recurse on the resulting instance with J 2 as the target. Any Q 2 that well-dominates J 2 in G \ Q 1 can be patched up with Q 1 to well-dominate J . Progress? Size of the new target is sufficiently smaller. (Warsaw) April 9, 2013
Generalizing the framework Y X J 2 S 2 Simply delete this part ( Q 1 ) and recurse on the resulting instance with J 2 as the target. Any Q 2 that well-dominates J 2 in G \ Q 1 can be patched up with Q 1 to well-dominate J . Progress? Size of the new target is sufficiently smaller. (Warsaw) April 9, 2013
Summary of the framework X S 2 S 1 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 S 2 S 1 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 S 2 S 1 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 S 2 S 1 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, 2 O ( k ) branches = ⇒ 2 O ( k 2 ) bound. (Warsaw) April 9, 2013
Summary of the framework X S 2 S 1 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, 2 O ( k ) branches = ⇒ 2 O ( k 2 ) bound. (Warsaw) April 9, 2013
Summary of the framework X S 2 S 1 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, 2 O ( k ) branches = ⇒ 2 O ( k 2 ) bound. (Warsaw) April 9, 2013
Summary of the framework X S 2 S 1 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, 2 O ( k ) branches = ⇒ 2 O ( k 2 ) bound. (Warsaw) April 9, 2013
Parity Multiway Cut Parity Multiway Cut Input: A Graph G = ( V , E ) , set T e of even terminals, set T o of odd 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 ∈ T o (t ∈ T e ) to T \ t? Even Multiway Cut if T o = ∅ , and Odd Multiway Cut if T e = ∅ and Multiway Cut if T e = T o . (Warsaw) April 9, 2013
Parity Multiway Cut Special case Odd/Even Multiway Cut NP complete even for 2 terminals. (Warsaw) April 9, 2013
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 6 k . (Warsaw) April 9, 2013
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 6 k . (Warsaw) April 9, 2013
Even Multiway Cut S Let S be a hypothetical solution. Observation: A component of G \ S contains at most 2 terminals. (Warsaw) April 9, 2013
Even Multiway Cut S Let S be a hypothetical solution. Observation: A component of G \ S contains at most 2 terminals. (Warsaw) April 9, 2013
Even Multiway Cut S Guess the partition of the terminals. (Warsaw) April 9, 2013
Even Multiway Cut S Guess the partition of the terminals. Fix one set in the partition (the red terminals). (Warsaw) April 9, 2013
Recommend
More recommend
