Approximation Algorithms for Polynomial-Expansion and Low ...r Fat convex objects in Rd with depth...
Transcript of Approximation Algorithms for Polynomial-Expansion and Low ...r Fat convex objects in Rd with depth...
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Approximation Algorithms forPolynomial-Expansion and
Low-Density Graphs
Sariel Har-Peled Kent Quanrud
University of Illinois at Urbana-Champaign
October 19, 2015
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Gameplan1. Pregame
- definition of problems - a hardness result
2. Low-density objects and graphs- basic properties - overview of results
3. Polynomial expansion- basic properties - overview of results
halftime!4. Two proofs
- independent set - dominating set
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Fat objects
f
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Fat objects
f
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Fat objects
f b
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Fat objects
f b
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Fat objects
vol(b ∩ f )vol(b)
= Ω(1)
f b
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Hitting set
Input: Set of points P , fat objects FOutput: The smallest cardinality subset ofP that pierces every object in F .
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Hitting set
Input: Set of points P , fat objects FOutput: The smallest cardinality subset ofP that pierces every object in F .
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Set cover
Input: Set of points P , fat objects FOutput: Find the smallest cardinalitysubset of F that covers every point in P .
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Set cover
Input: Set of points P , fat objects FOutput: Find the smallest cardinalitysubset of F that covers every point in P .
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Disks and Pseudo-disks
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Disks and Pseudo-disksSet Cover
NP-Hard Feder and Greene 1988
PTAS Mustafa, Raman and Ray 2014
Hitting Set
NP-Hard Feder and Greene 1988
PTAS Mustafa and Ray 2010
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Fat objects
Set cover
O(1) approx. for fat triangles of same sizeClarkson and Varadarajan 2007
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Fat objects
Set cover
O(log∗OPT) for fat objects in R2
Aronov, de Berg, Ezra and Sharir 2014
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Fat objects
Hitting set
O(log log OPT) for fat triangles of similarsize
Aronov, Ezra and Sharir 2010
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Fat, nearly equilateral triangles
Set coverI APX-Hard
Hitting setI APX-Hard
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Fat, nearly equilateral trianglesSet coverI APX-Hard
Hitting setI APX-Hard
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Fat, nearly equilateral trianglesSet coverI APX-Hard
Hitting setI APX-Hard
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2 3
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Fat, nearly equilateral trianglesSet coverI APX-Hard
Hitting setI APX-Hard
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2 3
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Fat, nearly equilateral trianglesSet coverI APX-Hard
Hitting setI APX-Hard
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Fat, nearly equilateral trianglesSet coverI APX-Hard
Hitting setI APX-Hard
1
2 3
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Fat, nearly equilateral trianglesSet coverI APX-Hard
Hitting setI APX-Hard
1
2 3
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2
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1
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Fat, nearly equilateral trianglesSet coverI APX-Hard
Hitting setI APX-Hard
1
2 3
4
56
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Fat, nearly equilateral trianglesSet coverI APX-Hard
Hitting setI APX-Hard
1
2 3
4
56
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Density
F has density ρ if any ball intersects ≤ ρ
objects with larger diameter.van der Stappen, Overmars, de Berg, Vleugels, 1998
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Density
F has density ρ if any ball intersects ≤ ρ
objects with larger diameter.van der Stappen, Overmars, de Berg, Vleugels, 1998
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Density
F has density ρ if any ball intersects ≤ ρ
objects with larger diameter.van der Stappen, Overmars, de Berg, Vleugels, 1998
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Density
F has density ρ if any ball intersects ≤ ρ
objects with larger diameter.van der Stappen, Overmars, de Berg, Vleugels, 1998
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Density
F has low density if ρ = O(1).van der Stappen, Overmars, de Berg, Vleugels, 1998
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Intersection graphs
F induces an intersection graph GFwith objects as vertices and edgesrepresenting overlap.
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Intersection graphs
F induces an intersection graph GFwith objects as vertices and edgesrepresenting overlap.
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Intersection graphs
F induces an intersection graph GFwith objects as vertices and edgesrepresenting overlap.
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Intersection graphs
A graph has low-density if induced bylow-density objects.
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Examples of low density
Interior disjoint disks have O(1) density.
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Examples of low density
Planar graphs have O(1)-density via CirclePacking Theorem.
Koebe, Andreev, Thurston
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Examples of low density
Planar graphs have O(1)-density via CirclePacking Theorem.
Koebe, Andreev, Thurston
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Examples of low density
Fat convex objects in Rd with depth k havedensity O(k2d).
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Examples of low density
r
Fat convex objects in Rd with depth k havedensity O(k2d).
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Examples of low density
r
Fat convex objects in Rd with depth k havedensity O(k2d).
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Examples of low density
r
Fat convex objects in Rd with depth k havedensity O(k2d).
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Examples of low density
rr
Fat convex objects in Rd with depth k havedensity O(k2d).
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Examples of low density
rr
Fat convex objects in Rd with depth k havedensity O(k2d).
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Examples of low density
rrr
Fat convex objects in Rd with depth k havedensity O(k2d).
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Additivity
red density = ρ1
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Additivity
blue density = ρ2
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Additivity
red density = ρ1blue density = ρ2
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Additivity
red density = ρ1blue density = ρ2+
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Additivity
red density = ρ1blue density = ρ2
total density ≤ ρ1 + ρ2
+
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Degeneracy
If F has density ρ, then the smallest objectintersects at most ρ− 1 other objects.
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Degeneracy
If F has density ρ, then the smallest objectintersects at most ρ− 1 other objects.
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Degeneracy
If F has density ρ, then the smallest objectintersects at most ρ− 1 other objects.
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SeparatorsFor k ≤ |F|, compute a sphere S thatI Strictly contains at least k − o(k)objects and at most k objects.
I Intersects O(ρ + ρ1/dk1−1/d) objects.
cR
r
3r
Miller, Teng, Thurston, and Vavasis, 1997;Smith and Wormald, 1998; Chan 2003
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Graph minors
A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.
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Minors of objects
G is a minor of F if it can be obtained bydeleting objects and taking unions ofoverlapping of objects.
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Minors of objects
G is a minor of F if it can be obtained bydeleting objects and taking unions ofoverlapping of objects.
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Minors of objects
G is a minor of F if it can be obtained bydeleting objects and taking unions ofoverlapping of objects.
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Minors of objects
G is a minor of F if it can be obtained bydeleting objects and taking unions ofoverlapping of objects.
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Minors of objects
G is a minor of F if it can be obtained bydeleting objects and taking unions ofoverlapping of objects.
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Minors of objects
G is a minor of F if it can be obtained bydeleting objects and taking unions ofoverlapping of objects.
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Large clique minors
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Shallow minors
A vertex in H corresponds to a connectedcluster of vertices in G.
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Shallow minors
H is a t-shallow minor if each clusterinduces a graph of radius t.
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Shallow minors
H is a 1-shallow minor if each clusterinduces a graph of radius 1.
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Shallow minors
H is a 2-shallow minor if each clusterinduces a graph of radius 2.
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Shallow minors of objects
Each object in the minor corresponds to acluster of objects in F .
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Shallow minors of objects
An object minor is a t-shallow minor ifthe intersection graph of each cluster hasradius ≤ t.
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Shallow minors of objects
An object minor is a 1-shallow minor ifthe intersection graph of each cluster hasradius ≤ 1.
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Shallow minors of objects
An object minor is a 2-shallow minor ifthe intersection graph of each cluster hasradius ≤ 2.
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Shallow minors of objects
An object minor is a 3-shallow minor ifthe intersection graph of each cluster hasradius ≤ 3.
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Shallow minors oflow-density objects
A t-shallow minor of objects withdensity ρ has density O(tO(d)ρ)
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Shallow minors oflow-density objects
A t-shallow minor of objects withdensity ρ has density O(tO(d)ρ)
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Shallow minors oflow-density objects
A t-shallow minor of objects withdensity ρ has density O(tO(d)ρ)
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Shallow minors oflow-density objects
A t-shallow minor of objects withdensity ρ has density O(tO(d)ρ)
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Shallow minors oflow-density objects
A t-shallow minor of objects withdensity ρ has density O(tO(d)ρ)
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Recap: low-density graphs are...
red density = ρ1blue density = ρ2
total density ≤ ρ1 + ρ2
+
(a) additive (b) degenerate (hence sparse)
cR
r
3r
(c) separable (d) kind of closed undershallow minors
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Main result: low-density
ρ = O(1): PTAS for hitting set, set cover,subset dominating set
ρ = polylog(n): QPTAS for same problems.No PTAS under ETH.
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Main result: low-density
density ρ O(1) polylog(n) unbounded
hardness NP-Hard No PTAS APX-Hard
algo PTAS QPTAS
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Main result: fat triangles
depthdensity ρ
O(1) polylog(n) unbounded
hardness NP-Hard No PTAS APX-Hard
algo PTAS QPTAS
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Shallow edge densityThe r-shallow density of a graph is themax edge density over all r-shallow minors.
aka “greatest reduced average density”Nešetřil and Ossona de Mendez, 2008
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Sparsity is not enough
Hide a clique by splitting the edges
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Sparsity is not enough
Hide a clique by splitting the edges
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ExpansionThe expansion of a graph is the r-shallowdensity as a function of r.
e.g. constant expansion, polynomialexpansion, exponential expansion
Nešetřil and Ossona de Mendez, 2008
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Examples of expansion
Planar graphs have constant expansion(Euler’s formula)
Minor-closed classes have constant expansion
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Sparsity is not enough
Constant degree expanders have exponentialexpansion
Wikipedia
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Low density ⇒polynomial expansion
Graphs with density ρ have polynomialexpansion f (r) = O(ρrd)
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Low density ⇒polynomial expansion
Graphs with density ρ have polynomialexpansion f (r) = O(ρrd)
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expansion examples
constantplanar graphs,
minor-closed families
polynomial low-density graphs
exponential expander graphs
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Recap: low-density graphs are...
red density = ρ1blue density = ρ2
total density ≤ ρ1 + ρ2
+
(a) additive (b) degenerate (hence sparse)
cR
r
3r
(c) separable (d) kind of closed undershallow minors
![Page 112: Approximation Algorithms for Polynomial-Expansion and Low ...r Fat convex objects in Rd with depth khave density O(k2d). Examples of low density r r Fat convex objects in Rd with depth](https://reader033.fdocuments.in/reader033/viewer/2022060905/60a02412adc81b1adc1c7181/html5/thumbnails/112.jpg)
Lexical product G •Kh
The lexical product G •Kh blows upeach vertex in G with the clique Kh.
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Lexical product G •Kh
V (G •K4) = V (G)× V (K4)
The lexical product G •Kh blows upeach vertex in G with the clique Kh.
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Lexical product G •Kh
V (G •K4) = V (G)× V (K4)
E(G •K4) =((a1, b1), (a2, b2))|a1 = a2))
The lexical product G •Kh blows upeach vertex in G with the clique Kh.
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Lexical product G •Kh
V (G •K4) = V (G)× V (K4)
E(G •K4) =((a1, b1), (a2, b2))|a1 = a2))
((a1, b1), (a2, b2))|(a1, a2) ∈ E(G)∪
The lexical product G •Kh blows upeach vertex in G with the clique Kh.
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Lexical product G •Kh
V (G •K4) = V (G)× V (K4)
E(G •K4) =((a1, b1), (a2, b2))|a1 = a2))
((a1, b1), (a2, b2))|(a1, a2) ∈ E(G)∪
If G has polynomial expansion, then G •Kh
has polynomial expansion.
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Small separatorsGraphs with subexponential expansion havesublinear separators.
Nešetřil and Ossona de Mendez (2008)
Why?
1. No Kh as an `-shallow minor implies aseparator of size O(n/` + 4`h2 log n).
Plotkin, Rao, and Smith (1994)
2. Small expansion => small clique minorsas a function of depth
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Small separatorsGraphs with subexponential expansion havesublinear separators.
Nešetřil and Ossona de Mendez (2008)
Why?
1. No Kh as an `-shallow minor implies aseparator of size O(n/` + 4`h2 log n).
Plotkin, Rao, and Smith (1994)
2. Small expansion => small clique minorsas a function of depth
![Page 119: Approximation Algorithms for Polynomial-Expansion and Low ...r Fat convex objects in Rd with depth khave density O(k2d). Examples of low density r r Fat convex objects in Rd with depth](https://reader033.fdocuments.in/reader033/viewer/2022060905/60a02412adc81b1adc1c7181/html5/thumbnails/119.jpg)
Small separatorsGraphs with subexponential expansion havesublinear separators.
Nešetřil and Ossona de Mendez (2008)
Why?
1. No Kh as an `-shallow minor implies aseparator of size O(n/` + 4`h2 log n).
Plotkin, Rao, and Smith (1994)
2. Small expansion => small clique minorsas a function of depth
![Page 120: Approximation Algorithms for Polynomial-Expansion and Low ...r Fat convex objects in Rd with depth khave density O(k2d). Examples of low density r r Fat convex objects in Rd with depth](https://reader033.fdocuments.in/reader033/viewer/2022060905/60a02412adc81b1adc1c7181/html5/thumbnails/120.jpg)
Small separatorsGraphs with subexponential expansion havesublinear separators.
Nešetřil and Ossona de Mendez (2008)
Why?
1. No Kh as an `-shallow minor implies aseparator of size O(n/` + 4`h2 log n).
Plotkin, Rao, and Smith (1994)
2. Small expansion => small clique minorsas a function of depth
![Page 121: Approximation Algorithms for Polynomial-Expansion and Low ...r Fat convex objects in Rd with depth khave density O(k2d). Examples of low density r r Fat convex objects in Rd with depth](https://reader033.fdocuments.in/reader033/viewer/2022060905/60a02412adc81b1adc1c7181/html5/thumbnails/121.jpg)
Shallow minors of polynomialexpansion
Shallow minors of graphs with polynomialexpansion have polynomial expansion.
(If H is an r1-shallow minor of G, then anr2-shallow minor of H is an (r1 · r2)-shallowminor of G, and poly(r1 · r2) = poly(r2).)
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Recap: polynomial expansion graphs are...
V (G •K4) = V (G)× V (K4)
E(G •K4) =((a1, b1), (a2, b2))|a1 = a2))
((a1, b1), (a2, b2))|(a1, a2) ∈ E(G)∪
FREE
(a) closed under lexicalproduct (b) degenerate
via separator for excludedshallow minors FREE
(c) separable (d) kind of closed undershallow minors
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Main result:polynomial expansion
I Graph G with polynomial expansionI PTAS for (subset) dominating setI Extensions: multiple demands, reach,connected dominating set, vertex cover.
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PTAS for independent set
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Recap: low-density graphs are...
red density = ρ1blue density = ρ2
total density ≤ ρ1 + ρ2
+
(a) additive (b) degenerate (hence sparse)
cR
r
3r
(c) separable (d) kind of closed undershallow minors
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Balanced separators
Input: G = (V,E) with n vertices.
Output: Partition V = X t S t Y s.t.(a) |X|, |Y | ≤ .99n.(b) |S| ≤ n.99.(c) No edges run between X and Y .
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Balanced separators
Input: G = (V,E) with n vertices.
Output: Partition V = X t S t Y s.t.(a) |X|, |Y | ≤ .99n.(b) |S| ≤ n.99.(c) No edges run between X and Y .
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Balanced separators
Input: G = (V,E) with n vertices.
Output: Partition V = X t S t Y s.t.(a) |X|, |Y | ≤ .99n.(b) |S| ≤ n.99.(c) No edges run between X and Y .
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Well-separated divisions
Input: G = (V,E) w/ n vertices, ε ∈ (0, 1)
Output: Cover V = C1∪C2∪ · · ·∪Cm s.t.(a) |Ci| = poly(1/ε) for all i(b) No edges between Ci \ Cj and Cj \ Ci(c)∑
i|Ci| ≤ (1 + ε)n
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Well-separated divisions
Input: G = (V,E) w/ n vertices, ε ∈ (0, 1)
Output: Cover V = C1∪C2∪ · · ·∪Cm s.t.(a) |Ci| = poly(1/ε) for all i(b) No edges between Ci \ Cj and Cj \ Ci(c)∑
i|Ci| ≤ (1 + ε)n
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Well-separated divisions
Input: G = (V,E) w/ n vertices, ε ∈ (0, 1)
Output: Cover V = C1∪C2∪ · · ·∪Cm s.t.(a) |Ci| = poly(1/ε) for all i(b) No edges between Ci \ Cj and Cj \ Ci(c)∑
i|Ci| ≤ (1 + ε)n
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Well-separated divisions
Input: G = (V,E) w/ n vertices, ε ∈ (0, 1)
Output: Cover V = C1∪C2∪ · · ·∪Cm s.t.(a) |Ci| = poly(1/ε) for all i(b) No edges between Ci \ Cj and Cj \ Ci(c)∑
i|Ci| ≤ (1 + ε)n
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Separators ⇒ divisions
⇓
Frederickson (1987)
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Local search
local-search(G = (V,E), ε)L← ∅, λ← poly(1/ε)
while there exists S ⊆ V s.t.(a) |S| ≤ λ
(b) L4S is an independent set(c) |L4S| > |L|
do L← L4Send whilereturn L
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Independent set: proof setupO: Optimal solutionL: λ-locally optimal sol’n for λ = poly(1/ε)
O t L has low density (by additivity)
C1, . . . , Cm: w.s.-division of O t L
Bi = Ci ∩(⋃
j 6=iCj), bi = |Bi|
Oi = (O ∩ Ci) \Bi, oi = |Oi|Li = (L ∩ Ci), ` = |Li|,
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Independent set: proof setupO: Optimal solutionL: λ-locally optimal sol’n for λ = poly(1/ε)
O t L has low density (by additivity)
C1, . . . , Cm: w.s.-division of O t L
Bi = Ci ∩(⋃
j 6=iCj), bi = |Bi|
Oi = (O ∩ Ci) \Bi, oi = |Oi|Li = (L ∩ Ci), ` = |Li|,
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Independent set: proof setupO: Optimal solutionL: λ-locally optimal sol’n for λ = poly(1/ε)
O t L has low density (by additivity)
C1, . . . , Cm: w.s.-division of O t L
Bi = Ci ∩(⋃
j 6=iCj), bi = |Bi|
Oi = (O ∩ Ci) \Bi, oi = |Oi|Li = (L ∩ Ci), ` = |Li|,
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Independent set: proof setupO: Optimal solutionL: λ-locally optimal sol’n for λ = poly(1/ε)
O t L has low density (by additivity)
C1, . . . , Cm: w.s.-division of O t L
Bi = Ci ∩(⋃
j 6=iCj), bi = |Bi|
Oi = (O ∩ Ci) \Bi, oi = |Oi|Li = (L ∩ Ci), ` = |Li|,
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Independent set: proof setupO: Optimal solutionL: λ-locally optimal sol’n for λ = poly(1/ε)
O t L has low density (by additivity)
C1, . . . , Cm: w.s.-division of O t L
Bi = Ci ∩(⋃
j 6=iCj), bi = |Bi|
Oi = (O ∩ Ci) \Bi, oi = |Oi|Li = (L ∩ Ci), ` = |Li|,
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PTAS for dominating set
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Recap: low-density graphs are...
red density = ρ1blue density = ρ2
total density ≤ ρ1 + ρ2
+
(a) additive (b) degenerate (hence sparse)
cR
r
3r
(c) separable (d) kind of closed undershallow minors
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Local search
local-search(G = (V,E), ε)L← V , λ← poly(1/ε)
while there exists S ⊆ V s.t.(a) |S| ≤ λ
(b) L4S is a dominating set(c) |L4S| < |L|
do L← L4Send whilereturn L
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Flowers
Glue an object in the dominating set withthe neighboring objects it dominates.
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Flowers
Head
Petal
Glue an object in the dominating set withthe neighboring objects it dominates.
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Flowers
Glue an object in the dominating set withthe neighboring objects it dominates.
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Flowers
Glue an object in the dominating set withthe neighboring objects it dominates.
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Dominating set: proof setupO = optimal solution, O = flowers of OL = locally-optimal sol’n, L = flowers of L
O ∪ L has low densityC1, . . . , Cm
: w.s.-division of O ∪ L
C1, . . . , Cm: centers of Ci for each iBi = Ci ∩
(⋃j 6=iCj
)Oi = O ∩ Ci, oi = |Oi|Li = L ∩ Ci, `i = |Li|
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Dominating set: proof setupO = optimal solution, O = flowers of OL = locally-optimal sol’n, L = flowers of L
O ∪ L has low density
C1, . . . , Cm
: w.s.-division of O ∪ L
C1, . . . , Cm: centers of Ci for each iBi = Ci ∩
(⋃j 6=iCj
)Oi = O ∩ Ci, oi = |Oi|Li = L ∩ Ci, `i = |Li|
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Dominating set: proof setupO = optimal solution, O = flowers of OL = locally-optimal sol’n, L = flowers of L
O ∪ L has low densityC1, . . . , Cm
: w.s.-division of O ∪ L
C1, . . . , Cm: centers of Ci for each iBi = Ci ∩
(⋃j 6=iCj
)Oi = O ∩ Ci, oi = |Oi|Li = L ∩ Ci, `i = |Li|
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Dominating set: proof setupO = optimal solution, O = flowers of OL = locally-optimal sol’n, L = flowers of L
O ∪ L has low densityC1, . . . , Cm
: w.s.-division of O ∪ L
C1, . . . , Cm: centers of Ci for each i
Bi = Ci ∩(⋃
j 6=iCj)
Oi = O ∩ Ci, oi = |Oi|Li = L ∩ Ci, `i = |Li|
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Dominating set: proof setupO = optimal solution, O = flowers of OL = locally-optimal sol’n, L = flowers of L
O ∪ L has low densityC1, . . . , Cm
: w.s.-division of O ∪ L
C1, . . . , Cm: centers of Ci for each iBi = Ci ∩
(⋃j 6=iCj
)Oi = O ∩ Ci, oi = |Oi|Li = L ∩ Ci, `i = |Li|
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Dominating set: proof setupO = optimal solution, O = flowers of OL = locally-optimal sol’n, L = flowers of L
O ∪ L has low densityC1, . . . , Cm
: w.s.-division of O ∪ L
C1, . . . , Cm: centers of Ci for each iBi = Ci ∩
(⋃j 6=iCj
)Oi = O ∩ Ci, oi = |Oi|Li = L ∩ Ci, `i = |Li|
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thanks!
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References I
G. N. Frederickson. Fast algorithms forshortest paths in planar graphs, withapplications. SIAM J. Comput., 16(6):1004–1022, 1987.
J. Nešetřil and P. Ossona de Mendez. Gradand classes with bounded expansion ii.algorithmic aspects. European J. Combin.,29(3):777–791, 2008.
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References II
S. Plotkin, S. Rao, and W.D. Smith. Shallowexcluded minors and improved graphdecompositions. In Proc. 5th ACM-SIAMSympos. Discrete Algs. (SODA), pages462–470, 1994.