Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex...
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Coloring Circular Arc GraphsRevisiting Tucker’s algorithm
Alex Angelopoulos
µ∏λ ∀
July 22, 2014
![Page 2: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/2.jpg)
Outline
Introduction
Analyzing Tucker’s algorithm
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 2/15
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The problem
Input: a family F of circular arcs
χ(F ) = 3
Output: is there a proper coloring with ≤ k colors?what is the minimum k s.t. F has a proper coloring?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 3/15
![Page 4: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/4.jpg)
The problem
Input: a family F of circular arcs
χ(F ) = 3
Output: is there a proper coloring with ≤ k colors?what is the minimum k s.t. F has a proper coloring?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 3/15
![Page 5: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/5.jpg)
The problem
Input: a family F of circular arcs
χ(F ) = 3
Output: is there a proper coloring with ≤ k colors?what is the minimum k s.t. F has a proper coloring?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 3/15
![Page 6: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/6.jpg)
Some quantities
L(F ) =3
l(F ) = 4
= ω(F )
u Load of F : L
u circular-cover: l
u max. clique of F : ω(as usual)
u We also discretize anduse the -at most- 2|F |points defining thearcs.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 4/15
![Page 7: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/7.jpg)
Some quantities
L(F ) =3
l(F ) = 4
= ω(F )
u Load of F : L
u circular-cover: l
u max. clique of F : ω(as usual)
u We also discretize anduse the -at most- 2|F |points defining thearcs.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 4/15
![Page 8: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/8.jpg)
Some quantities
L(F ) =3
l(F ) = 4
= ω(F )
u Load of F : L
u circular-cover: l
u max. clique of F : ω(as usual)
u We also discretize anduse the -at most- 2|F |points defining thearcs.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 4/15
![Page 9: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/9.jpg)
Some quantities
L(F ) =3
l(F ) = 4
= ω(F )
u Load of F : L
u circular-cover: l
u max. clique of F : ω(as usual)
u We also discretize anduse the -at most- 2|F |points defining thearcs.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 4/15
![Page 10: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/10.jpg)
Some quantities
L(F ) =3
l(F ) = 4
= ω(F )
u Load of F : L
u circular-cover: l
u max. clique of F : ω(as usual)
u We also discretize anduse the -at most- 2|F |points defining thearcs.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 4/15
![Page 11: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/11.jpg)
Results
Theorem 1 (Tucker (1975)).
Let F be a family of circular arcs with load L = L(F ) andcircular-cover l = l(F ). If l(F ) ≥ 4, then
⌊32L⌋
colors suffice toproperly color F .
u This is actually a 2-approximation algorithm.
u Tucker, [4] conjecture that χ(F ) ≤ 32ω(F ).
u Karapetian (1980) proves the above.
u Garey et al. (1980) show NP-completeness for Circular Arc
Color
u Many exact algos for subfamilies of graphs (≥ O(|F |1.5)).
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 5/15
![Page 12: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/12.jpg)
Results
Theorem 1 (Tucker (1975)).
Let F be a family of circular arcs with load L = L(F ) andcircular-cover l = l(F ). If l(F ) ≥ 4, then
⌊32L⌋
colors suffice toproperly color F .
u This is actually a 2-approximation algorithm.
u Tucker, [4] conjecture that χ(F ) ≤ 32ω(F ).
u Karapetian (1980) proves the above.
u Garey et al. (1980) show NP-completeness for Circular Arc
Color
u Many exact algos for subfamilies of graphs (≥ O(|F |1.5)).
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 5/15
![Page 13: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/13.jpg)
Results
Theorem 1 (Tucker (1975)).
Let F be a family of circular arcs with load L = L(F ) andcircular-cover l = l(F ). If l(F ) ≥ 4, then
⌊32L⌋
colors suffice toproperly color F .
u This is actually a 2-approximation algorithm.
u Tucker, [4] conjecture that χ(F ) ≤ 32ω(F ).
u Karapetian (1980) proves the above.
u Garey et al. (1980) show NP-completeness for Circular Arc
Color
u Many exact algos for subfamilies of graphs (≥ O(|F |1.5)).
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 5/15
![Page 14: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/14.jpg)
Results
Theorem 1 (Tucker (1975)).
Let F be a family of circular arcs with load L = L(F ) andcircular-cover l = l(F ). If l(F ) ≥ 4, then
⌊32L⌋
colors suffice toproperly color F .
u This is actually a 2-approximation algorithm.
u Tucker, [4] conjecture that χ(F ) ≤ 32ω(F ).
u Karapetian (1980) proves the above.
u Garey et al. (1980) show NP-completeness for Circular Arc
Color
u Many exact algos for subfamilies of graphs (≥ O(|F |1.5)).
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 5/15
![Page 15: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/15.jpg)
Results
More recently:
Theorem 2 (Valencia-Pabon (2003)).
Consider F with load L(F ) and circular-cover l(F ) ≥ 5. Then⌈l−1l−2
L⌉
colors suffice to color F , bound being tight.
That’s this presentation about!
It is based exactly on the algorithm proposed by Tucker, [4].
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 6/15
![Page 16: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/16.jpg)
Results
More recently:
Theorem 2 (Valencia-Pabon (2003)).
Consider F with load L(F ) and circular-cover l(F ) ≥ 5. Then⌈l−1l−2
L⌉
colors suffice to color F , bound being tight.
That’s this presentation about!
It is based exactly on the algorithm proposed by Tucker, [4].
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 6/15
![Page 17: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/17.jpg)
Results
More recently:
Theorem 2 (Valencia-Pabon (2003)).
Consider F with load L(F ) and circular-cover l(F ) ≥ 5. Then⌈l−1l−2
L⌉
colors suffice to color F , bound being tight.
That’s this presentation about!
It is based exactly on the algorithm proposed by Tucker, [4].
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 6/15
![Page 18: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/18.jpg)
Outline
Introduction
Analyzing Tucker’s algorithm
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 7/15
![Page 19: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/19.jpg)
Tucker’s algorithm
u Select p so that L(F ) arcscontain it
u Assign color #1 to the arcwhich extends at least onthe counterclockwise sideof p
u Move clockwise, assigncurrent color to the firstarc to begin after theprevious ends
u Unless it is not possible,so use next color
p
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 8/15
![Page 20: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/20.jpg)
Tucker’s algorithm
u Select p so that L(F ) arcscontain it
u Assign color #1 to the arcwhich extends at least onthe counterclockwise sideof p
u Move clockwise, assigncurrent color to the firstarc to begin after theprevious ends
u Unless it is not possible,so use next color
p
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 8/15
![Page 21: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/21.jpg)
Tucker’s algorithm
u Select p so that L(F ) arcscontain it
u Assign color #1 to the arcwhich extends at least onthe counterclockwise sideof p
u Move clockwise, assigncurrent color to the firstarc to begin after theprevious ends
u Unless it is not possible,so use next color
p
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 8/15
![Page 22: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/22.jpg)
Tucker’s algorithm
u Select p so that L(F ) arcscontain it
u Assign color #1 to the arcwhich extends at least onthe counterclockwise sideof p
u Move clockwise, assigncurrent color to the firstarc to begin after theprevious ends
u Unless it is not possible,so use next color
p
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 8/15
![Page 23: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/23.jpg)
Tucker’s algorithm
u Select p so that L(F ) arcscontain it
u Assign color #1 to the arcwhich extends at least onthe counterclockwise sideof p
u Move clockwise, assigncurrent color to the firstarc to begin after theprevious ends
u Unless it is not possible,so use next color
p
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 8/15
![Page 24: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/24.jpg)
Tucker’s algorithm
u Select p so that L(F ) arcscontain it
u Assign color #1 to the arcwhich extends at least onthe counterclockwise sideof p
u Move clockwise, assigncurrent color to the firstarc to begin after theprevious ends
u Unless it is not possible,so use next color
p
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 8/15
![Page 25: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/25.jpg)
Notations needed
t
A1
A2
A3
F1F2
u starting point: t
u first arc colored with i:Ai
u arcs colored until kth
round: Fk
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 9/15
![Page 26: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/26.jpg)
Notations needed
t
A1
A2
A3
F1F2
u starting point: t
u first arc colored with i:Ai
u arcs colored until kth
round: Fk
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 9/15
![Page 27: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/27.jpg)
Notations needed
t
A1
A2
A3
F1F2
u starting point: t
u first arc colored with i:Ai
u arcs colored until kth
round: Fk
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 9/15
![Page 28: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/28.jpg)
Notations needed
t
A1
A2
A3
F1
F2
u starting point: t
u first arc colored with i:Ai
u arcs colored until kth
round: Fk
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 9/15
![Page 29: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/29.jpg)
Notations needed
t
A1
A2
A3
F1
F2
u starting point: t
u first arc colored with i:Ai
u arcs colored until kth
round: Fk
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 9/15
![Page 30: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/30.jpg)
Properties
A1
A2
A3
Base of induction
1. Ai intersects Ai−1or else no new color is needed
2. L(F \ Fi) ≤ L(F )− iOK, verify again later*
3. For 1 ≤ i ≤ l− 2, Fi
is colored with at mosti+ 1 colors by theAlgorithm...Proof?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 10/15
![Page 31: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/31.jpg)
Properties
A1
A2
A3
Base of induction
1. Ai intersects Ai−1or else no new color is needed
2. L(F \ Fi) ≤ L(F )− iOK, verify again later*
3. For 1 ≤ i ≤ l− 2, Fi
is colored with at mosti+ 1 colors by theAlgorithm...Proof?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 10/15
![Page 32: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/32.jpg)
Properties
A1
A2
A3
Base of induction
1. Ai intersects Ai−1or else no new color is needed
2. L(F \ Fi) ≤ L(F )− iOK, verify again later*
3. For 1 ≤ i ≤ l− 2, Fi
is colored with at mosti+ 1 colors by theAlgorithm...
Proof?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 10/15
![Page 33: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/33.jpg)
Properties
A1
A2
A3
Base of induction
1. Ai intersects Ai−1or else no new color is needed
2. L(F \ Fi) ≤ L(F )− iOK, verify again later*
3. For 1 ≤ i ≤ l− 2, Fi
is colored with at mosti+ 1 colors by theAlgorithm...Proof?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 10/15
![Page 34: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/34.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
t
Af3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse tthese may be the same arc
u Now think aboutAi+1 and Af
i+1Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 35: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/35.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
tAf
3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse t
these may be the same arc
u Now think aboutAi+1 and Af
i+1Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 36: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/36.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
t
Af3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse tthese may be the same arc
u Now think aboutAi+1 and Af
i+1Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 37: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/37.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
tAf
3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse tthese may be the same arc
u Now think aboutAi+1 and Af
i+1
Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 38: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/38.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
tAf
3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse tthese may be the same arc
u Now think aboutAi+1 and Af
i+1Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 39: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/39.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
tAf
3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse tthese may be the same arc
u Now think aboutAi+1 and Af
i+1Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 40: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/40.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
tAf
3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse tthese may be the same arc
u Now think aboutAi+1 and Af
i+1Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 41: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/41.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
tAf
3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse tthese may be the same arc
u Now think aboutAi+1 and Af
i+1Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 42: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/42.jpg)
Proving Property 3
1 ≤ i ≤ l− 2
tAf
3
A3
A1
A2
A 3=Af3
A4
Af4
|{A2, A3, ..., Ai, Afi }| = i+ 1
There is a circular cover of size ≤ l− 1
Contradiction!
Suppose Fi−1 is properlycolored with i colors. Let:
u Ai the first to getcolor i in round i− 1
u Afi the last of round
i− 1 to traverse tthese may be the same arc
u Now think aboutAi+1 and Af
i+1Same color except...
In this case:A2, A3, ..., Ai, A
fi
cover the circle!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 11/15
![Page 43: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/43.jpg)
Recap
Property 2. L(F \ Fl−2) ≤ L− (l − 2)
Property 3. Fl−2 are properly colored with l − 1 colors.
But how to use some induction here?
Maintain little arcs (p, p+ 1) to create a constant load of L(F )around the circle. Neither χ(F ) nor the Algorithm’s output ischanged! *Now check again Property 2!
Now: F ′ = F \ Fl−2 has lF ′ ≥ lF≥ 5, so use induction!...
u L rounds, every l− 2 rounds need at most l− 1 colors
u ⇒⌈l−1l−2
L⌉
is the output of the algorithm.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 12/15
![Page 44: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/44.jpg)
Recap
Property 2. L(F \ Fl−2) ≤ L− (l − 2)
Property 3. Fl−2 are properly colored with l − 1 colors.
But how to use some induction here?
Maintain little arcs (p, p+ 1) to create a constant load of L(F )around the circle. Neither χ(F ) nor the Algorithm’s output ischanged! *Now check again Property 2!
Now: F ′ = F \ Fl−2 has lF ′ ≥ lF≥ 5, so use induction!...
u L rounds, every l− 2 rounds need at most l− 1 colors
u ⇒⌈l−1l−2
L⌉
is the output of the algorithm.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 12/15
![Page 45: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/45.jpg)
Recap
Property 2. L(F \ Fl−2) ≤ L− (l − 2)
Property 3. Fl−2 are properly colored with l − 1 colors.
But how to use some induction here?
Maintain little arcs (p, p+ 1) to create a constant load of L(F )around the circle. Neither χ(F ) nor the Algorithm’s output ischanged! *Now check again Property 2!
Now: F ′ = F \ Fl−2 has lF ′ ≥ lF≥ 5, so use induction!...
u L rounds, every l− 2 rounds need at most l− 1 colors
u ⇒⌈l−1l−2
L⌉
is the output of the algorithm.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 12/15
![Page 46: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/46.jpg)
Recap
Property 2. L(F \ Fl−2) ≤ L− (l − 2)
Property 3. Fl−2 are properly colored with l − 1 colors.
But how to use some induction here?
Maintain little arcs (p, p+ 1) to create a constant load of L(F )around the circle. Neither χ(F ) nor the Algorithm’s output ischanged! *Now check again Property 2!
Now: F ′ = F \ Fl−2 has lF ′ ≥ lF≥ 5, so use induction!...
u L rounds, every l− 2 rounds need at most l− 1 colors
u ⇒⌈l−1l−2
L⌉
is the output of the algorithm.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 12/15
![Page 47: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/47.jpg)
Recap
Property 2. L(F \ Fl−2) ≤ L− (l − 2)
Property 3. Fl−2 are properly colored with l − 1 colors.
But how to use some induction here?
Maintain little arcs (p, p+ 1) to create a constant load of L(F )around the circle. Neither χ(F ) nor the Algorithm’s output ischanged! *Now check again Property 2!
Now: F ′ = F \ Fl−2 has lF ′ ≥ lF≥ 5, so use induction!...
u L rounds, every l− 2 rounds need at most l− 1 colors
u ⇒⌈l−1l−2
L⌉
is the output of the algorithm.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 12/15
![Page 48: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/48.jpg)
Recap
Property 2. L(F \ Fl−2) ≤ L− (l − 2)
Property 3. Fl−2 are properly colored with l − 1 colors.
But how to use some induction here?
Maintain little arcs (p, p+ 1) to create a constant load of L(F )around the circle. Neither χ(F ) nor the Algorithm’s output ischanged! *Now check again Property 2!
Now: F ′ = F \ Fl−2 has lF ′ ≥ lF≥ 5, so use induction!...
u L rounds, every l− 2 rounds need at most l− 1 colors
u ⇒⌈l−1l−2
L⌉
is the output of the algorithm.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 12/15
![Page 49: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/49.jpg)
Recap
Property 2. L(F \ Fl−2) ≤ L− (l − 2)
Property 3. Fl−2 are properly colored with l − 1 colors.
But how to use some induction here?
Maintain little arcs (p, p+ 1) to create a constant load of L(F )around the circle. Neither χ(F ) nor the Algorithm’s output ischanged! *Now check again Property 2!
Now: F ′ = F \ Fl−2 has lF ′ ≥ lF≥ 5, so use induction!...
u L rounds, every l− 2 rounds need at most l− 1 colors
u ⇒⌈l−1l−2
L⌉
is the output of the algorithm.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 12/15
![Page 50: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/50.jpg)
Some final interaction
u What if every arc in F spans at most n/k “points” of the circle?
l ≥ k+ 1⇒ SOL =⌈
kk−1
L⌉
u What if every arc spans more than a semi-circle?
The circular arc graph is complete!
u To show tightness, Valencia-Pabon uses a result of Stahl, [3] regardingr-tuple colorings...
Major open problem: better than 32 -approximation?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 13/15
![Page 51: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/51.jpg)
Some final interaction
u What if every arc in F spans at most n/k “points” of the circle?
l ≥ k+ 1⇒ SOL =⌈
kk−1
L⌉
u What if every arc spans more than a semi-circle?
The circular arc graph is complete!
u To show tightness, Valencia-Pabon uses a result of Stahl, [3] regardingr-tuple colorings...
Major open problem: better than 32 -approximation?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 13/15
![Page 52: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/52.jpg)
Some final interaction
u What if every arc in F spans at most n/k “points” of the circle?
l ≥ k+ 1⇒ SOL =⌈
kk−1
L⌉
u What if every arc spans more than a semi-circle?
The circular arc graph is complete!
u To show tightness, Valencia-Pabon uses a result of Stahl, [3] regardingr-tuple colorings...
Major open problem: better than 32 -approximation?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 13/15
![Page 53: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/53.jpg)
Some final interaction
u What if every arc in F spans at most n/k “points” of the circle?
l ≥ k+ 1⇒ SOL =⌈
kk−1
L⌉
u What if every arc spans more than a semi-circle?
The circular arc graph is complete!
u To show tightness, Valencia-Pabon uses a result of Stahl, [3] regardingr-tuple colorings...
Major open problem: better than 32 -approximation?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 13/15
![Page 54: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/54.jpg)
Some final interaction
u What if every arc in F spans at most n/k “points” of the circle?
l ≥ k+ 1⇒ SOL =⌈
kk−1
L⌉
u What if every arc spans more than a semi-circle?
The circular arc graph is complete!
u To show tightness, Valencia-Pabon uses a result of Stahl, [3] regardingr-tuple colorings...
Major open problem: better than 32 -approximation?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 13/15
![Page 55: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/55.jpg)
Some final interaction
u What if every arc in F spans at most n/k “points” of the circle?
l ≥ k+ 1⇒ SOL =⌈
kk−1
L⌉
u What if every arc spans more than a semi-circle?
The circular arc graph is complete!
u To show tightness, Valencia-Pabon uses a result of Stahl, [3] regardingr-tuple colorings...
Major open problem: better than 32 -approximation?
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 13/15
![Page 56: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/56.jpg)
That’s all folks!
Thank you!
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• The end 14/15
![Page 57: Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex Angelopoulos Q 8 July 22, 2014](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b0833d07f8b9a51508b86ec/html5/thumbnails/57.jpg)
Bibliography I
[1] M. R. Garey, D. S. Johnson, G. L. Miller, and C. H.Papadimitriou. The complexity of coloring circular arcs andchords. SIAM J. Alg. Disc. Meth., 1(2):216–227, June 1980.
[2] I. Karapetian. Coloring of arc graphs. Akad. Nauk Armyan. SSRDokl., 70:306–311, 1980.
[3] S. Stahl. n-tuple colorings and associated graphs. Journal ofComb. Theory, Series B, 20(2):185 – 203, 1976.
[4] A. Tucker. Coloring a family of circular arcs. SIAM J. Appl.Math., 29:493–502, 1975.
[5] M. Valencia-Pabon. Revisiting tucker’s algorithm to color circulararc graphs. SIAM J. Comput., 32(4):1067–1072, 2003.
Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• The end 15/15