connected dominating set and short cycles
DESCRIPTION
TRANSCRIPT
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Connected Dominating Set and Short Cycles
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Connected Dominating Set and Short Cycles
Joint work with Geevarghese Philip, Venkatesh Ramanand Saket Saurabh
The Institute of Mathematical Sciences, Chennai
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Talk Outline
Polynomial KernelsNo Polynomial Kernels, but FPT
W-hard
e impact of excluding short cycles.
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Talk Outline
Polynomial KernelsNo Polynomial Kernels, but FPT
W-hard
e impact of excluding short cycles.
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Talk Outline
Polynomial KernelsNo Polynomial Kernels, but FPT
W-hard
Girth: at least seven
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Talk Outline
Polynomial Kernels (> 7)No Polynomial Kernels, but FPT
W-hard
Girth: five or six
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Talk Outline
Polynomial Kernels (> 7)No Polynomial Kernels, but FPT (5, 6)
W-hard
Girth: at most four
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Talk Outline
Polynomial Kernels (> 7)No Polynomial Kernels, but FPT (5, 6)
W-hard (6 4)
Girth: at least seven
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Talk Outline
Polynomial Kernels (> 7)No Polynomial Kernels, but FPT (5, 6)
W-hard (6 4)
e longer the cycles we exclude, the easier the problem becomes to solve.
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Talk Outline
Polynomial Kernels (> 7)No Polynomial Kernels, but FPT (5, 6)
W-hard (6 4)
e longer the cycles we exclude, the easier the problem becomes to solve.
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Part 1
Why Its Usefulto not have short cycles
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Consider the problem of domination:
Find a subset of at most k vertices S such that every v ∈ V is either in S orhas a neighbor in S.
Let us consider graphs of girth at least five.We begin by examining vertices of degree more than k.
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Consider the problem of domination:
Find a subset of at most k vertices S such that every v ∈ V is either in S orhas a neighbor in S.
Let us consider graphs of girth at least five.We begin by examining vertices of degree more than k.
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Consider the problem of domination:
Find a subset of at most k vertices S such that every v ∈ V is either in S orhas a neighbor in S.
Let us consider graphs of girth at least five.
We begin by examining vertices of degree more than k.
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Consider the problem of domination:
Find a subset of at most k vertices S such that every v ∈ V is either in S orhas a neighbor in S.
Let us consider graphs of girth at least five.We begin by examining vertices of degree more than k.
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Who dominates the neighbors of the red vertex?
Does there exist a dominating set of size at most k that does not includethe red vertex?
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Who dominates the neighbors of the red vertex?Does there exist a dominating set of size at most k that does not include
the red vertex?
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Who dominates the neighbors of the red vertex?Clearly, it is not the case that each one dominates itself.
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Who dominates the neighbors of the red vertex?Clearly, it is not the case that each one dominates itself.
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Either a vertex from within the neighborhood dominates at least two ofvertices...
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or a vertex from “outside” the neighborhood dominates at least two of thevertices...
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High Degree Vertices
On graphs that have girth at least five:
Any vertex of degree more than k belongs to any dominating set of size atmost k.
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Notice that the number of reds is at most k and the number of blues is atmost k2.
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Greens in the neighborhood of a blue vertex cannot have a common redneighbor.
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us, for every blue vertex, there are at most k green vertices.
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vertices that have been dominated that have no edges into blue are irrelevant.
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So now all is bounded, and we have a happy ending: a kernel on at most
O(k+ k2 + k3)
vertices.
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Part 2
FPT Algorithmsand Polynomial Kernels
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In the connected dominating set situation, we have to backtrack a smallway in the story so far to see what fails, and how badly.
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High Degree Vertices
On graphs that have girth at least five:
connected
Any vertex of degree more than k belongs to any dominating set of size atmost k.
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High Degree Vertices
On graphs that have girth at least five:
connectedAny vertex of degree more than k belongs to any dominating set of size at
most k.
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Notice that the number of reds is at most k and the number of blues isagain at most k2.
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Greens in the neighborhood of a blue vertex cannot have a common redneighbor.
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us, for every blue vertex, there are at most k green vertices.
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vertices that have been dominated that have no edges into blue are irrelevant.
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vertices that have been dominated that have no edges into blue are irrelevant.
Not any more...
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any connected dom set contains a minimal dom set
that resides in the bounded part of the graph.
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Now we know what to do:
Guess the minimal dominating set.
Extend it to a connected dominating set by an application of a SteinerTree algorithm.
is is evidently FPT. What about kernels?
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Now we know what to do:
Guess the minimal dominating set.
Extend it to a connected dominating set by an application of a SteinerTree algorithm.
is is evidently FPT. What about kernels?
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Now we know what to do:
Guess the minimal dominating set.
Extend it to a connected dominating set by an application of a SteinerTree algorithm.
is is evidently FPT. What about kernels?
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Now we know what to do:
Guess the minimal dominating set.
Extend it to a connected dominating set by an application of a SteinerTree algorithm.
is is evidently FPT.
What about kernels?
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Now we know what to do:
Guess the minimal dominating set.
Extend it to a connected dominating set by an application of a SteinerTree algorithm.
is is evidently FPT. What about kernels?
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It turns out that if G did not admit cycles of length six and less, then thenumber of green vertices can be bounded.
And we will show (later) that on graphs of girth five, connecteddominating set is unlikely to admit polynomial kernels.
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It turns out that if G did not admit cycles of length six and less, then thenumber of green vertices can be bounded.
And we will show (later) that on graphs of girth five, connecteddominating set is unlikely to admit polynomial kernels.
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Low Degree Vertices
If there is blue pendant vertex, make its neighbor red.
If there is a green vertex that is pendant, delete it.
If there is a blue vertex that is isolated, say no.
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no pair 'sees' more than one vertex.
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So now the green vertices are bounded:
• At most O(k3) with at least one neighbor in the blues,
• At most O(k2) with only red neighbors,
• At most O(k2) with no blue neighbors and at least one whiteneighbor.
is gives us a O(k3) vertex kernel.
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So now the green vertices are bounded:
• At most O(k3) with at least one neighbor in the blues,
• At most O(k2) with only red neighbors,
• At most O(k2) with no blue neighbors and at least one whiteneighbor.
is gives us a O(k3) vertex kernel.
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The Hard Part
Infeasibility of Polynomial Kernelizationand W-hardness
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Introducing...
F C C
NP-complete F FPT F Compositional
Ready for reduction!
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Introducing...
F C C
NP-complete F FPT F Compositional
Ready for reduction!
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Introducing...
F C C
NP-complete F FPT F Compositional
Ready for reduction!
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Introducing...
F C C
NP-complete F FPT F Compositional
Ready for reduction!
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Promise: Every vertex has degree at most one intoany color class, and every color class is independent.
Question: Does there exist a colorful tree?
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Add global vertices (with guards) to each of the color classes. subdividethe edges,
add a global vertex for the newly added vertices (with guards).
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Between a pair of color classes:
• subdivide the edges,• add a global vertex for the newly added vertices (with guards).
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Between a pair of color classes:• subdivide the edges,
• add a global vertex for the newly added vertices (with guards).
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Between a pair of color classes:• subdivide the edges,• add a global vertex for the newly added vertices
(with guards).
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Between a pair of color classes:• subdivide the edges,• add a global vertex for the newly added vertices (with guards).
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For vertices that don’t look inside the neighboring color class, add a path oflength two to the global vertex. add a global vertex for the newly added
vertices (with guards).
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Here is (1/k2)th of the full picture. subdivide the edges,add a global vertex for the newly added vertices (with guards).
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Notice that we have ended up with a bipartite graph... subdivide the edges,add a global vertex for the newly added vertices (with guards).
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....with a cycle of length six. subdivide the edges,add a global vertex for the newly added vertices (with guards).
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Suppose we have a colorful tree in the source instance.
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Consider the same subset.
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(along with the subdivided vertices along the edges of the tree)
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is is a connected subset, but not dominating.
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To fix this, add all the global vertices to this set.
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ere are k+(k2
)such vertices.
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Now we have a dominating set, but it’s not connected any more!
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Let’s look more closely.
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e vertices global to color classes are connected to the tree.
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Some of the other vertices are connected too!
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Each of the rest corresponds to a non-edge in the tree,
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for which we add the path of length two to the picture.
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e size of the connected dominating set thus obtained is:
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e size of the connected dominating set thus obtained is:
2(k2
)+ 2k
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e size of the connected dominating set thus obtained is:
2(k2
)+ 2k
Two vertices for every pair of original vertices in the tree.(A global vertex, and a neighbor.)
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e size of the connected dominating set thus obtained is:
2(k2
)+ 2k
Two vertices for every original vertex in the tree:itself, and the corresponding global vertex.
![Page 89: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/89.jpg)
And that works out to...
2(k2
)+ 2k
= k2 + k
![Page 90: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/90.jpg)
Let S be a connected dominating set of size at most:(k
2
)+
(k
2
)+ k+ k
![Page 91: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/91.jpg)
Let S be a connected dominating set of size at most:(k
2
)+
(k
2
)+ k+ k
Of course, global vertices are forced in S, because of the guard vertices.
![Page 92: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/92.jpg)
Let S be a connected dominating set of size at most:(k
2
)+
(k
2
)+ k+ k
Since S is a connected subset, each of these vertices have at least oneneighbor in S.
![Page 93: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/93.jpg)
Let S be a connected dominating set of size at most:(k
2
)+
(k
2
)+ k+ k
Since S is a connected subset, each of these vertices have at least oneneighbor in S.
Because of the budget, each of them have...
![Page 94: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/94.jpg)
Let S be a connected dominating set of size at most:(k
2
)+
(k
2
)+ k+ k
Since S is a connected subset, each of these vertices have exactly oneneighbor in S.
Because of the budget, each of them have...
![Page 95: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/95.jpg)
us, the dominating set picks exactly one vertex from each color class.
• Neglect the global vertices and “subdivision vertices” of degree one, tobe left with a connected subtree with subdivided edges.
• is can be easily pulled back to a colorful tree of the original graph.
![Page 96: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/96.jpg)
us, the dominating set picks exactly one vertex from each color class.
• Neglect the global vertices and “subdivision vertices” of degree one, tobe left with a connected subtree with subdivided edges.
• is can be easily pulled back to a colorful tree of the original graph.
![Page 97: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/97.jpg)
us, the dominating set picks exactly one vertex from each color class.
• Neglect the global vertices and “subdivision vertices” of degree one, tobe left with a connected subtree with subdivided edges.
• is can be easily pulled back to a colorful tree of the original graph.
![Page 98: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/98.jpg)
e W-hardness result.
![Page 99: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/99.jpg)
G
A dominating set
instance
![Page 100: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/100.jpg)
G
We begin by
making a copy of the graph.
![Page 101: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/101.jpg)
G G
![Page 102: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/102.jpg)
en, for every edge in G, we stretch it across the copies.
![Page 103: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/103.jpg)
en, for every edge in G, we stretch it across the copies.
![Page 104: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/104.jpg)
Finally, add a new vertex to each copy,with one of them global to the other copy.
![Page 105: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/105.jpg)
Any old dominating set + new global vertex= new connected dominating set.
![Page 106: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/106.jpg)
Any new dominating set contains the newly addedglobal vertex WLOG.
![Page 107: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/107.jpg)
Consider now the vertices of thenew connected dominating set.
![Page 108: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/108.jpg)
Observe that they form a dominating set of G.
![Page 109: Connected Dominating Set and Short Cycles](https://reader033.vdocuments.net/reader033/viewer/2022051513/547c6c115906b5a4798b4700/html5/thumbnails/109.jpg)
Takk