Approximation Algorithms for Polynomial-Expansion and Low ...r Fat convex objects in Rd with depth...

Approximation Algorithms forPolynomial-Expansion and

Low-Density Graphs

Sariel Har-Peled Kent Quanrud

University of Illinois at Urbana-Champaign

October 19, 2015

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

Fat objects

vol(b ∩ f )vol(b)

= Ω(1)

Hitting set

Input: Set of points P , fat objects FOutput: The smallest cardinality subset ofP that pierces every object in F .

Hitting set

Input: Set of points P , fat objects FOutput: The smallest cardinality subset ofP that pierces every object in F .

Set cover

Input: Set of points P , fat objects FOutput: Find the smallest cardinalitysubset of F that covers every point in P .

Set cover

Input: Set of points P , fat objects FOutput: Find the smallest cardinalitysubset of F that covers every point in P .

Disks and Pseudo-disks

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

Fat objects

Set cover

O(1) approx. for fat triangles of same sizeClarkson and Varadarajan 2007

Fat objects

Set cover

O(log∗OPT) for fat objects in R2

Aronov, de Berg, Ezra and Sharir 2014

Fat objects

Hitting set

O(log log OPT) for fat triangles of similarsize

Aronov, Ezra and Sharir 2010

Fat, nearly equilateral triangles

Set coverI APX-Hard

Hitting setI APX-Hard

Fat, nearly equilateral trianglesSet coverI APX-Hard

Density

F has density ρ if any ball intersects ≤ ρ

objects with larger diameter.van der Stappen, Overmars, de Berg, Vleugels, 1998

Density

F has low density if ρ = O(1).van der Stappen, Overmars, de Berg, Vleugels, 1998

Intersection graphs

F induces an intersection graph GFwith objects as vertices and edgesrepresenting overlap.

Intersection graphs

A graph has low-density if induced bylow-density objects.

Examples of low density

Interior disjoint disks have O(1) density.

Planar graphs have O(1)-density via CirclePacking Theorem.

Koebe, Andreev, Thurston

Planar graphs have O(1)-density via CirclePacking Theorem.

Koebe, Andreev, Thurston

Fat convex objects in Rd with depth k havedensity O(k2d).

Additivity

red density = ρ1

Additivity

blue density = ρ2

Additivity

red density = ρ1blue density = ρ2

Additivity

red density = ρ1blue density = ρ2+

Additivity

total density ≤ ρ1 + ρ2

Degeneracy

If F has density ρ, then the smallest objectintersects at most ρ− 1 other objects.

Degeneracy

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.

Miller, Teng, Thurston, and Vavasis, 1997;Smith and Wormald, 1998; Chan 2003

Graph minors

A minor of G is a graph H obtained bycontracting edges, deleting edges, anddeleting vertices.

Graph minors

Minors of objects

G is a minor of F if it can be obtained bydeleting objects and taking unions ofoverlapping of objects.

Minors of objects

Large clique minors

Shallow minors

A vertex in H corresponds to a connectedcluster of vertices in G.

Shallow minors

H is a t-shallow minor if each clusterinduces a graph of radius t.

Shallow minors

H is a 1-shallow minor if each clusterinduces a graph of radius 1.

Shallow minors

H is a 2-shallow minor if each clusterinduces a graph of radius 2.

Shallow minors of objects

Each object in the minor corresponds to acluster of objects in F .

An object minor is a t-shallow minor ifthe intersection graph of each cluster hasradius ≤ t.

An object minor is a 1-shallow minor ifthe intersection graph of each cluster hasradius ≤ 1.

Shallow minors oflow-density objects

A t-shallow minor of objects withdensity ρ has density O(tO(d)ρ)

Recap: low-density graphs are...

(a) additive (b) degenerate (hence sparse)

(c) separable (d) kind of closed undershallow minors

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.

Main result: low-density

density ρ O(1) polylog(n) unbounded

hardness NP-Hard No PTAS APX-Hard

algo PTAS QPTAS

Main result: fat triangles

depthdensity ρ

O(1) polylog(n) unbounded

hardness NP-Hard No PTAS APX-Hard

algo PTAS QPTAS

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

Sparsity is not enough

Hide a clique by splitting the edges

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

Examples of expansion

Planar graphs have constant expansion(Euler’s formula)

Minor-closed classes have constant expansion

Constant degree expanders have exponentialexpansion

Wikipedia

Low density ⇒polynomial expansion

Graphs with density ρ have polynomialexpansion f (r) = O(ρrd)

Low density ⇒polynomial expansion

Graphs with density ρ have polynomialexpansion f (r) = O(ρrd)

expansion examples

constantplanar graphs,

minor-closed families

polynomial low-density graphs

exponential expander graphs

Lexical product G •Kh

The lexical product G •Kh blows upeach vertex in G with the clique Kh.

V (G •K4) = V (G)× V (K4)

E(G •K4) =((a1, b1), (a2, b2))|a1 = a2))

V (G •K4) = V (G)× V (K4)

E(G •K4) =((a1, b1), (a2, b2))|a1 = a2))

((a1, b1), (a2, b2))|(a1, a2) ∈ E(G)∪

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.

Small separatorsGraphs with subexponential expansion havesublinear separators.

Nešetřil and Ossona de Mendez (2008)

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

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).)

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)∪

(a) closed under lexicalproduct (b) degenerate

via separator for excludedshallow minors FREE

Main result:polynomial expansion

I Graph G with polynomial expansionI PTAS for (subset) dominating setI Extensions: multiple demands, reach,connected dominating set, vertex cover.

PTAS for independent set

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 .

Balanced separators

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

Separators ⇒ divisions

Frederickson (1987)

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

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|,

Bi = Ci ∩(⋃

j 6=iCj), bi = |Bi|

Bi = Ci ∩(⋃

j 6=iCj), bi = |Bi|

Bi = Ci ∩(⋃

j 6=iCj), bi = |Bi|

Bi = Ci ∩(⋃

j 6=iCj), bi = |Bi|

PTAS for dominating set

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

Flowers

Glue an object in the dominating set withthe neighboring objects it dominates.

Flowers

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|

O ∪ L has low density

C1, . . . , Cm

(⋃j 6=iCj

C1, . . . , Cm: centers of Ci for each i

Bi = Ci ∩(⋃

j 6=iCj)

Oi = O ∩ Ci, oi = |Oi|Li = L ∩ Ci, `i = |Li|

(⋃j 6=iCj

thanks!

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.

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.

Approximation Algorithms for Polynomial-Expansion and Low ...r Fat convex objects in Rd with depth...

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