Transcript Bidimensionality and Approximation Algorithms Mohammad T. Hajiaghayi

Bidimensionality and
Approximation Algorithms
r
r
Mohammad T.
Hajiaghayi
UMD
Dealing with Hard
Network 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
Overview
 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 Graphs
 A graph G has a minor H if
H can be formed by removing
and contracting edges of G
G
delete
*
H
minor of G
contract
• Otherwise, if G exclude H as a minor is
called an H-minor-free graph
• For example, planar graphs are both
K3,3-minor-free and K5-minor-free
Graph Minor Theory
[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 property
can be decided in polynomial time
▪ For fixed graph H, graphs minor-excluding H
have a special structure: drawings on
bounded-genus surfaces + “extra features”
Treewidth
[GM2—Robertson & Seymour 1986]
 Treewidth of a graph is the smallest
possible width of a tree decomposition
 Tree decomposition spreads
out each vertex as a
connected subtree of a
common tree, such that
adjacent vertices have
Tree
Graph
decomposition
overlapping subtrees
▪ Width = maximum overlap − 1
(width 3)
 Treewidth 1  tree; 2  series-parallel; …
Treewidth 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
O(treewidth)
2
 Computable in 2
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 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
O(treewidth)
2
 Computable in 2
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 singlecrossing-minor-free graphs [EDD,MTH,NN,PR,DMT]
 O(|V(H)|^2)-approximable in nO(1) time in H-minorfree graphs [Feige, Hajiaghayi, Lee 2004]
Bidimensionality (version 1)
 Parameter k is minor-bidimensional if
▪ Closed under minors:
k does not increase
when deleting or
contracting edges
v
and
w
v
w
delete
vw
contract
▪ Large on grids:
For the r  r grid, k = Ω(r2)
and more generally Ω(f(r))
r
r
Example 1: Vertex Cover
 k = minimum number of vertices required
to cover every edge (on either endpoint)
cover
v
w
v
w
v
w
v
w
 Closed under minors:
v
w
v
w
delete
 still a cover
(only fewer edges)
vw
contract
 still a cover,
possibly 1 smaller
Example 1: Vertex Cover
 k = minimum number of vertices required
to cover every edge (on either endpoint)
cover
v
w
v
w
v
w
v
w
 Large on grids:
▪ Matching of size Ω(r2)
▪ Every edge must be covered
by a different vertex
r
r
Bidimensionality (version 2)
 Parameter k is contractionbidimensional if
▪ Closed under contractions:
k does not increase
when contracting edges
and
▪ Large on a grid-like graph:
For naturally triangulated
r  r grid graphs, k = Ω(r2)
v
w
vw
contract
Example 2: Dominating Set
 k = minimum number of vertices required
to cover every vertex or its neighbor
v
cover
w
u
v
w
u
v
w
u
…
v
w
u
 Large on grids:
▪ Ω(r2) vertex-disjoint cycles
▪ Every cycle must be covered
by a different vertex
r
r
Example 2: Dominating Set
 k = minimum number of vertices required
to cover every vertex or its neighbor
v
w
cover
v
u
w
u
v
w
u
…
v
w
u
 Closed under contraction but not minor:
v
w
v
w
delete
Not necessarily
a cover anymore
vw
contract
 still a cover,
possibly 1 smaller
Contraction-Bidimensional Problems








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
vw
contract
Bidimensional  Relate
Parameter & Treewidth
&
 Theorem 1: If a parameter k is
bidimensional, then it satisfies
parameter-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
 very large k
[minor theory]
[bidimensional]
Bidimensional 
Subexponential 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) time
in 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 
Subexponential FPT
v
w
v
w
u
 Corollary 1: Vertex cover and feedback
vertex set have subexponential fixedparameter 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]
Bidimensional  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  PTAS
v
w
v
w
u
 Corollary 3: Vertex cover and feedback
vertex set have PTASs
in 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
Consequence: Separator Theorem
 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 Hminor-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 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 (cont’d)
• The maximum number of levels is at most log3/2 n
• Say C is the union of all separator (ignored) vertices.
• 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)
Polynomial-Time
Approximation 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-Time
Approximation 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-Time
Approximation 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, Rdominating 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
SIMPLIFYING
DECOMPOSITIONS
Graph Decomposition
Separator
Decomposition
Small separator
Small pieces
…
…
…
…
… … … …
[Lipton & Tarjan 1980; …]
Simplifying
Decomposition
Large interaction
Simple pieces (e.g.
bounded treewidth)
Simplifying Graph Decomposition
[Demaine, Hajiaghayi, Kawarabayashi, SODA 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
et al. 2004; Demaine, Hajiaghayi, Kawarabayashi, FOCS’05]
[DeVos
Example: Graph Coloring
 Chromatic number: Use fewest colors
to 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 Coloring
[Demaine, Hajiaghayi, Kawarabayashi 2005/2010]
 2-approximation for chromatic number
in odd-H-minor-free graphs
General
using decomposition into two
graphs:
Inapprox.
bounded-treewidth pieces:
within n1−ε
unless
ZPP = NP
Simplifying Graph Decompositions
[DeVos et al. 2004; Demaine, Hajiaghayi, Kawarabayashi 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 Under
Contractions but not Deletions









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 Decomposition
[Demaine, Hajiaghayi, Kawarabayashi, 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], apexminor-free [Demaine, Hajiaghayi, Kawarabayashi 2009]
Applications
 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 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 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 TSP
 Corollary: PTAS for minimum-weight
c-edge-connected submultigraph
in 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 Algorithmic
Applications: k-cut
 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 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 treewidth
of anything by O(1)
▪ Vortices similar
[Demaine, Hajiaghayi, Mohar 2007]
Radial Coloring for Bounded Genus
 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 Paths
have Bounded Treewidth
Contraction Decomposition
[Demaine, Hajiaghayi, Kawarabayashi 2011]
 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
 Seems a powerful tool for approximation &
fixed-parameter algorithms
 Let’s find more applications!
IMPROVING
GRAPH MINORS
Graph Minors[Robertson&Seymour 1983–2004]
in ≥ 20 papers
Nonconstructive
Graph Minors
 Theorem: Every H-minorfree graph can be written
as a tree of graphs joined
along 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]
Constructive
Graph Minors
 Theorem: Every H-minorfree graph can be written
as a tree of graphs joined
along 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 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 Bidimensionality
 Nontrivial weights
▪ Min-weight k disjoint paths?
 Directed networks (with Rajesh)
k=3
▪ 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
Thanks for your attention…
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