wg qcolorable
DESCRIPTION
TRANSCRIPT
![Page 1: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/1.jpg)
Parameterized Algorithms for the
Max q-Colorable Induced Subgraph problem
on Perfect Graphs
Neeldhara Misra, Fahad Panolan, Ashutosh Rai,Venkatesh Raman, and Saket Saurabh
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The Problem
Given a graphG, find the largest subgraph that isq-colorable.
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The Problem
Given a graphG, find the largest subgraph that is1-colorable.
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The Problem
Given a graphG, find the largest subgraph that isan independent set.
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The Problem
Given a graphG, find the largest subgraph that is2-colorable.
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The Problem
Given a graphG, find the largest subgraph that isan induced bipartite subgraph.
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Our Motivation
Yannakakis and Gavril [IPL 1987] showed that this problem is NP-complete evenon split graphs if q is part of input, but gave an nO(q) algorithm on chordal
graphs for fixed q.
An immediate natural question is: What is the status of this question inparametrized complexity?
In other words: Does this problem have an algorithm with running timef(q) · p(n) or f(q, ℓ) · p(n, q)? Here ℓ is the size of the subgraph we are
looking for.
![Page 8: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/8.jpg)
Our Motivation
Yannakakis and Gavril [IPL 1987] showed that this problem is NP-complete evenon split graphs if q is part of input, but gave an nO(q) algorithm on chordal
graphs for fixed q.
An immediate natural question is: What is the status of this question inparametrized complexity?
In other words: Does this problem have an algorithm with running timef(q) · p(n) or f(q, ℓ) · p(n, q)? Here ℓ is the size of the subgraph we are
looking for.
![Page 9: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/9.jpg)
Our Motivation
Yannakakis and Gavril [IPL 1987] showed that this problem is NP-complete evenon split graphs if q is part of input, but gave an nO(q) algorithm on chordal
graphs for fixed q.
An immediate natural question is: What is the status of this question inparametrized complexity?
In other words: Does this problem have an algorithm with running timef(q) · p(n) or f(q, ℓ) · p(n, q)? Here ℓ is the size of the subgraph we are
looking for.
![Page 10: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/10.jpg)
We first present a randomized algorithm with running time:
f(ℓ) · [Time to enumerate maximal independent sets] · p(n, q),
to find an induced q-colorable subgraph of size at least ℓ.
![Page 11: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/11.jpg)
We first present a randomized algorithm with running time:
f(ℓ) · [Time to enumerate maximal independent sets] · p(n, q),
to find an induced q-colorable subgraph of size at least ℓ.
![Page 12: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/12.jpg)
We first present a randomized algorithm with running time:
f(ℓ) · [Time to enumerate maximal independent sets] · p(n, q),
to find an induced q-colorable subgraph of size at least ℓ.
Notice that we do not expect an algorithm with running time:
f(ℓ) · p(n, q),
since the problem isW[1]-hard on general graphs even when q = 1.
![Page 13: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/13.jpg)
We first present a randomized algorithm with running time:
f(ℓ) · [Time to enumerate maximal independent sets] · p(n, q),
to find an induced q-colorable subgraph of size at least ℓ.
This algorithm is reasonable for graph classes where maximal independent setscan be enumerated efficiently, for example, co-chordal and split graphs.
The problem remains NP-complete even when restricted to these graph classes.
![Page 14: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/14.jpg)
We first present a randomized algorithm with running time:
f(ℓ) · [Time to enumerate maximal independent sets] · p(n, q),
to find an induced q-colorable subgraph of size at least ℓ.
We also establish the non-existence of a polynomial kernels for the problemwhen parameterized by solution size alone.
(Under standard complexity-theoretic assumptions.)
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The Algorithm
Enumerate all maximal independent sets, and create an auxiliary graph that hasone vertexmi for every MIS.
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The Algorithm
Enumerate all maximal independent sets, and create an auxiliary graph that hasone vertexmi for every MIS.
![Page 17: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/17.jpg)
The Algorithm
Enumerate all maximal independent sets, and create an auxiliary graph that hasone vertexmi for every MIS.
![Page 18: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/18.jpg)
The Algorithm
Enumerate all maximal independent sets, and create an auxiliary graph that hasone vertexmi for every MIS.
![Page 19: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/19.jpg)
The Algorithm
Enumerate all maximal independent sets, and create an auxiliary graph that hasone vertexmi for every MIS.
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The Algorithm
Introduce vertices corresponding to V(G) and introduce the edge (v,mi) ifv ∈ mi .
a
b
b
cc
d
d
e
e
f
f
a
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The Algorithm
Introduce vertices corresponding to V(G) and introduce the edge (v,mi) ifv ∈ mi .
a
b
b
cc
d
d
e
e
f
f
a
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The Algorithm
Introduce vertices corresponding to V(G) and introduce the edge (v,mi) ifv ∈ mi .
a
b
b
cc
d
d
e
e
f
f
a
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The Algorithm
Introduce vertices corresponding to V(G) and introduce the edge (v,mi) ifv ∈ mi .
a
b
b
cc
d
d
e
e
f
f
a
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The Algorithm
Introduce vertices corresponding to V(G) and introduce the edge (v,mi) ifv ∈ mi .
a
b
b
cc
d
d
e
e
f
f
a
![Page 25: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/25.jpg)
The Algorithm
Introduce vertices corresponding to V(G) and introduce the edge (v,mi) ifv ∈ mi .
a
b
b
cc
d
d
e
e
f
f
a
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The Algorithm
Color the vertices of the graph with ℓ colors, uniformly at random. Consider thecolor classes in the auxiliary graph.
a
b
b
cc
d
d
e
e
f
f
a
MaximalIndependent
Sets
Partitioninto
L classesA random coloring of G
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The Algorithm
Now contract all the color classes into singletons, and find a dominating set ofsize at most q.
a
b
b
cc
d
d
e
e
f
f
a
MaximalIndependent
Sets
Partitioninto
L classesA random coloring of G
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The Algorithm
Clearly, if there is a dominating set, then we have found q-colorable subgraph ofsize at least ℓ.
a
b
b
cc
d
d
e
e
f
f
a
MaximalIndependent
Sets
Partitioninto
L classesA random coloring of G
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The Algorithm
To compute the dominating set, make the “MIS vertices” a clique and run SteinerTree with the V as the terminals.
a
b
b
cc
d
d
e
e
f
f
a
MaximalIndependent
Sets
Partitioninto
L classesA random coloring of G
Make this a clique
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The Algorithm
When we randomly color the vertices, each vertex in a possible solution will getdifferent colors with probability:
ℓ!
ℓℓ⩾ e−ℓ
Once we have a nice coloring, clearlyeverything else works out fine.
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The Algorithm
When we randomly color the vertices, each vertex in a possible solution will getdifferent colors with probability:
ℓ!
ℓℓ⩾ e−ℓ
Once we have a nice coloring, clearlyeverything else works out fine.
![Page 32: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/32.jpg)
The Algorithm
When we randomly color the vertices, each vertex in a possible solution will getdifferent colors with probability:
ℓ!
ℓℓ⩾ e−ℓ
Once we have a nice coloring, clearlyeverything else works out fine.
![Page 33: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/33.jpg)
The Algorithm
We then present a randomized algorithm with running time:
f(ℓ) · [Time to find a maximum sized independent sets] · p(n, q),
to find an induced q-colorable subgraph of size at least ℓ.
This algorithm is good for perfect graphs where maximum independent set canbe found in polynomial time.
![Page 34: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/34.jpg)
The Algorithm
We then present a randomized algorithm with running time:
f(ℓ) · [Time to find a maximum sized independent sets] · p(n, q),
to find an induced q-colorable subgraph of size at least ℓ.
This algorithm is good for perfect graphs where maximum independent set canbe found in polynomial time.
![Page 35: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/35.jpg)
The Algorithm
A graphH is q colorable if and only if its vertex set can be partitioned into qindependent sets.
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The Algorithm
• Randomly color the vertices ofG with {1, . . . , q}.• Let Vi be the set of vertices with color i.• Find a maximum sized independent set Ii ⊆ G[Vi].• Return I = ∪q
i=1Ii.
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The Algorithm
Suppose we have a solutionW =⊎q
i=1Wi. HereWi is an independentsolution and
∑q1=1Wi| = ℓ.
When we randomly color the vertices, a vertex inWi gets color i
1
qℓ
Once we have a nice coloring, clearlyeverything else works out fine.
![Page 38: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/38.jpg)
The Algorithm
Suppose we have a solutionW =⊎q
i=1Wi. HereWi is an independentsolution and
∑q1=1Wi| = ℓ.
When we randomly color the vertices, a vertex inWi gets color i
1
qℓ
Once we have a nice coloring, clearlyeverything else works out fine.
![Page 39: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/39.jpg)
The Algorithm
Suppose we have a solutionW =⊎q
i=1Wi. HereWi is an independentsolution and
∑q1=1Wi| = ℓ.
When we randomly color the vertices, a vertex inWi gets color i
1
qℓ
Once we have a nice coloring, clearlyeverything else works out fine.
![Page 40: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/40.jpg)
The Algorithm
Suppose we have a solutionW =⊎q
i=1Wi. HereWi is an independentsolution and
∑q1=1Wi| = ℓ.
When we randomly color the vertices, a vertex inWi gets color i
1
qℓ
Once we have a nice coloring, clearlyeverything else works out fine.
![Page 41: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/41.jpg)
Kernelization Algorithm
A parameterized problem is said to admit a polynomial kernel if every instance(I, k) can be reduced in polynomial time to an equivalent instance (I ′, k ′) with
both size and parameter value bounded by a polynomial in k.
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No Kernel Results
Finally, we show that (under standard complexity-theoretic assumptions) theproblem does not admit a polynomial kernel on split and perfect graphs in the
following sense:
(a) On split graphs, we do not expect a polynomial kernel if q is a part of theinput.
(b) On perfect graphs, we do not expect a polynomial kernel even for fixedvalues of q ⩾ 2.
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No Kernel Results
Finally, we show that (under standard complexity-theoretic assumptions) theproblem does not admit a polynomial kernel on split and perfect graphs in the
following sense:
(a) On split graphs, we do not expect a polynomial kernel if q is a part of theinput.
(b) On perfect graphs, we do not expect a polynomial kernel even for fixedvalues of q ⩾ 2.
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Composition
A OR-composition algorithm for a parameterized problem Π ⊆ Σ∗ × N is analgorithm
that receives as input a sequence ((x1, k), ..., (xt, k)), with
(xi, k) ∈ Σ∗ × N for each 1 ⩽ i ⩽ t,
uses time polynomial in∑t
i=1 |xi|+ k, and outputs (y, k ′) ∈ Σ∗ × N
with (a) (y, k ′) ∈ Π ⇐⇒ (xi, k) ∈ Π for some 1 ⩽ i ⩽ t and (b) k ′ ispolynomial in k.
A parameterized problem is or OR-compositional if there is a compositionalgorithm for it.
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Composition
A OR-composition algorithm for a parameterized problem Π ⊆ Σ∗ × N is analgorithm that receives as input a sequence ((x1, k), ..., (xt, k)), with
(xi, k) ∈ Σ∗ × N for each 1 ⩽ i ⩽ t,
uses time polynomial in∑t
i=1 |xi|+ k, and outputs (y, k ′) ∈ Σ∗ × N
with (a) (y, k ′) ∈ Π ⇐⇒ (xi, k) ∈ Π for some 1 ⩽ i ⩽ t and (b) k ′ ispolynomial in k.
A parameterized problem is or OR-compositional if there is a compositionalgorithm for it.
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Composition
A OR-composition algorithm for a parameterized problem Π ⊆ Σ∗ × N is analgorithm that receives as input a sequence ((x1, k), ..., (xt, k)), with
(xi, k) ∈ Σ∗ × N for each 1 ⩽ i ⩽ t,
uses time polynomial in∑t
i=1 |xi|+ k, and outputs (y, k ′) ∈ Σ∗ × N
with (a) (y, k ′) ∈ Π ⇐⇒ (xi, k) ∈ Π for some 1 ⩽ i ⩽ t and (b) k ′ ispolynomial in k.
A parameterized problem is or OR-compositional if there is a compositionalgorithm for it.
![Page 47: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/47.jpg)
Composition
A OR-composition algorithm for a parameterized problem Π ⊆ Σ∗ × N is analgorithm that receives as input a sequence ((x1, k), ..., (xt, k)), with
(xi, k) ∈ Σ∗ × N for each 1 ⩽ i ⩽ t,
uses time polynomial in∑t
i=1 |xi|+ k, and outputs (y, k ′) ∈ Σ∗ × N
with (a) (y, k ′) ∈ Π ⇐⇒ (xi, k) ∈ Π for some 1 ⩽ i ⩽ t and (b) k ′ ispolynomial in k.
A parameterized problem is or OR-compositional if there is a compositionalgorithm for it.
![Page 48: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/48.jpg)
Composition
A OR-composition algorithm for a parameterized problem Π ⊆ Σ∗ × N is analgorithm that receives as input a sequence ((x1, k), ..., (xt, k)), with
(xi, k) ∈ Σ∗ × N for each 1 ⩽ i ⩽ t,
uses time polynomial in∑t
i=1 |xi|+ k, and outputs (y, k ′) ∈ Σ∗ × N
with (a) (y, k ′) ∈ Π ⇐⇒ (xi, k) ∈ Π for some 1 ⩽ i ⩽ t and (b) k ′ ispolynomial in k.
A parameterized problem is or OR-compositional if there is a compositionalgorithm for it.
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The Composition
The Composition Algorithm
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The Composition
.. G1
.
G2
.
G3
.
G4
.G5
.
G6
.
G7
.
G8
A spillover across two instances could still be a problem.
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The Composition
.. G1
.
G2
.
G3
.
G4
.
G6
.
G7
.
G8
.G5
A spillover across two instances could still be a problem.
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The Composition
..
G2
.
G3
.
G6
.
G8
. G1
.
G4
.G5
.
G7
A spillover across two instances could still be a problem.
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The Composition
.. G1
.
G2
.
G3
.
G4
.G5
.
G6
.
G7
.
G8
A spillover across two instances could still be a problem.
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The Composition
..
G2
.
G3
.
G6
.
G8
.
G7
. G1
.
G4
.G5
A spillover across two instances could still be a problem.
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The Composition
..
G2
.
G3
.
G6
.
G8
.G5
.
G7
. G1
.
G4
A spillover across two instances could still be a problem.
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The Composition
What is this gadget trying to achieve?If the gadget contributes six vertices to any induced bipartite subgraph, then:
At most two inner vertices participate in the solution.
..
xij
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wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
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The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
At most two inner vertices participate in the solution.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
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The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
At most two inner vertices participate in the solution.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 59: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/59.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
At most two inner vertices participate in the solution.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 60: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/60.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
At most two inner vertices participate in the solution.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
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dij
![Page 61: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/61.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
At most two inner vertices participate in the solution.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 62: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/62.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
At most two inner vertices participate in the solution.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 63: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/63.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
All four outer vertices participate in the solution.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 64: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/64.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
Inner vertices from two levels do not participate together.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 65: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/65.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
Inner vertices from two levels do not participate together.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 66: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/66.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
Inner vertices from two levels do not participate together.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 67: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/67.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
Inner vertices from two levels do not participate together.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 68: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/68.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
Inner vertices from two levels do not participate together.
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 69: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/69.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
A valid contribution, therefore, is either …
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
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cij
.
dij
![Page 70: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/70.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
A valid contribution, therefore, is either this
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 71: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/71.jpg)
The Composition
What is this gadget trying to achieve?
If the gadget contributes six vertices to any induced bipartite subgraph, then:
A valid contribution, therefore, is either or this:
..
xij
.
wij
.
yij
.
zij
.
aij
.
bij
.
cij
.
dij
![Page 72: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/72.jpg)
The Composition
.......................................................
..
G0 : 000
...................
G1 : 001
...................
G2 : 010
...................
G3 : 011
...................
G4 : 100
...................
G5 : 101
...................
G6 : 110
...................
G7 : 111
..................
.
—log t—
.
2k
![Page 73: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/73.jpg)
The Composition
........................................................
.
G0 : 000
...................
G1 : 001
...................
G2 : 010
...................
G3 : 011
...................
G4 : 100
...................
G5 : 101
...................
G6 : 110
...................
G7 : 111
..................
.
—log t—
.
2k
![Page 74: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/74.jpg)
The Composition
.......................................................
.
.
G0 : 000
..................
.
G1 : 001
...................
G2 : 010
...................
G3 : 011
...................
G4 : 100
...................
G5 : 101
...................
G6 : 110
...................
G7 : 111
..................
.
—log t—
.
2k
![Page 75: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/75.jpg)
The Composition
.......................................................
..
G0 : 000
..................
.
G1 : 001
..................
.
G2 : 010
...................
G3 : 011
...................
G4 : 100
...................
G5 : 101
...................
G6 : 110
...................
G7 : 111
..................
.
—log t—
.
2k
![Page 76: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/76.jpg)
The Composition
.......................................................
..
G0 : 000
...................
G1 : 001
..................
.
G2 : 010
..................
.
G3 : 011
...................
G4 : 100
...................
G5 : 101
...................
G6 : 110
...................
G7 : 111
..................
.
—log t—
.
2k
![Page 77: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/77.jpg)
The Composition
.......................................................
..
G0 : 000
...................
G1 : 001
...................
G2 : 010
..................
.
G3 : 011
..................
.
G4 : 100
...................
G5 : 101
...................
G6 : 110
...................
G7 : 111
..................
.
—log t—
.
2k
![Page 78: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/78.jpg)
The Composition
.......................................................
..
G0 : 000
...................
G1 : 001
...................
G2 : 010
...................
G3 : 011
..................
.
G4 : 100
..................
.
G5 : 101
...................
G6 : 110
...................
G7 : 111
..................
.
—log t—
.
2k
![Page 79: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/79.jpg)
The Composition
.......................................................
..
G0 : 000
...................
G1 : 001
...................
G2 : 010
...................
G3 : 011
...................
G4 : 100
..................
.
G5 : 101
..................
.
G6 : 110
...................
G7 : 111
..................
.
—log t—
.
2k
![Page 80: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/80.jpg)
The Composition
.......................................................
..
G0 : 000
...................
G1 : 001
...................
G2 : 010
...................
G3 : 011
...................
G4 : 100
...................
G5 : 101
..................
.
G6 : 110
..................
.
G7 : 111
..................
.
—log t—
.
2k
![Page 81: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/81.jpg)
The Composition
.......................................................
..
G0 : 000
...................
G1 : 001
...................
G2 : 010
...................
G3 : 011
...................
G4 : 100
...................
G5 : 101
...................
G6 : 110
..................
.
G7 : 111
...................
—log t—
.
2k
![Page 82: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/82.jpg)
k −→ k+ 12k · log t
...can be shown to be a polynomial in k without loss of generality.
![Page 83: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/83.jpg)
k −→ k+ 12k · log t
...can be shown to be a polynomial in k without loss of generality.
![Page 84: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/84.jpg)
The Forward Direction
![Page 85: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/85.jpg)
Suppose someGi has a bipartite subgraph on k vertices.
Consider the binary representation of i. Pick the six vertices from the 2k log tgadgets thatGi is non-adjacent to.
![Page 86: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/86.jpg)
Suppose someGi has a bipartite subgraph on k vertices.
Consider the binary representation of i. Pick the six vertices from the 2k log tgadgets thatGi is non-adjacent to.
![Page 87: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/87.jpg)
........................................................
G5 : 101
..................
![Page 88: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/88.jpg)
......................................
G5 : 101
![Page 89: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/89.jpg)
The Reverse Direction
![Page 90: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/90.jpg)
Suppose the composed instance has an induced bipartite subgraph S on at leastk+ 12k · log t vertices.
Recall that each of the (2k log t) gadgets can contribute at most six verticeseach.
Consider:
(S ∩Gi) for 1 ⩽ i ⩽ t.
If S ∩Gi is non-empty for exactly one instanceGi, then we are done, becausethen we automatically have that |S ∩Gi| ⩾ k.
![Page 91: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/91.jpg)
Suppose the composed instance has an induced bipartite subgraph S on at leastk+ 12k · log t vertices.
Recall that each of the (2k log t) gadgets can contribute at most six verticeseach.
Consider:
(S ∩Gi) for 1 ⩽ i ⩽ t.
If S ∩Gi is non-empty for exactly one instanceGi, then we are done, becausethen we automatically have that |S ∩Gi| ⩾ k.
![Page 92: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/92.jpg)
Suppose the composed instance has an induced bipartite subgraph S on at leastk+ 12k · log t vertices.
Recall that each of the (2k log t) gadgets can contribute at most six verticeseach.
Consider:
(S ∩Gi) for 1 ⩽ i ⩽ t.
If S ∩Gi is non-empty for exactly one instanceGi, then we are done, becausethen we automatically have that |S ∩Gi| ⩾ k.
![Page 93: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/93.jpg)
Suppose the composed instance has an induced bipartite subgraph S on at leastk+ 12k · log t vertices.
Recall that each of the (2k log t) gadgets can contribute at most six verticeseach.
Consider:
(S ∩Gi) for 1 ⩽ i ⩽ t.
If S ∩Gi is non-empty for exactly one instanceGi, then we are done, becausethen we automatically have that |S ∩Gi| ⩾ k.
![Page 94: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/94.jpg)
We also know that S cannot be shared by three of the instances, since that wouldinduce a triangle.
Let us now address the only remaining case: what if S intersectsGi andGj
non-trivially, for some 1 ⩽ i ̸= j ⩽ t?
![Page 95: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/95.jpg)
We also know that S cannot be shared by three of the instances, since that wouldinduce a triangle.
Let us now address the only remaining case: what if S intersectsGi andGj
non-trivially, for some 1 ⩽ i ̸= j ⩽ t?
![Page 96: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/96.jpg)
If i ̸= j, then the bit vectors corresponding to i and j are also different.
Let b be an index (1 ⩽ b ⩽ log t) where i and j differ.
![Page 97: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/97.jpg)
If i ̸= j, then the bit vectors corresponding to i and j are also different.
Let b be an index (1 ⩽ b ⩽ log t) where i and j differ.
![Page 98: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/98.jpg)
........................................................
G5 : 101 andG7 : 111
........................
![Page 99: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/99.jpg)
These 2k gadgets, corresponding to the index b, cannot contribute to S at all,being global to vertices inGi andGj.
![Page 100: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/100.jpg)
......................................
G5 : 101 andG7 : 111
............
![Page 101: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/101.jpg)
The remaining gadgets can contribute at most:
[2k · (log t− 1)]6+ 4(2k) = 12k log t− 4k.
ThusGi andGj together must account for at least 5k vertices between them.
This implies that at least one of them must be contributing at least k vertices,and this leads us to our conclusion.
![Page 102: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/102.jpg)
The remaining gadgets can contribute at most:
[2k · (log t− 1)]6+ 4(2k) = 12k log t− 4k.
ThusGi andGj together must account for at least 5k vertices between them.
This implies that at least one of them must be contributing at least k vertices,and this leads us to our conclusion.
![Page 103: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/103.jpg)
The remaining gadgets can contribute at most:
[2k · (log t− 1)]6+ 4(2k) = 12k log t− 4k.
ThusGi andGj together must account for at least 5k vertices between them.
This implies that at least one of them must be contributing at least k vertices,and this leads us to our conclusion.
![Page 104: Wg qcolorable](https://reader033.vdocuments.net/reader033/viewer/2022051411/547bd1ed5906b559798b4658/html5/thumbnails/104.jpg)
Danke!