Minors & Algorithms

ErikDemaine

M.I.T.

Minors

• 𝐻𝐻 is a minor of 𝐺𝐺 if𝐺𝐺 can reach 𝐻𝐻 via edge deletions edge contractions

v w

v wdelete

vwcontract

𝐾𝐾3,3 deletecontract

is a minor of

Graph Minors [Robertson & Seymour 1983–]

Wagner’s Conjecture[Robertson & Seymour 2004]• Minor relation is a well-quasi-ordering:

Every infinite graph sequence 𝐺𝐺1,𝐺𝐺2,𝐺𝐺3, …has some 𝐺𝐺𝑖𝑖 a minor of 𝐺𝐺𝑗𝑗 (with 𝑖𝑖 < 𝑗𝑗)

…or equivalently…• Every minor-closed graph family is

characterized by a finite set of excluded minors Proof by contradiction: Infinite excluded minors

would have a minor relation among them In some sense necessarily nonconstructive

[Friedman, Robertson, Seymour 1987]

Wagner’s Conjecture[Robertson & Seymour 2004]• Every minor-closed graph family is

characterized by a finite set of excluded minors• Examples: Forests: 𝐾𝐾3 Outerplanar: 𝐾𝐾4 & 𝐾𝐾2,3

Series-parallel: 𝐾𝐾4 Planar: 𝐾𝐾5 & 𝐾𝐾3,3

[Wagner’s 1937] Linklessly embeddable:

Petersen family [Robertson & Seymour 1995]

Algorithms via Minors

• Theorem: Polynomial test for a fixed minor 𝐻𝐻 𝑓𝑓 𝐻𝐻 𝑛𝑛3 [GM XII, 1995] 𝑓𝑓 𝐻𝐻 𝑛𝑛2 [Kawarabayashi, Kobayashi, Reed 2012]

• Corollary: Any minor-closed graph property has a polynomial-time decision algorithm But algorithm is existential:

in general, don’t know whichexcluded minors to test for

Warning:Graph Minor Constants• Example: Planar minor testing [GM5 1986],

analyzed by David Johnson [1987] Running time = 𝑂𝑂 𝑛𝑛3

Lead constant ≈ 222𝑥𝑥

, 𝑥𝑥 = 222⋅⋅⋅

222⋅⋅⋅

…

”

“

12𝑉𝑉 𝐻𝐻

Fixed-Parameter Algorithmsvia Minors• Graph parameter is minor-closed if it only

decreases when deleting or contracting Examples: vertex cover, feedback vertex set

• Corollary: Any minor-closed graph parameter has an 𝑓𝑓 𝑘𝑘 𝑛𝑛2 algorithm for deciding ≤ 𝑘𝑘 “Fixed-parameter tractable” But a different, unknown algorithm for each 𝑘𝑘 Not really a (uniform) fixed-parameter algorithm[Fellows & Langston 1988]

Fixed-Parameter Algorithmsvia Minors• Theorem: Any minor-closed graph parameter nonzero on some planar graph, at least the sum over connected components in a

disconnected graph, and fixed-parameter tractable with respect to treewidth

can be solved in 𝑓𝑓 𝑘𝑘 𝑛𝑛 by a known algorithm[Demaine & Hajiaghayi 2007]

What Are 𝑯𝑯-minor-free Graphs Like?

• Bounded average degree (& degeneracy):𝑂𝑂 𝑉𝑉 𝐻𝐻 log 𝑉𝑉 𝐻𝐻

[Kostochka 1982; Thomason 1984]

• 𝑂𝑂 𝑛𝑛 treewidth [Grohe 2003]• ⇒ 𝑂𝑂 𝑛𝑛 balanced separators

(like Lipton-Tarjan planar separator theorem)[Alon, Seymour, Thomas 1990]

• Kinda like planar graphs?

𝑯𝑯-Minor-Free Graphs[Robertson & Seymour 2003]

Structure Theorem:• 𝐻𝐻-minor-free graph looks like a tree of

“almost embeddable” graphs: Base graph drawn on

surface of genus 𝑓𝑓(𝐻𝐻) 𝑓𝑓(𝐻𝐻) vortex faces

filled with graphsof pathwidth 𝑓𝑓(𝐻𝐻) 𝑓𝑓(𝐻𝐻) apex vertices

connected to anything

𝑯𝑯-Minor-Free Graphs[Robertson & Seymour 2003]

Structure Theorem:• 𝐻𝐻-minor-free graph looks like a tree of

“almost embeddable” graphs

Structure Algorithms:• 𝑛𝑛𝑓𝑓 𝐻𝐻 [Demaine, Hajiaghayi,

Kawarabayashi — FOCS 2005]

• 𝑂𝑂 𝑛𝑛3 [Kawarabayashi &Wollan — STOC 2011]

• 𝑂𝑂 𝑛𝑛2 [Grohe, Kawarabayashi,Reed — SODA 2013]

Grid Minors vs. Treewidth

• Every 𝐻𝐻-minor-free graph of treewidth ≥𝑓𝑓 𝐻𝐻 𝑟𝑟 has an 𝑟𝑟 × 𝑟𝑟 grid minor[Demaine & Hajiaghayi 2005] Previous bounds exponential in 𝑟𝑟 (and 𝐻𝐻) [GM5, …]

𝑓𝑓 𝐻𝐻 = |𝑉𝑉 𝐻𝐻 |𝑂𝑂 |𝐸𝐸 𝐻𝐻 | [Kawarabayashi & Kobayashi 2012]

• Every graph of treewidth ≥ 𝑐𝑐1 𝑟𝑟𝑐𝑐2 has an 𝑟𝑟 × 𝑟𝑟grid minor [Chekuri & Chuzhoy 2013] 𝑐𝑐2 > 2 necessary [Robertson, Seymour, Thomas 1994]

Previous bounds exponential in 𝑟𝑟 [RST94]

• Almost-embeddable graphs: Remove apices by deletion Contract each vortex to a vertex Pull apart each handle

[Demaine, Fomin, Hajiaghayi, Thilikos 2004]

Planar graph has large grid minor [RST94]

• Tree takes max of treewidths, so onealmost-embeddable graph has full treewidth But may not be 𝐻𝐻-minor-free from tree joins “Approximate” graph to make minor of original

graph ⇒ 𝐻𝐻-minor-free [Demaine & Hajiaghayi 2005]

Grid Minors vs. Treewidthin 𝑯𝑯-Minor-Free Graphs

Graph Minor Hierarchy

𝐻𝐻-minor-free

apex-minor-free

bounded genus

planar

Apex-Minor-Free Graphs

• Bounded local treewidth =radius-𝑟𝑟 neighborhood around every vertex 𝑣𝑣has treewidth ≤ 𝑓𝑓 𝑟𝑟 Examples: planar and bounded-genus graphs

• Minor-closed graph family hasbounded local treewidth ⇔excludes an apex graph =minor of planar graph + one vertex[Eppstein 2000] Call such graphs apex-minor-free

𝑟𝑟 = 1𝑟𝑟 = 2

𝑣𝑣

𝑟𝑟 = 1𝑟𝑟 = 2

𝑣𝑣

Apex-Minor-Free Graphs

• Structure Theorem:Apices attach only to vortices,allowing bounded-treewidth vortices[Demaine, Hajiaghayi, Kawarabayashi 2009] Additive +2 approximation for chromatic number

(extending from bounded genus [Thomassen 1997])• Bounded local treewidth 𝑓𝑓 𝑟𝑟 = 22𝑂𝑂 𝑟𝑟

[Eppstein 2000] 𝑓𝑓 𝑟𝑟 = 2𝑂𝑂 𝑟𝑟 [Demaine & Hajiaghayi 2004a] 𝑓𝑓 𝑟𝑟 = 𝑂𝑂 𝑟𝑟 [Demaine & Hajiaghayi 2004b]

[EPTASs: 222𝑂𝑂 ⁄1 𝜀𝜀

𝑛𝑛𝑂𝑂 1 → 2𝑂𝑂 ⁄1 𝜀𝜀 𝑛𝑛𝑂𝑂 1 ]

odd-𝐻𝐻-minor-free

Beyond Graph Minors

𝐻𝐻-minor-free

apex-minor-free

bounded genus

planar

Minors

• Equivalently, 𝐻𝐻 is a minor of 𝐺𝐺 if Every vertex 𝑣𝑣 in 𝐻𝐻 has a corresp. tree 𝑇𝑇𝑣𝑣 in 𝐺𝐺 Any edge 𝑣𝑣,𝑤𝑤 in 𝐻𝐻 has a corresponding edge

connecting 𝑇𝑇𝑣𝑣 to 𝑇𝑇𝑤𝑤 in 𝐺𝐺

is a minor of

K3,3

v w

v wdelete

vwcontract

deletecontract

“model”

Odd Minors

• 𝐻𝐻 is an odd minor of 𝐺𝐺 if we canalso 2-color the vertices in 𝐺𝐺 so that Tree edges are bichromatic 𝑇𝑇𝑣𝑣-to-𝑇𝑇𝑤𝑤 edges are monochromatic

K3,3

v w

v wdelete

vwcontract

deletecontract

is an oddminor of

Odd Minors

• 𝐻𝐻 is an odd minor of 𝐺𝐺 if we canalso 2-color the vertices in 𝐺𝐺 so that Tree edges are bichromatic 𝑇𝑇𝑣𝑣-to-𝑇𝑇𝑤𝑤 edges are monochromatic

v w

v wdelete

vwcontract

is an oddminor of

K3,3 deletecontract

Odd Minors

• 𝐻𝐻 is an odd minor of 𝐺𝐺 if we canalso 2-color the vertices in 𝐺𝐺 so that Tree edges are bichromatic 𝑇𝑇𝑣𝑣-to-𝑇𝑇𝑤𝑤 edges are monochromatic

K3,3 deletecontract

v w

v wdelete

vwcontract

is an oddminor of

Odd-Minor-Free Graphs[Demaine, Hajiaghayi, Kawarabayashi 2010]

• Structure Theorem: Every odd-𝐻𝐻-minor-free graph can be written as a tree of graphs joined along 𝑓𝑓(𝐻𝐻)-size cliques, where each term is Bounded-genus graph

+ 𝑓𝑓(𝐻𝐻) vortices+ 𝑓𝑓 𝐻𝐻 apices(joined at only 3 surface vertices) Bipartite graph

+ 𝑓𝑓(𝐻𝐻) apices(joined at only 1 bipartite vertex)

• 𝑛𝑛𝑓𝑓 𝐻𝐻 algorithm 𝑓𝑓 𝐻𝐻 𝑛𝑛𝑂𝑂 1 simpler decomp. [Tazari 2012]

any problem satisfying:

Bidimensionality Overview

closed under contraction [& deletion]

OPT costly on 𝑟𝑟 × 𝑟𝑟 grid

minor 𝐻𝐻

subexponential fixed-parameter

algorithm: 2𝑂𝑂 OPT 𝑛𝑛𝑂𝑂 1

efficient PTAS: 𝑓𝑓 𝜀𝜀 𝑛𝑛𝑂𝑂 1 time

linear kernelization: 𝑛𝑛 → 𝑂𝑂(OPT)

… on 𝐻𝐻-minor-free graphs

History of Subexponential FPT

• In the early days, most fixed-parameter algorithms ran in 2𝑂𝑂 𝑘𝑘𝑐𝑐 𝑛𝑛𝑂𝑂 1 time, for 𝑐𝑐 ≥ 1

• Natural question: Is 2𝑜𝑜 𝑘𝑘 𝑛𝑛𝑂𝑂 1 possible?

History of Subexponential FPT

• Vertex Cover: choose smallest set of verticesto cover every edge (on either endpoint)

• Vertex Cover can be kernelized to ≤ 2𝑘𝑘 vertices• Lipton & Tarjan’s separator approach [1980]

solves Planar Independent Set in 2𝑂𝑂 𝑛𝑛 time Min. Vertex Cover + Max. Independent Set = |𝑉𝑉|

• ⇒ 𝑛𝑛𝑂𝑂 1 + 2𝑂𝑂 𝑘𝑘 time

v w v w v w v wcover

History of Subexponential FPT

• What about problems like these? Dominating Set: Cover all vertices

with 𝑘𝑘 vertex neighborhoods Feedback Vertex Set: Cover cycles with 𝑘𝑘 vertices Long Path: Is there a simple path of length ≥ 𝑘𝑘?

• No linear (or even polynomial) kernel was known for these problems then

• We now know that some, e.g. Long Path,have no polynomial kernel (unless NP ⊆

⁄coNP poly), even for planar graphs[Bodlaender, Downey, Fellows, Hermelin 2009]

History of Subexponential FPT

• 2𝑂𝑂 𝑘𝑘 𝑛𝑛𝑂𝑂 1 for planar Dominating Set[Alber, Bodlaender, Fernau, Kloks, Niedermeier 2000]

• Same approach extended to Dominating Set variations, Feedback Vertex Set, etc.[Alber et al. 2002; Kanj & Perković 2002; Fomin & Thilikos2003; Alber, Fernau, Niedermeier 2004; Chang, Kloks, Lee 2001; Kloks, Lee, Liu 2002; Gutin, Kloks, Lee 2001]

any problem satisfying:

Bidimensionality Overview

closed under contraction [& deletion]

OPT costly on 𝑟𝑟 × 𝑟𝑟 grid

minor 𝐻𝐻

parameter-treewidth

bound

subexponential fixed-parameter

algorithm

efficient PTAS

linear kernel

… on 𝐻𝐻-minor-free graphs

Bidimensionality (version 1)[Demaine, Fomin, Hajiaghayi, Thilikos 2004]

• Parameter 𝑘𝑘 = 𝑘𝑘(𝐺𝐺) is bidimensional if Closed under minors:𝑘𝑘 only decreaseswhen deleting orcontracting edges

and

Large on grids:For the 𝑟𝑟 × 𝑟𝑟 grid, 𝑘𝑘 = Ω 𝑟𝑟2 r

r

v w

v wdelete

vwcontract

Example 1: Vertex Cover

• 𝑘𝑘 = minimum number of vertices required to cover every edge (on either endpoint)

• Closed under minors:⇒ still a cover(only fewer edges)

⇒ still a cover,possibly 1 smaller

v w v w v w v wcover

v w

v wdelete

vwcontract

Example 1: Vertex Cover

• 𝑘𝑘 = minimum number of vertices required to cover every edge (on either endpoint)

• Large on grids: Matching of size Ω 𝑟𝑟2

Every edge must be coveredby a different vertex

v w v w v w v wcover

r

r

Example 2: Feedback Vertex Set

• 𝑘𝑘 = minimum number of vertices required to cover every cycle (on some vertex)

• Closed under minors:⇒ still a cover(only break cycles)

⇒ still a cover,possibly 1 smaller

v w

u

v w

u

v w

u

v w

u…

v w

v wdelete

vwcontract

cover

Example 2: Feedback Vertex Set

• 𝑘𝑘 = minimum number of vertices required to cover every cycle (on some vertex)

• Large on grids: Ω 𝑟𝑟2 vertex-disjoint cycles Every cycle must be covered

by a different vertexr

r

v w

u

v w

u

v w

u

v w

u…

cover

Bidimensional ⇒ RelateParameter & Treewidth• Theorem 1: If a parameter 𝑘𝑘 is

bidimensional, then it satisfiesparameter-treewidth bound

treewidth = 𝑂𝑂 𝑘𝑘in 𝐻𝐻-minor-free graphs[Demaine, Fomin, Hajiaghayi, Thilikos 2004;Demaine & Hajiaghayi 2005]

• Proof sketch:Treewidth 𝑤𝑤 ⇒ Ω 𝑤𝑤 × Ω 𝑤𝑤 grid minor

⇒ 𝑘𝑘 = Ω 𝑤𝑤2 [bidimensional]

&

Bidimensional ⇒Subexponential FPT• Theorem 2: If a parameter 𝑘𝑘 is bidimensional, and fixed-parameter tractable on graphs of

bounded treewidth: ℎ treewidth 𝑛𝑛𝑂𝑂(1) timethen it has a subexponential fixed-parameter algorithm, running in ℎ 𝑘𝑘 𝑛𝑛𝑂𝑂 1 time,in 𝐻𝐻-minor-free graphs

• Proof sketch: If (approx.) treewidth = 𝜔𝜔 𝑘𝑘 , answer NO Else run bounded-treewidth algorithm

&

any problem satisfying:

Bidimensionality Overview

closed under contraction [& deletion]

OPT costly on 𝑟𝑟 × 𝑟𝑟 grid

minor 𝐻𝐻

parameter-treewidth

bound

subexponential fixed-parameter

algorithm

efficient PTAS

linear kernel

… on 𝐻𝐻-minor-free graphs

any problem satisfying:

Bidimensionality Overview

closed under contraction [& deletion]

OPT costly on 𝑟𝑟 × 𝑟𝑟 grid

minor 𝐻𝐻

parameter-treewidth

bound

subexponential fixed-parameter

algorithm

efficient PTAS

linear kernel

… on 𝐻𝐻-minor-free graphs

Bidimensional ⇒Linear Kernel• Theorem 4: If a parameter is bidimensional, satisfies the “separation property”, and has “finite integer index”,

then it has a linear kernelin 𝐻𝐻-minor-free graphs[Fomin, Lokshtanov, Saurabh, Thilikos 2009]

&

𝑛𝑛k k

𝑂𝑂(𝑘𝑘)poly

Bidimensional ⇒Linear Kernel• Protrusion:

• Idea: Kernelize protrusions

rest ofgraphtreewidth

𝑂𝑂 1cut

rest ofgraphsize

𝑂𝑂 1cut

Bidimensionality (version 2)[Demaine, Fomin, Hajiaghayi, Thilikos 2004]

• Parameter 𝑘𝑘 is contraction-bidimensional if Closed under contractions:𝑘𝑘 only decreaseswhen contracting edges

and

Large on gridoids:For 𝑟𝑟 × 𝑟𝑟 “grid-like graphs”,𝑘𝑘 = Ω 𝑟𝑟2oTriangulated + few extra edges

r

r

v w

vwcontract

Bidimensionality (version 3)[Fomin, Golovach, Thilikos 2009]

• Parameter 𝑘𝑘 is contraction-bidimensional if Closed under contractions:𝑘𝑘 only decreaseswhen contracting edges

and

Large on 𝚪𝚪 graphs:For naturally triangulated𝑟𝑟 × 𝑟𝑟 grid graphs, 𝑘𝑘 = Ω 𝑟𝑟2

v w

vwcontract

Contraction-Bidimensional Problems• Minimum maximal matching• Face cover (planar graphs)• Dominating set• Edge dominating set• 𝑟𝑟-dominating set• Connected … dominating set• Unweighted TSP tour• Chordal completion (fill-in)

v w

vwcontract

Contraction-Bidimensional ⇒ Good

• Theorem 6: As before, obtain parameter-treewidth bound subexponential FPT efficient PTASs linear kernel

for contraction-bidimensional parametersin any graph family excluding an apex minor

[Demaine, Fomin, Hajiaghayi, Thilikos 2004;Demaine & Hajiaghayi 2005 & 2005;Fomin, Golovach, Thilikos 2009;Fomin, Lokshtanov, Saurabh, Thilikos 2009]

Separators via Bidimensionality

• Number of vertices is minor-bidimensional• Parameter-treewidth bound⇒ treewidth = 𝑂𝑂 𝑘𝑘 = 𝑂𝑂 𝑛𝑛

• Corollary: For any fixed graph 𝐻𝐻, every𝐻𝐻-minor-free graph has treewidth 𝑂𝑂 𝑛𝑛and hence separator of size 𝑂𝑂 𝑛𝑛[Alon, Seymour, Thomas 1990; Grohe 2003]

Local Treewidth viaBidimensionality• Diameter is a contraction-bidimensional

parameter except Ω 𝑟𝑟 , not Ω 𝑟𝑟2 , in 𝑟𝑟 × 𝑟𝑟 grid• ⇒ Treewidth = 𝑂𝑂 diameter

in apex-minor-free graphs• ⇒ Treewidth of radius-𝑟𝑟 neighborhood = 𝑂𝑂 𝑟𝑟

[Demaine & Hajiaghayi 2004]

𝑟𝑟 = 1𝑟𝑟 = 2

𝑣𝑣

General Graphs via Bidimensionality

• Theorem: Any minor-closed graph parameter nonzero on some planar graph 𝑋𝑋, at least the sum over connected components in a

disconnected graph, and fixed-parameter tractable with respect to treewidth

can be solved in 𝑓𝑓 𝑘𝑘 𝑛𝑛 by a known algorithm[Demaine & Hajiaghayi 2007]

• Proof sketch: Treewidth > 𝑐𝑐1(𝑥𝑥 𝑘𝑘)𝑐𝑐2 ⇒ 𝑥𝑥 𝑘𝑘 × (𝑥𝑥 𝑘𝑘) grid minor⇒ 𝑘𝑘2 of 𝑥𝑥 × 𝑥𝑥 grid minors, with 𝑋𝑋 minor⇒ OPT ≥ 𝑘𝑘2 ⇒ answer NO

Beyond Apex-Minor-Free

• Conjecture: Algorithms for contraction-bidimensional problems generalize to𝐻𝐻-minor-free graphs, even for anon-apex graph 𝐻𝐻 True for e.g. dominating set subexponential FPT

(but parameter-treewidth bound false)[Demaine, Fomin, Hajiaghayi, Thilikos 2004] Stronger form of contraction-

bidimensionality works[Fomin, Golovach, Thilikos 2009]

New York Times

Beyond 𝑯𝑯-Minor-Free Graphs• Conjecture: Most of bidimensionality theory

generalizes to fixed powers of𝐻𝐻-minor-free graphs, e.g., map graphs Treewidth Ω 𝑟𝑟7⇒ 𝑟𝑟 × 𝑟𝑟 grid minor[Demaine, Hajiaghayi,Kawarabayashi 2009] Subexponential FPT for

dominating set in map graphs[Demaine, Fomin, Hajiaghayi, Thilikos 2003] Subexponential FPT & EPTASs for map graphs

by “cleaning” cliques [Fomin, Lokshtanov, 2012]

Beyond 𝑯𝑯-Minor-Free Graphs

• Excluded topological minors Linear kernel for (connected) dominating set

[Fomin, Lokshtanov, Saurabh, Thilikos 2013] General framework for linear kernels

[Langer, Reidl, Rossmanith, Sikdar 2012]

• Directed graphs Subexponential fixed-parameter algorithms for e.g.

Directed Hamiltonian Path via bidimensionality[Dorn, Fomin, Lokshtanov, Raman, Saurabh 2010] PTASs?

Beyond: Subset Problems

• What if only some of the nodes are “critical”?• Steiner tree, subset TSP, etc. have PTASs

up to bounded-genus graphs[Borradaile, Mathieu, Klein 2007;Borradaile, Demaine, Tazari 2009]

• Steiner forest has PTAS in planar graphs[Bateni, Hajiaghayi, Marx 2010]

• Wanted: General framework

Contractions not Well-Quasi-Ordered [Demaine, Hajiaghayi, Kawarabayashi 2009]

• No 𝐾𝐾2,𝑘𝑘 can be contracted into 𝐾𝐾2,𝑗𝑗

• Well-quasi-ordered for triangulated planar graphs 2-connected outerplanar graphs trees

• Minor-closed ⇒ finite excluded contractions

…

Simplifying Graph Decomposition[Demaine, Hajiaghayi, Kawarabayashi 2005; DeVos et al. 2004]

• 𝐻𝐻-minor-free graphs can have vertices or edges partitioned into 𝒌𝒌 pieces such that deleting any one piece results in bounded treewidth For PTAS, set 𝑘𝑘 ≈ 1/𝜀𝜀 𝑛𝑛𝑓𝑓 𝐻𝐻 algorithm 𝑓𝑓 𝐻𝐻 𝑛𝑛𝑂𝑂 1 algorithm

[Tazari 2012]

• Previously known for planar graphs [Baker 1994]

apex-minor-free [Eppstein 2000]

Simplifying Graph Decomposition[Demaine, Hajiaghayi, Kawarabayashi 2010]

• Odd-𝐻𝐻-minor-free graphs can have their vertices or edges partitioned into 2 piecessuch that deleting any one piece results in bounded treewidth 𝑛𝑛𝑓𝑓 𝐻𝐻 algorithm 𝑓𝑓 𝐻𝐻 𝑛𝑛𝑂𝑂 1 algorithm

[Tazari 2012]

Contraction Decomposition[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Theorem: 𝐻𝐻-minor-free graphs can have their edges partitioned into 𝒌𝒌 pieces such that contracting any one piece results in bounded treewidth Polynomial-time algorithm Previously known for

o planar [Klein 2005, 2006]

o bounded-genus [Demaine, Hajiaghayi, Mohar 2007]

o apex-minor-free[Demaine, Hajiaghayi, Kawarabayashi 2009]

Applications ofContraction Decomposition• PTAS for Traveling Salesman Problem

in weighted 𝐻𝐻-minor-free graphs Decade-old problem by [Grohe]

• PTAS for minimum-size𝑐𝑐-edge-connected submultigraph

• Fixed-parameter algorithm for𝒌𝒌-cut & bisection

[Demaine, Hajiaghayi, Kawarabayashi 2011]

Fixed-Parameter Algorithmsvia Contraction Decomposition

• Bisection: Cut graph into equal halveswith ≤ 𝑘𝑘 edges between them

• FPT in 𝐻𝐻-minor-free graphs: Closed under contraction

(but not minors) Contraction decomposition with 𝑘𝑘 + 1 layers

avoids OPT in some contraction Vertices become weighted Can still solve weighted bounded-treewidth ⇒ Solve in 2 �𝑂𝑂 𝑘𝑘 𝑛𝑛 + 𝑛𝑛𝑂𝑂 1 time

Fixed-Parameter Algorithmsvia Contraction Decomposition

• 𝒌𝒌-cut: Remove fewest edges to makeat least 𝑘𝑘 connected components

• FPT in 𝐻𝐻-minor-free graphs: Average degree 𝑐𝑐𝐻𝐻 = 𝑂𝑂 𝐻𝐻 lg𝐻𝐻 ⇒ OPT ≤ 𝑐𝑐𝐻𝐻 𝑘𝑘 ⇒ Contraction decomposition with 𝑐𝑐𝐻𝐻 𝑘𝑘 + 1 layers

avoids OPT in some contraction ⇒ Solve in 2 �𝑂𝑂 𝑘𝑘 𝑛𝑛 + 𝑛𝑛𝑂𝑂 1 time

• Generalization to arbitrary graphs [Kawarabayashi & Thorup 2011]

Radial Coloring for Bounded Genus[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Start with tight precolored edges: Treewidth of radius-𝑟𝑟 neighborhood is 𝑓𝑓(𝑟𝑟) 𝑐𝑐 = 𝑂𝑂(1) induced connected components

• Color edges at radial distance 𝑟𝑟with color 𝑟𝑟 mod 𝑘𝑘

Radial Coloring for Bounded Genus[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Contract color 𝑖𝑖 Each connected

component →articulation point Split these apart →

blobs, connected in DAG• Outdegree > 1 ⇒ split• Indegree > 1 ⇒ rejoin ⇒ Indegree ≤ 𝑐𝑐 + 𝑔𝑔

(initial component or handle)

Radial Coloring for Bounded Genus[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Split nonroot blob into≤ 𝑐𝑐 + 𝑔𝑔 chunksby closest BFS seed

• Each chunk hasradial diameter 𝑂𝑂 𝑘𝑘

• Nonroot blob has radial diameter, sotreewidth 𝑂𝑂 𝑘𝑘 𝑐𝑐 + 𝑔𝑔 [Eppstein 2000]

• Blob DAG = tree + ≤ 𝑔𝑔 extra edges Tree of 1-sums doesn’t affect treewidth Add extra edges to all bags ⇒ +𝑂𝑂 𝑔𝑔

Contraction Decomp.for 𝑯𝑯-Minor-Free

• 𝐻𝐻-minor-free graph = “tree”of “almost-embeddable graphs” [GM16]

• Each almost-embeddable graph has contraction decomposition: Bounded genus done Apices easy:

increase treewidthof anything by 𝑂𝑂(1) Vortices similar[Demaine, Hajiaghayi, Mohar 2007]

Hard Part of the Problem[Demaine, Hajiaghayi, Mohar 2007]

• Tree “joins” between two graphs areclique sums: Attach at clique of same size Delete any desired edges

𝐺𝐺2𝐺𝐺1 1

2

34

5

𝐾𝐾5 𝐾𝐾5𝐺𝐺 = 𝐺𝐺1 ⊕ 𝐺𝐺2

1

2

34

5

destroyscontraction

property

Solution Idea[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Clique sums either: [Graph Minors] Vortices ↔ apices

⇒ no contraction necessary 𝐾𝐾3 in bounded-genus part

(hard part)

𝐺𝐺2𝐺𝐺1 1

2

34

5

𝐾𝐾5 𝐾𝐾5𝐺𝐺 = 𝐺𝐺1 ⊕ 𝐺𝐺2

1

2

34

5

Solution Idea[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Find a path in child 𝐺𝐺2 to simulate the effectof each deleted edge in 𝐾𝐾3 Can make pieces 𝐺𝐺𝑖𝑖 3-edge-connected

• Need these paths to all contract together,by putting them all in piece 1

a

b c

Solution Idea[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Claim: Precoloring 𝑂𝑂(1) shortest pathsdoesn’t hurt contraction decomposition

• Shortcut paths to shortest paths(in radial graph ≈ primal + dual) Two paths 𝑎𝑎 → 𝑏𝑏; one path 𝑐𝑐 → 𝑎𝑎-𝑏𝑏 path

a

b c

𝑶𝑶 𝟏𝟏 Shortest Paths are Tight[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Theorem: Treewidth of radius-𝑟𝑟 neighborhood of 𝑂𝑂 1 shortest paths is 𝑓𝑓 𝑟𝑟

• If not, large gridwithin theneighborhood

• Intuition: Thenthere should bea shortcut:2𝑟𝑟 ≪ 𝑓𝑓(𝑟𝑟) Contradiction

𝑶𝑶 𝟏𝟏 Shortest Paths are Tight[Demaine, Hajiaghayi, Kawarabayashi 2011]

• Theorem: Treewidth of radius-𝑟𝑟 neighborhood of 𝑂𝑂 1 shortest paths is 𝑓𝑓 𝑟𝑟 Path is 𝑟𝑟-dive intoΘ 𝑟𝑟 × Θ 𝑟𝑟 wall ⇒ 𝑟𝑟-rainbow By Menger’s

Theorem, get Θ 𝑟𝑟“cross paths” Build Ω 𝑟𝑟 × Ω 𝑟𝑟

wall ≥ 2 𝑟𝑟 awayfrom shortest path Repeat for each path

Minors & Algorithms

ErikDemaine

M.I.T.