Bidimensionality and Bidimensionality and Approximation Approximation

AlgorithmsAlgorithmsMohammad T. Mohammad T.

HajiaghayiHajiaghayiUMDUMDr

r

Dealing with Hard Dealing with Hard Network Design ProblemsNetwork Design Problems

Main (theoretical) approaches to solve NP-hard problems:▪ Special instances: Planar graphs (fiber networks in

ground), etc.▪ Approximation algorithms (PTAS):

Within a factor C of the optimal solution (PTAS if C= 1+ ε for arbitrary constant ε)▪ Fixed-parameter algorithms:

Parameterize problem by parameter P(typically, the cost of the optimal solution)and aim for f(P) nO(1) (or even f(P) + nO(1))

We consider all above in Bidimentionality and aim for general algorithmic frameworks

OverviewOverview For any network design problem in a large class

(“bidimensional”)▪ Vertex cover, dominating set, connected dominating

set, r-dominating set, feedback vertex set, TSP, k-cut, Steiner tree, Steiner forest, multiway cut,…

In broad classes of networks generalizing planar networks (most “minor-closed” graph families)

We Obtain (in a series of more than 25 papers):▪ Strong combinatorial properties▪ Fixed-parameter algorithms

◦ Often subexponential: 2O(√k) nO(1) where k=|OPT|▪ Approximation algorithms

◦ Often PTASs (1+ ε approx): f(1/ε) nO(1)

Beyond Planar GraphsBeyond Planar Graphs A graph G has a minor H if

H can be formed by removing and contracting edges of G

delete

contract

Hminor of G

G*

• Otherwise, if G exclude H as a minor is called an H-minor-free graph• For example, planar graphs are bothK3,3-minor-free and K5-minor-free

Graph Minor TheoryGraph Minor Theory[Robertson & Seymour 1984–2004][Robertson & Seymour 1984–2004] Seminal series of ≥ 20 papers Powerful results on excluded minors:

▪Every minor-closed graph property(preserved when taking minors)has a finite set of excluded minors[Wagner’s Conjecture]

▪Every minor-closed graph propertycan be decided in polynomial time

▪For fixed graph H, graphs minor-excluding H have a special structure: drawings onbounded-genus surfaces + “extra features”

TreewidthTreewidth[GM2—Robertson & Seymour 1986][GM2—Robertson & Seymour 1986] Treewidth of a graph is the smallest

possible width of a tree decomposition Tree decomposition spreads

out each vertex as aconnected subtree of acommon tree, such thatadjacent vertices haveoverlapping subtrees▪Width = maximum overlap − 1

Treewidth 1 tree; 2 series-parallel; …

GraphTree

decomposition

(width 3)

Treewidth BasicsTreewidth Basics Many fast algorithms for NP-hard problems on

graphs of small treewidth▪ Typical running time: 2O(treewidth) nO(1)

Computing treewidth is NP-hard Computable in 22O(treewidth) n time, including

a tree decomposition [Bodlaender 1996] O(1)-approximable in 2O(treewidth) nO(1) time,

including a tree decomposition [Amir 2001] O(√lg opt)-approximable in nO(1) time

[Feige, Hajiaghayi, Lee 2004] (using a new framework for vertex separators based on embedding with minimum average distortion into line)

Treewidth BasicsTreewidth Basics Many fast algorithms for NP-hard problems

on graphs of small treewidth▪ Typical running time: 2O(treewidth) nO(1)

Computing treewidth is NP-hard Computable in 22O(treewidth) n time, including

a tree decomposition [Bodlaender 1996] O(1)-approximable in 2O(treewidth) nO(1) time,

including a tree decomposition [Amir 2001] 1.5-approximation for planar graphs and

single-crossing-minor-free graphs [EDD,MTH,NN,PR,DMT]

O(|V(H)|^2)-approximable in nO(1) time in H-minor-free graphs [Feige, Hajiaghayi, Lee 2004]

Bidimensionality (version 1)Bidimensionality (version 1)

Parameter k is minor-bidimensional if▪Closed under minors:

k does not increasewhen deleting orcontracting edges

and▪Large on grids:

For the r r grid, k = Ω(r2) and more generally Ω(f(r))

r

r

v w

v wdelete

vwcontract

Example 1: Vertex CoverExample 1: Vertex Cover

k = 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 wdelete

vwcontract

v w v w v w v wcover

Example 1: Vertex CoverExample 1: Vertex Cover

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

Large on grids:▪Matching of size Ω(r2)▪Every edge must be covered

by a different vertex

v w v w v w v wcover

r

r

Bidimensionality (version 2)Bidimensionality (version 2)

Parameter k is contraction-bidimensional if▪Closed under contractions:

k does not increasewhen contracting edges

and▪Large on a grid-like graph:

For naturally triangulatedr r grid graphs, k = Ω(r2)

v w

vwcontract

Example 2: Dominating SetExample 2: Dominating Set

k = minimum number of vertices required to cover every vertex or its neighbor

Large on grids:▪Ω(r2) vertex-disjoint cycles▪Every cycle must be covered

by a different vertex

v wcover u

v wu

v wu

v wu

…

r

r

Example 2: Dominating SetExample 2: Dominating Set

k = minimum number of vertices required to cover every vertex or its neighbor

Closed under contraction but not minor:

Not necessarily a cover anymore

still a cover,possibly 1 smaller

v w

v wdelete

vwcontract

v wcover u

v wu

v wu

v wu

…

Contraction-Bidimensional Contraction-Bidimensional ProblemsProblems

Minimum maximal matching Face cover (planar graphs) Dominating set Edge dominating set R-dominating set Connected … dominating set Unweighted TSP tour Chordal completion (fill-in)

v w

vwcontract

Bidimensional Bidimensional Relate RelateParameter & TreewidthParameter & Treewidth

Theorem 1: If a parameter k isbidimensional, then it satisfiesparameter-treewidth bound

treewidth = O(√k)in any graph family excluding some minor[Demaine, Fomin, Hajiaghayi, Thilikos, JACM 2005;

Demaine & Hajiaghayi, Combinatorica 2010] Proof sketch:

Large treewidth very large grid [minor theory] very large k [bidimensional]

&

Bidimensional Bidimensional Subexponential FPTSubexponential FPT

Theorem 2: If a parameter k is▪ bidimensional, and▪ fixed-parameter tractable on graphs of bounded

treewidth: h(treewidth) nO(1) time then it has a subexponential fixed-parameter

algorithm: h(√k) nO(1) timein any graph family excluding some minor▪ Typically 2O(√k) nO(1) time (h(w) = 2O(w))

[Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine & Hajiaghayi 2005]

Proof sketch:Run bounded-treewidth algorithm (tw = O(√k)) [If (approx.) treewidth is large, answer NO]

&

Bidimensional Bidimensional Subexponential FPTSubexponential FPT

Corollary 1: Vertex cover and feedback vertex set have subexponential fixed-parameter algorithms: 2O(√k) nO(1) time in any graph family excluding some minor[Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine & Hajiaghayi 2005]

▪Previously known for vertex cover (and some other problems) on planar graphs [Alber et al. 2002; Kanj & Perković 2002; Fomin & Thilikos 2003; Alber, Fernau, Niedermeier 2004; Chang, Kloks, Lee 2001; Kloks, Lee, Liu 2002; Gutin, Kloks, Lee 2001]

v wv w

u

Bidimensional Bidimensional PTAS PTAS

Theorem 3: If a parameter is▪bidimensional,▪fixed-parameter tractable on graphs of

bounded treewidth: h(treewidth) nO(1) time,▪O(1)-approximable in polynomial time, and▪satisfies the “separation property”

then it has an PTAS:(1+ε)-approximation in h(O(1/ε)) nO(1) time in any graph family excluding some minor [Demaine & Hajiaghayi, SODA’05]

&

Bidimensional Bidimensional PTAS PTAS

Corollary 3: Vertex cover and feedback vertex set have PTASsin any graph family excluding some minor[Demaine & Hajiaghayi 2005]▪Previously known for vertex cover (and

many, many other problems) on planar graphs

▪E.g., feedback vertex set result is new,even for planar graphs

v wv w

u

Consequence: Separator Consequence: Separator TheoremTheorem Theorem: [Demaine, Fomin, Hajiaghayi, Thilikos

2004; Demaine & Hajiaghayi 2005]For every bidimensional parameter P, treewidth(G) ≤ √P(G)

Apply to P(G) = number of vertices in G Corollary: For any fixed graph H, every H-

minor-free graph has treewidth O(√8n)[Alon, Seymour, Thomas 1990; Grohe 2003]

Corollary: 1/3-2/3 separators, size O(√n)(A vertex set whose removal leaves no

component of size greater than 2n/3)

Application to Independent Application to Independent Set (Lipton-Tarjan 1980)Set (Lipton-Tarjan 1980)

Independent Set: a set of vertices with no edges in between

Note that OPT is at least n/4 since planar graphs are 4-colorable

For PTAS break each component of greater than εn (=log n) and ignore separator vertices

Solve each component individually and take their union as the final solution

Consider a laminar family: level 0 are leaves

Application to Ind. Set Application to Ind. Set (cont’d) (cont’d)

• Note that l<= n/ ((3/2)i-1 ε n) since the size of a level I component is at least (3/2)i-1 ε n• Let ε= log n/n, so εn= log n (and thus we can solve each component individually in 2log n= n time)• So the total number of ignored vertices is at most n/ (√ log n)< ε n/4<= ε OPT (In each component we are not worse than OPT)

• The maximum number of levels is at most log3/2 n• Say C is the union of all separator (ignored) vertices.

Polynomial-TimePolynomial-TimeApproximation SchemesApproximation Schemes Separator approach [Lipton & Tarjan 1980]

gives PTASs only when OPT (after kernelization) can be lower bounded in terms of n (typically, OPT = Ω(n))▪Examples: Various forms of TSP

[Grigni, Koutsoupias, Papadimitriou 1995; Arora, Grigni, Karger, Klein, Woloszyn 1998; Grigni 2000; Grigni & Sissokho 2002]

Parameter-treewidth bounds give separators in terms of OPT, not n

Polynomial-TimePolynomial-TimeApproximation SchemesApproximation Schemes Theorem: [Demaine & Hajiaghayi 2005]

(1+ε)-approximation with running time h(O(1/ε)) nO(1) for any bidimensional optimization problem that is▪Computable in h(treewidth(G)) nO(1)

▪Solution on disconnected graph = union of solutions of each connected component

▪Given solution to G − C, can compute solution to G at an additional cost of ± O(|C|)

▪Solution S of G induced on connected component X of G − C has size |S X| ± O(|C|)

Polynomial-TimePolynomial-TimeApproximation SchemesApproximation Schemes Corollary: [Demaine & Hajiaghayi 2005]

▪PTAS in H-minor-free graphs for feedback vertex set, face cover, vertex cover, minimum maximal matching, and related vertex-removal problems

▪PTAS in apex-minor-free graphs for dominating set, edge dominating set, R-dominating set, connected … dominating set, clique-transversal set

No PTAS previously known for, e.g., feedback vertex set or connected dominating set, even in planar graphs

SIMPLIFYINGSIMPLIFYINGDECOMPOSITIONSDECOMPOSITIONS

Graph DecompositionGraph Decomposition

Separator Decomposition

Simplifying Decomposition

Small separator Large interactionSmall pieces Simple pieces (e.g.

bounded treewidth)

… … … ……

…

…

…

[Lipton & Tarjan 1980; …]

Simplifying Graph Simplifying Graph DecompositionDecomposition[Demaine, Hajiaghayi, Kawarabayashi, SODA [Demaine, Hajiaghayi, Kawarabayashi, SODA 2010]2010] Theorem: Odd H-minor-free graphs can have

their vertices or edges partitioned into two pieces such that each induced graph has bounded treewidth

▪ Previously for planar graphs [Baker 1994],apex-minor-free [Eppstein 2000], H-minor-free [DeVos et al. 2004; Demaine, Hajiaghayi, Kawarabayashi, FOCS’05]

Example: Graph ColoringExample: Graph Coloring

Chromatic number: Use fewest colorsto color the vertices of a graph such that no two equal colors connected by an edge▪Classic NP-hard problem▪Inapproximable within n1−ε unless ZPP = NP

Martin Gardner,April 1, 1975

Example: Graph ColoringExample: Graph Coloring[Demaine, Hajiaghayi, Kawarabayashi [Demaine, Hajiaghayi, Kawarabayashi 2005/2010]2005/2010] 2-approximation for chromatic

numberin odd-H-minor-free graphsusing decomposition into twobounded-treewidth pieces:

General graphs:Inapprox. within n1−ε unlessZPP = NP

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

Generalization to k pieces:H-minor-free graphs can have their vertices or edges partitioned into k pieces such that deleting any one piece results in bounded treewidth▪Useful for PTASs for minor-closed properties

(where k ~ 1/ε)▪(Not true for odd-minor)▪Application: e.g. PTAS for MaxCut

Many Problems Closed Many Problems Closed Under Contractions but not Under Contractions but not DeletionsDeletions Dominating set Edge dominating set R-dominating set Connected … dominating set Face cover (planar graphs) Minimum maximal matching Chordal completion (fill-in) Traveling Salesman Problem …

Contraction DecompositionContraction Decomposition[Demaine, Hajiaghayi, Kawarabayashi, [Demaine, Hajiaghayi, Kawarabayashi, STOC’11]STOC’11] Theorem: H-minor-free graphs can have

their edges partitioned into k pieces such that contracting any one piece results in bounded treewidth

▪Polynomial-time algorithm▪Previously known for planar [Klein 2005, 2006],

bounded-genus [Demaine, Hajiaghayi, Mohar 2007], apex-minor-free [Demaine, Hajiaghayi, Kawarabayashi 2009]

ApplicationsApplications Lots of applications via a general theorem,e.g.

Corollary 1: PTAS for Traveling Salesman Problem in weighted H-minor-free graphs[Demaine, Hajiaghayi, Kawarabayashi 2011] solving an open problem of [Grohe 2001]

Coroallary 2: Fixed-Parameter Algorithm for k-cut and Bisection on planar graphs and H-minor-free graphs [Demaine, Hajiaghayi, Kawarabayashi 2011] solving an open problem of [Downey, Estivill-Castro, Fellows 2003]

Application to TSPApplication to TSP

Corollary: PTAS for Traveling Salesman Problem in weighted H-minor-free graphs[Demaine, Hajiaghayi, Kawarabayashi 2011]▪Existing bounded-treewidth algorithm

[Dorn, Fomin, Thilikos 2006]▪Existing spanner [Grigni, Sissokho 2002]▪Decontraction:

Euler tour(cost ≤ 2 weight)

+ perfect matching on odd-degree vxs (cost

≤ weight)

Graph TSP HistoryGraph TSP History PTAS for unweighted planar

[Grigni, Koutsoupias, Papadimitriou 1995] PTAS for weighted planar

[Arora, Grigni, Karger, Klein, Woloszyn 1998] Linear PTAS for weighted planar [Klein 2005] QPTAS (n(1/ε) O(log log n) time) for weighted

bounded-genus / unweighted H-minor-free [Grigni 2000]

PTAS for weighted bounded genus[Demaine, Hajiaghayi, Mohar 2007]

PTAS for unweighted apex-minor-free [Demaine, Hajiaghayi, Kawarabayashi 2009]

PTAS for weighted H-minor-free [DHK 2011]

Application Beyond TSPApplication Beyond TSP Corollary: PTAS for minimum-weight

c-edge-connected submultigraphin H-minor-free graphs[Demaine, Hajiaghayi, Kawarabayashi 2011]

Previous results:▪PTASs for 2-edge-connected in planar graphs

[Klein 2005] (linear)[Berger, Czumaj, Grigni, Zhao 2005][Czumaj, Grigni, Sissokho, Zhao 2004]

▪PTAS for c-edge-connected in bounded-genus graphs [Demaine, Hajiaghayi, Mohar 2007]

Fixed-Parameter Fixed-Parameter Algorithmic Applications: k-Algorithmic Applications: k-cutcut k-cut: Remove fewest edges to make

at least k connected components FPT in H-minor-free graphs:

▪Average degree cH = O(H lg H )▪ OPT ≤ cH k▪ Contraction decomposition with cH k + 1

layers avoids OPT in some contraction▪ Solve in 2Õ(k) n + nO(1) time

Generalization to arbitrary graphs [Kawarabayashi & Thorup 2011]

√‾‾‾

Proof SketchProof Sketch

H-minor-free graph = “tree” of“almost-embeddable graphs” [Graph Minors]

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

increase treewidthof anything by O(1)

▪Vortices similar[Demaine, Hajiaghayi, Mohar 2007]

Radial Coloring for Bounded Radial Coloring for Bounded GenusGenus Color edge at radial distance r as r mod

k▪Radial graph ≈ primal graph + dual graph

Any k consecutive layers have bounded treewidth, provided first k do

Neighborhoods of Shortest Neighborhoods of Shortest Paths have Bounded Paths have Bounded TreewidthTreewidth

Contraction DecompositionContraction Decomposition[Demaine, Hajiaghayi, Kawarabayashi [Demaine, Hajiaghayi, Kawarabayashi 2011]2011] Theorem: H-minor-free graphs can have

their edges partitioned into k piecessuch that contracting any one piece results in bounded treewidth▪Polynomial-time algorithm

Seems a powerful tool for approximation & fixed-parameter algorithms

Let’s find more applications!

IMPROVINGIMPROVINGGRAPH MINORSGRAPH MINORS

Graph MinorsGraph Minors[Robertson&Seymour 1983–[Robertson&Seymour 1983–2004]2004] in in ≥≥ 20 papers 20 papers

…

…

NonconstructiveNonconstructiveGraph MinorsGraph Minors Theorem: Every H-minor-

free graph can be writtenas a tree of graphs joinedalong f(H)-size cliques▪Each term is a graph that can be almost

embedded into a bounded-genus surface(f(H) “vortices” and “apices”)

[GM16: Robertson & Seymour 2003]

ConstructiveConstructiveGraph MinorsGraph Minors Theorem: Every H-minor-

free graph can be writtenas a tree of graphs joinedalong f(H)-size cliques▪Computable in nf(H) time

[Demaine, Hajiaghayi, Kawarabayashi, FOCS 2005]

▪Weaker form in f(H) nO(1) time[Dawar, Grohe, Kreutzer 2007]

Grid MinorsGrid Minors

Every H-minor-free graph of treewidth≥ f(H) r has an r r grid minor[Demaine & Hajiaghayi, SODA 2005, Combinatorica 2010]▪Previous bounds exponential in r and H

[GM5—Robertson & Seymour 1986;Robertson, Seymour, Thomas 1994; Reed 1997;Diestel, Jensen, Gorbunov, Thomassen 1999]

Open: What is f(H)?▪Ω(√|V(H)| lg |V(H)|)▪Conjecture: |V(H)|O(1) or even O(|V(H)|)

Beyond BidimensionalityBeyond Bidimensionality Nontrivial weights

▪Min-weight k disjoint paths? Directed networks (with Rajesh)

▪Useful notion of treewidth? Subset problems (with Rajesh and

Marek)▪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: A more general framework

k = 3

50

Thanks for your attention…Thanks for your attention…