A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: From Undirected to Directed

A Quest for Subexponential Time Parameterized
Algorithms for Planar-k-Path: From Undirected to
Directed Graphs
Saket Saurabh1
1Institute for Mathematical Sciences, Chennai, India,
and University of Bergen, Norway.
NMI Workshop on Complexity Theory, IITGN, Saturday 5th
November, 2016
(Slides by D. Marx, M. Pilipzuck and S. Saurabh)
1

Main message
NP-hard problems become easier on planar graphs
and geometric objects, and usually exactly by a
square root factor.
Planar graphs Geometric objects
2

Better exponential algorithms
Most NP-hard problems (e.g., 3-Coloring, Independent Set,
Hamiltonian Cycle, Steiner Tree, etc.) remain NP-hard on
planar graphs,1 so what do we mean by “easier”?
1
Notable exception: Max Cut is in P for planar graphs.
3

Better exponential algorithms
Most NP-hard problems (e.g., 3-Coloring, Independent Set,
Hamiltonian Cycle, Steiner Tree, etc.) remain NP-hard on
planar graphs,1 so what do we mean by “easier”?
The running time is still exponential, but signiﬁcantly smaller:
2O(n)
) 2O(
p
n)
nO(k)
) nO(
p
k)
2O(k)
· nO(1)
) 2O(
p
k)
· nO(1)
1
Notable exception: Max Cut is in P for planar graphs.
3

Overview
Chapter 1:
Subexponential algorithms using treewidth.
Chapter 2:
Grid minors and bidimensionality.
Chapter 3:
Beyond bidimensionality:
Finding bounded-treewidth solutions.
Chapter 4:
Pattern Coverage.
4

Chapter 1: Subexponential algorithms using treewidth
Treewidth is a measure of “how treelike the graph is.”
We need only the following basic facts:
Treewidth
1 If a graph G has treewidth k, then many classical NP-hard
problems can be solved in time 2O(k) · nO(1) or
2O(k log k) · nO(1) on G.
2 A planar graph on n vertices has treewidth O(
p
n).
5

Treewidth — a measure of “tree-likeness”
Tree decomposition: Vertices are arranged in a tree structure
satisfying the following properties:
1 If u and v are neighbors, then there is a bag containing both
of them.
2 For every v, the bags containing v form a connected subtree.
Width of the decomposition: largest bag size 1.
treewidth: width of the best decomposition.
dcb
a
e f g h
g, hb, e, fa, b, c
d, f , gb, c, f
c, d, f
6

Treewidth — a measure of “tree-likeness”
Tree decomposition: Vertices are arranged in a tree structure
satisfying the following properties:
1 If u and v are neighbors, then there is a bag containing both
of them.
2 For every v, the bags containing v form a connected subtree.
Width of the decomposition: largest bag size 1.
treewidth: width of the best decomposition.
hgfe
a
b c d
g, hb, e, fa, b, c
d, f , gb, c, f
c, d, f
A subtree communicates with the outside world
only via the root of the subtree.
6

Finding tree decompositions
Various algorithms for ﬁnding optimal or approximate tree
decompositions if treewidth is w:
optimal decomposition in time 2O(w3) · n
[Bodlaender 1996].
4-approximate decomposition in time 2O(w) · n2
[Robertson and Seymour].
5-approximate decomposition in time 2O(w) · n
[Bodlaender et al. 2013].
O(
p
log w)-approximation in polynomial time
[Feige, Hajiaghayi, Lee 2008].
As we are mostly interested in algorithms with running time
2O(w) · nO(1), we may assume that we have a decomposition.
7

3-Coloring and tree decompositions
Theorem
Given a tree decomposition of width w, 3-Coloring can be
solved in time 3w · wO(1) · n.
Bx : vertices appearing in node x.
Vx : vertices appearing in the subtree rooted at x.
For every node x and coloring c : Bx !
{1, 2, 3}, we compute the Boolean value
E[x, c], which is true if and only if c can
be extended to a proper 3-coloring of Vx .
Claim:
We can determine E[x, c] if all the values are
known for the children of x.
c, d, f
b, c, f d, f , g
a, b, c b, e, f g, h
bcf=T bcf=F
bcf=T bcf=F
. . . . . .
8

Subexponential algorithm for 3-Coloring
Theorem [textbook dynamic programming]
3-Coloring can be solved in time 2O(w) · nO(1) on graphs of
treewidth w.
+
Theorem [Robertson and Seymour]
A planar graph on n vertices has treewidth O(
p
n).
9

Theorem [textbook dynamic programming]
treewidth w.
+
p
n).
+
Corollary
3-Coloring can be solved in time 2O(
p
n) on planar graphs.
textbook algorithm + combinatorial bound
+
subexponential algorithm
9

Theorem
treewidth w.
+
p
n).
+
Corollary
p
textbook algorithm + combinatorial bound
+
subexponential algorithm
10

Lower bounds
Corollary
p
Two natural questions:
Can we achieve this running time on general graphs?
Can we achieve even better running time (e.g., 2O( 3p
n)) on
planar graphs?
11

Lower bounds
Corollary
p
Two natural questions:
Can we achieve this running time on general graphs?
Can we achieve even better running time (e.g., 2O( 3p
n)) on
planar graphs?
P 6= NP is not a suﬃciently strong hypothesis: it is compatible with
3SAT being solvable in time 2O(n1/1000) or even in time nO(log n).
We need a stronger hypothesis!
11

Exponential Time Hypothesis (ETH)
Hypothesis introduced by Impagliazzo, Paturi, and Zane:
Exponential Time Hypothesis (ETH) [consequence of]
There is no 2o(n)-time algorithm for n-variable 3SAT.
Note: current best algorithm is 1.30704n [Hertli 2011].
Note: an n-variable 3SAT formula can have m = ⌦(n3) clauses.
12

Are there algorithms that are subexponential in the size n + m of
the 3SAT formula?
12

Are there algorithms that are subexponential in the size n + m of
the 3SAT formula?
Sparsiﬁcation Lemma [Impagliazzo, Paturi, Zane 2001]
There is a 2o(n)-time algorithm for n-variable 3SAT.
m
There is a 2o(n+m)-time algorithm for n-variable m-clause 3SAT.
12

Lower bounds based on ETH
ETH + Sparsiﬁcation Lemma
There is no 2o(n+m)-time algorithm for n-variable m-clause 3SAT.
The textbook reduction from 3SAT to 3-Coloring:
3SAT formula
n variables
m clauses
)
Graph G
O(n + m) vertices
O(n + m) edges
Corollary
Assuming ETH, there is no 2o(n) algorithm for 3-Coloring on an
n-vertex graph G.
13

ETH + Sparsiﬁcation Lemma
There is no 2o(n+m)-time algorithm for n-variable m-clause 3SAT.
The textbook reduction from 3SAT to 3-Coloring:
3SAT formula
n variables
m clauses
)
Graph G
O(m) vertices
O(m) edges
(we can assume n = O(m))
Corollary
Assuming ETH, there is no 2o(n) algorithm for 3-Coloring on an
n-vertex graph G.
13

Transfering bounds
There are polynomial-time reductions from, say, 3-Coloring to
many other problems such that the reduction increases the number
of vertices by at most a constant factor.
Consequence: Assuming ETH, there is no 2o(n) time algorithm on
n-vertex graphs for
Independent Set
Clique
Dominating Set
Vertex Cover
Hamiltonian Path
Feedback Vertex Set
. . .
14

What about 3-Coloring on planar graphs?
The textbook reduction from 3-Coloring to Planar
3-Coloring uses a “crossover gadget” with 4 external connectors:
In every 3-coloring of the gadget, opposite external connectors
have the same color.
Every coloring of the external connectors where the opposite
vertices have the same color can be extended to the whole
gadgets.
If two edges cross, replace them with a crossover gadget.
15

The reduction from 3-Coloring to Planar 3-Coloring
introduces O(1) new edge/vertices for each crossing.
A graph with m edges can be drawn with O(m2) crossings.
3SAT formula
n variables
m clauses
)
Graph G
O(m) vertices
O(m) edges
)
Planar graph G0
O(m2) vertices
O(m2) edges
Corollary
Assuming ETH, there is no 2o(
p
n) algorithm for 3-Coloring on
an n-vertex planar graph G.
(Essentially observed by [Cai and Juedes 2001])
16

Summary of Chapter 1
Streamlined way of obtaining tight upper and lower bounds for
planar problems.
Upper bound:
Standard bounded-treewidth algorithm + treewidth bound on
planar graphs give 2O(
p
n) time subexponential algorithms.
Lower bound:
Textbook NP-hardness proof with quadratic blow up + ETH
rule out 2o(
p
n) algorithms.
Works for Hamiltonian Cycle, Vertex Cover,
Independent Set, Feedback Vertex Set, Dominating
Set, Steiner Tree, . . .
17

Chapter 2: Grid minors and bidimensionality
More refined analysis of the running time: we express the running
time as a function of input size n and a parameter k.
Definition
A problem is fixed-parameter tractable (FPT) parameterized by
k if it can be solved in time f (k) · nO(1) for some computable
function f .
Examples of FPT problems:
Finding a vertex cover of size k.
Finding a feedback vertex set of size k.
Finding a path of length k.
Finding k vertex-disjoint triangles.
. . .
Note: these four problems have 2O(k) · nO(1) time algorithms, which
is best possible on general graphs.
18

W[1]-hardness
Negative evidence similar to NP-completeness. If a problem is
W[1]-hard, then the problem is not FPT unless FPT=W[1].
Some W[1]-hard problems:
Finding a clique/independent set of size k.
Finding a dominating set of size k.
Finding k pairwise disjoint sets.
. . .
For these problems, the exponent of n has to depend on k
(the running time is typically nO(k)).
19

Subexponential parameterized algorithms
What kind of upper/lower bounds we have for f (k)?
For most problems, we cannot expect a 2o(k) · nO(1) time
algorithm on general graphs.
(As this would imply a 2o(n) algorithm.)
For most problems, we cannot expect a 2o(
p
k) · nO(1) time
algorithm on planar graphs.
(As this would imply a 2o(
p
n) algorithm.)
20

Subexponential parameterized algorithms
What kind of upper/lower bounds we have for f (k)?
For most problems, we cannot expect a 2o(k) · nO(1) time
algorithm on general graphs.
(As this would imply a 2o(n) algorithm.)
For most problems, we cannot expect a 2o(
p
k) · nO(1) time
algorithm on planar graphs.
(As this would imply a 2o(
p
n) algorithm.)
However, 2O(
p
k) · nO(1) algorithms do exist for several
problems on planar graphs, even for some W[1]-hard problems.
Quick proofs via grid minors and bidimensionality.
[Demaine, Fomin, Hajiaghayi, Thilikos 2004]
Next: subexponential parameterized algorithm for k-Path.
20

Minors
Deﬁnition
Graph H is a minor of G (H  G) if H can be obtained from G by
deleting edges, deleting vertices, and contracting edges.
deleting uv
vu w
u v
contracting uv
Note: length of the longest path in H is at most the length of the
longest path in G.
21

Planar Excluded Grid Theorem
Theorem [Robertson, Seymour, Thomas 1994]
Every planar graph with treewidth at least 5k has a k ⇥ k grid
minor.
Note: for general graphs, treewidth at least k100 (now even k19) or
so guarantees a k ⇥ k grid minor [Chekuri and Chuzhoy 2013] and
[Chuzhoy 2016]!
22

Bidimensionality for k-Path
Observation: If the treewidth of a planar graph G is at least 5
p
k
) It has a
p
k ⇥
p
k grid minor (Planar Excluded Grid Theorem)
) The grid has a path of length at least k.
) G has a path of length at least k.
23

p
k
) It has a
p
k ⇥
p
We use this observation to ﬁnd a path of length at least k on
planar graphs:
If treewidth w of G is at least 5
p
k:
we answer “there is a path of length at
least k.”
If treewidth w of G is less than 5
p
k,
then we can solve the problem in time
2O(w) · nO(1) = 2O(
p
k) · nO(1).
23

p
k
) It has a
p
k ⇥
p
We use this observation to ﬁnd a path of length at least k on
planar graphs:
Set w := 5
p
k.
Find an O(1)-approximate tree
decomposition.
If treewidth is at least w: we can
answer “there is a path of length at
least k.”
If we get a tree decomposition of
width O(w), then we can solve the
problem in time
2O(w)
· nO(1)
= 2O(
p
k)
· nO(1)
.
23

Bidimensionality
Deﬁnition
A graph invariant x(G) is minor-bidimensional if
x(G0)  x(G) for every minor G0 of G, and
If Gk is the k ⇥ k grid, then x(Gk) ck2
(for some constant c > 0).
Examples: minimum vertex cover, length of the longest path,
feedback vertex set are minor-bidimensional.
24

Tight bounds for minor-bidimensional planar problems.
Upper bound:
Standard bounded-treewidth algorithm + planar excluded grid
theorem give 2O(
p
k) · nO(1) time FPT algorithms.
Lower bound:
Textbook NP-hardness proof with quadratic blow up + ETH
rule out 2o(
p
n) time algorithms ) no 2o(
p
k) · nO(1) time
algorithm.
Variant of theory works for contraction-bidimensional problems,
e.g., Independent Set, Dominating Set.
25

Limits of bidimensionality?
Perhaps the most basic problem:
Subgraph Isomorphism
Given graphs H and G, decide if G has a subgraph isomorphic to
H.
26

H.
Theorem [Eppstein 1999]
Subgraph Isomorphism for planar graphs can be solved in time
2O(k log k) · n for k := |V (H)|.
26

H.
Question already asked several time over the last decades:
Does the square root phenomenon appear for Subgraph
Isomorphism on planar graphs?
26

H.
Assuming ETH, there is no 2o(k/ log k)nO(1) time algorithm for
general patterns.
[Bodlaender, Nederlof, van der Zanden; ICALP’16)]
26

H.
Assuming ETH, there is no 2o(k/ log k)nO(1) time algorithm for
general patterns.
[Bodlaender, Nederlof, van der Zanden; ICALP’16)]
There is a 2O(k/ log k)nO(1) time (randomized) algorithm for
connected patterns.
[At the end of the talk – hopefully!]
26

Limits of bidimensionality? [Counting problems]
Counting is harder than decision:
Counting version of easy problems:
not clear if they remain easy.
Counting version of hard problems:
not clear if we can keep the same running time.
27

Limits of bidimensionality? [Counting problems]
Counting is harder than decision:
Counting version of easy problems:
not clear if they remain easy.
Counting version of hard problems:
not clear if we can keep the same running time.
Working on counting problems is fun:
You can revisit fundamental, “well-understood” problems.
Requires a new set of lower bound techniques.
Requires new algorithmic techniques.
27

Bidimensionality and counting
Does not work for counting k-paths in a planar graph:
If treewidth w is O(
p
k): can be solved in time
2O(w)nO(1) = 2O(
p
k)nO(1) using dynamic programming.
If treewidth w is larger than c
p
k: answer is positive, but how
much exactly?
28

p
2O(w)nO(1) = 2O(
p
p
much exactly?
What about counting vertex covers of size k in a planar graph?
28

p
2O(w)nO(1) = 2O(
p
p
much exactly?
Works for counting vertex covers of size k in a planar graph:
p
2O(w)nO(1) = 2O(
p
p
k: answer is 0.
28

Counting k-matching
Counting matchings can be signiﬁcantly harder than ﬁnding a
matching!
Counting perfect matchings is #P-hard [Valiant 1979].
Counting matchings of size k is #W[1]-hard
[Curticapean 2013], [Curticapean and M. 2014].
Counting matchings of size k is FPT in planar graphs.
[Frick 2004]
Open question: Is there a 2O(
p
k) · nO(1) algorithm for counting k
matchings in planar graphs?
29

Counting k-matching
Counting matchings can be signiﬁcantly harder than ﬁnding a
matching!
Counting perfect matchings in planar graphs is
polynomial-time solvable.
[Kasteleyn 1961], [Temperley and Fischer 1961].
Corollary: we can count matchings covering n k vertices in
time nO(k)
. . . but (assuming ETH) there is no f (k)no(k/ log k) time
algorithm [Curticapean and Xia 2015].
29

Chapter 3: Finding bounded-treewidth solutions
So far, we have exploited that the input has bounded treewidth
and used standard algorithms.
30

So far, we have exploited that the input has bounded treewidth
and used standard algorithms.
Change of viewpoint:
In many cases, we have to exploit instead that the solution has
bounded treewidth.
30

So far the way we have used treewidth is to ﬁnd something (e.g.,
Hamiltonian cycle) in a large bounded-treewidth graph:
31

But we can also ﬁnd small bounded-treewidth objects in an arbitrary
large graph.
G
H
Theorem [Alon, Yuster, Zwick 1994]
Given a graph H and weighted graph G, we can ﬁnd a minimum
weight subgraph of G isomorphic to H in time 2O(|V (H)|) · nO(tw(H)).
31

But we can also find small bounded-treewidth objects in an arbitrary
large graph.
G
H
Theorem [Alon, Yuster, Zwick 1994]
Given a graph H and weighted graph G, we can find a minimum
weight subgraph of G isomorphic to H in time 2O(|V (H)|) · nO(tw(H)).
If the problem can be formulated as finding a graph of treewidth
O(
p
k), then we get an nO(
p
k) time algorithm.
31

Examples
Several examples:
Planar k-Terminal Cut
Improvement from nO(k) to 2O(k) · nO(
p
k).
Planar Strongly Connected Subgraph
Improvement from nO(k) to 2O(k log k) · nO(
p
k).
Subset TSP with k cities in a planar graph
Improvement from 2O(k) · nO(1) to 2O(
p
k log k) · nO(1).
Rectilinear Steiner Tree with n points in a plane.
Improvement from 2O(n) · nO(1) to 2O(
p
n log n) · nO(1).
32

TSP
TSP
Input: A set T of cities and a distance function d on T
Output: A tour on T with minimum total distance
Theorem [Held and Karp]
TSP with k cities can be solved in time 2k · nO(1).
Dynamic programming:
Let x(v, T0) be the minimum length of path from vstart to v
visiting all the cities T0 ✓ T.
33

Subset TSP on planar graphs
Assume that the cities correspond to a subset T of a planar graph
and distance is measured in this planar graph.
34

Can be solved in time 2O(
p
n).
Can be solved in time 2k · nO(1).
Question: Can we solve it in time 2O(
p
k) · nO(1)?
34

Theorem [Klein and Marx]
Subset TSP for k cities in a (unit-weight) planar graph can be
solved in time 2O(
p
k log k) · nO(1).
34

TSP and treewidth
We wanted to formulate the problem as ﬁnding a low
treewidth subgraph.
A cycle has treewidth 2, is this of any help?
Problem:
We have to remember the subset of cities visited by the partial tour
(2k possibilities).
35

c-change TSP
c-change operation: removing c steps of the tour and
connecting the resulting c paths in some other way.
A solution is c-OPT if no c-change can improve it.
We can ﬁnd a c-OPT solution in kO(c) · D time, where D is
the maximum distance (if distances are integers).
36

The treewidth bound
Consider the union of an optimum solution and a 4-OPT solution
as a graph on k vertices:
Lemma
The union of an optimum solution and a 4-OPT solution has
treewidth O(
p
k) [some techincal details omitted].
37

The treewidth bound
Lemma
treewidth O(
p
The union has separators of size O(
p
k).
In each component, the set of cities visited by the optimum
solution is nice: it is the same as what O(
p
k) segments of the
4-OPT tour visited (kO(
p
k) possibilities).
38

The treewidth bound
Lemma
treewidth O(
p
The union has separators of size O(
p
k).
In each component, the set of cities visited by the optimum
solution is nice: it is the same as what O(
p
k) segments of the
4-OPT tour visited (kO(
p
k) possibilities).
Use board for few intuition!
38

Steiner Tree
Steiner Tree
Input: A graph G, a set T of n terminal vertices, a weight function
w : E(G) ! Q 0
Parameter: |T|
Question: A minimum weight Steiner tree for T in G.
39

Steiner Tree
Dreyfus-Wagner gave an algorithm with running time
3n · log W · |V (G)|O(1). W is the maximum weight on an edge.
40

Steiner Tree
In 2013, Nederlof et. al. gave an alternative algorithm with
running time 2n · |V (G)|O(1) · W .
40

Steiner Tree
In 2013, Nederlof et. al. gave an alternative algorithm with
running time 2n · |V (G)|O(1) · W .
Approximation algorithm: Byrka et. al. gave an algorithm with
approximation factor ln(4) + ✏.
40

Rectilinear Steiner Tree
Input: A set T of n terminal points, recdistG or the taxi-
cab/rectilinear metric.
Parameter: |T| = n
Question: A minimum length network connecting any two points
in T.
41

t2
t1
t4 t3
Figure: The set T
42

t2
t1
t4 t3
Figure: A network connecting T
43

Result
Theorem
Rectilinear Steiner Tree can be solved in time 2O(
p
n log n)nO(1).
44

Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
grid graph.
45

grid graph.
Step 2: Find a 2-approximate solution bS for Rectilinear Steiner
Tree.
45

grid graph.
Tree.
Step 3: Show that there exists an optimal solution Sopt such
that the subgraph S = bS [ Sopt has a “useful” structure.
45

grid graph.
Tree.
Step 4: Use a “ghost” structure to do dynamic programming
to solve the problem.
45

t2
t1
t4 t3
47

t2
t1
t4 t3
Figure: The Hanan grid.
48

Step 1: Graphic Interpretation
Input: A set T of n terminal points, the Hanan grid G of T and
recdistG .
Parameter: |T| = n
Question: A minimum Steiner tree for T in G.
A Hanan grid with n terminals has at most n2 vertices in total.
49

Step 2: Shortest path RST
t2
t1
t4 t3
50

t2
t1
t4 t3
51

t2
t1
t4 t3
52

Shortest path RST
Can ensure that at each iterative step, at most one bend
vertex is created. Total of n bend vertices.
53

Shortest path RST
Can ensure that at each iterative step, at most one bend
vertex is created. Total of n bend vertices.
The tree is constructed in polynomial time.
53

grid graph.
Tree.
54

Hanan grid.
Tree.
55

Hanan grid.
Step 2: Find a Shortest Path RST bS.
56

Hanan grid.
that the subgraph S = bS [ Sopt has bounded treewidth.
57

Hanan grid.
that the subgraph S = bS [ Sopt has bounded treewidth.
Step 4: Use a “ghost” tree decomposition to do dynamic
programming to solve the problem.
58

Step 3: Bounded treewidth subgrid
Let Sopt be the optimal solution that has minimum number of
bends.
Sopt has 3n bend vertices.
59

bends.
bS has n bend vertices.
59

bends.
bS has n bend vertices.
There are n terminal vertices.
59

Attempt 1:
S = bS [ Sopt is a planar graph.
60

Attempt 1:
Has treewidth at most
p
|V (S)|.
60

Attempt 1:
Has treewidth at most
p
|V (S)|.
In this way, bound on treewidth as bad as O(n).
60

minor.
61

minor.
Suppose S has treewidth at most ↵
p
n. Then there is a
9
p
n ⇥ 9
p
n grid as a minor.
61

62

63

64

65

Bounded treewidth subgrid
P147 P258 P369
P123
P456
P789
66

The subgraph S = bS [ Sopt has treewidth 41
p
n.
67

The subgraph S = bS [ Sopt has treewidth 41
p
n.
Potential tree decomposition with bounded treewidth.
67

Step 4: Treewidth Dynamic programming
If we knew the tree decomposition, we can do bounded
treewidth DP.
68

If we knew the tree decomposition, we can do bounded
treewidth DP.
We don’t know S.
68

Upper graph
Lower graph
Bag
• •
• •
••
•
•
• • • •
•
69

Steiner Tree Treewidth DP
The important step in the algorithm for Steiner Tree is to compute
the following information:
70

the following information: for each bag Xt, X ✓ Xt and a partition
P = (P1, . . . , Pq) of X, the value c[t, X, P] is the minimum weight
of a subgraph F of St with the following properties:
70

1 F has exactly q connected components C1, . . . , Cq such that
; 6= Pi = Xt V (Ci ) for all i 2 [q]. That is, P corresponds to
connected components of F.
70

2 Xt V (F) = X. That is, the vertices of Xt X are untouched
by F.
70

2 Xt V (F) = X. That is, the vertices of Xt X are untouched
by F.
3 T Vt ✓ V (F). That is, all the terminal vertices in St belong
to F.
70

In our case, we deﬁne types. Bags of a tree decomposition '
types.A type is a tuple (Y , Y 0 ✓ Y , P, T0) such that the following
holds.
71

holds.
(i) Y is a subset of V (G) of size at most q = 41
p
n + 2.
(ii) P is a partition of Y 0.
71

holds.
(i) Y is a subset of V (G) of size at most q = 41
p
n + 2.
(ii) P is a partition of Y 0.
(iii) There exists a set of components C1, . . . , Cq in bS Y such
that T0 = T Y 0 [
Sq
i=1 V (Ci ) .
71

There are at most 2
p
n log n types. This helps to design a dynamic
programming for the ﬁnal step of the algorithm.
72

Parameterized problems where bidimensionality does not work.
Upper bounds:
Algorithms based on ﬁnding a bounded-treewidth subgraph.
Treewidth bound is problem-speciﬁc:
k-Terminal Cut: dual solution has O(k) branch vertices.
Planar Strongly Connected Subgraph: solution
consists of O(k) paths/strips.
Subset TSP on planar graphs: the union of an optimum
solution and a 4-OPT solution has treewidth O(
p
k).
Rectilinear Steiner Tree on planar graphs: the union of
an optimum solution and an approximation solution has
treewidth O(
p
k).
Lower bounds:
To rule out f (k) · no(
p
k) time algorithms, we have to prove
W[1]-hardness by reduction from Grid Tiling.
73

Moving towards directed graphs.
74

Grid minors and treewidth
Grid Minor Theorem Robertson,
Seymour; GM V
There is a function f such that tw(G) f (k) implies the existence
of a k ⇥ k grid minor in G.
75

Seymour; GM V
Upper bound: f (k) 2 ˜O(k19) (Chuzhoy ’15)
75

Seymour; GM V
Lower bound: f (k) 2 ⌦(k2 log k)
75

Seymour; GM V
Minor-free: f (k) 2 ⇥(k) for graphs excluding a ﬁxed minor.
75

Seymour; GM V
Planar: tw(G) > 4.5k + 1 implies a k ⇥ k grid minor.
75

Seymour; GM V
Planar: tw(G) > 4.5k + 1 implies a k ⇥ k grid minor.
H-minor-free: tw(G) > cH · k implies a k ⇥ k grid minor.
75

Bidimensionality
k-Path: Is there a simple path on k vertices in a graph?
76

Bidimensionality
Goal: 2O(
p
k) · n algorithm for planar graphs.
76

Bidimensionality
Goal: 2O(
p
Fact 1: k-Path can be solved in time 2O(w) · n on a tree
decomposition of width w.
76

Bidimensionality
Goal: 2O(
p
Fact 1: k-Path can be solved in time 2O(w) · n on a tree
decomposition of width w.
Fact 2: If there is
p
k ⇥
p
k grid minor, then there is a k-path.
76

Algorithm
Approximate treewidth.

Algorithm
If tw 100
p
k then there is a
p
k ⇥
p
k grid minor, so also a
k-path.

Algorithm
If tw 100
p
k then there is a
p
k ⇥
p
k grid minor, so also a
k-path.
Otherwise we have a tree decomposition of width O(
p
k).
Apply dynamic programming.
77

Bidimensionality: drawbacks
The argument is elegant, but very delicate.
78

Assumption “large grid minor ) certain answer” is very strong.
78

Directed k-Path: simple path on k vertices in a
directed graph.
78

directed graph.
A large grid minor in the underlying undirected graph tells
little about directed k-paths.
78

directed graph.
Similar problem for searching for k-path of maximum weight,
or a cycle of length exactly k, ...
78

directed graph.
Can such problems be solved in time 2
˜O(
p
k) · nc on planar
graphs?
78

directed graph.
˜O(
p
k) · nc on planar
graphs?
Until Recently, no better algorithms were known than
2O(k)
· nc
inherited from the general setting.
78

directed graph.
˜O(
p
k) · nc on planar
graphs?
2O(k)
· nc
General: Let a k-pattern be a vertex subset P such that
|P|  k and G[P] is connected.
78

directed graph.
˜O(
p
k) · nc on planar
graphs?
2O(k)
· nc
We are looking with k-patterns with some prescribed property.
78

directed graph.
˜O(
p
k) · nc on planar
graphs?
2O(k)
· nc
We are looking with k-patterns with some prescribed property.
Idea: Find a small family of subgraphs of small treewidth that
cover every pattern.
78

Pattern coverage
Main result
Suppose G is a planar graph on n vertices, and suppose k is a
positive integer. Then there exists a family F of subsets of V (G)
such that:
(a) |F|  2
Õ(
p
k) · nc for some constant c;
(b) For each A 2 F, we have tw(G[A])  Õ(
p
k);
(c) For every k-pattern P, there is some A 2 F such that P ✓ A.
Moreover, F can be constructed in randomized time 2
Õ(
p
k) · nc.
79

Pattern coverage
Main result
such that:
(a) |F|  2
˜O(
p
p
k);
˜O(
p
k) · nc.
Directed k-Path algorithm:
79

Pattern coverage
Main result
such that:
(a) |F|  2
˜O(
p
p
k);
˜O(
p
k) · nc.
Construct F.
79

Pattern coverage
Main result
such that:
(a) |F|  2
Õ(
p
p
k);
Õ(
p
k) · nc.
Construct F.
For each A 2 F, try to find a k-path in G[A] by DP.
79

Pattern coverage
Main result
such that:
(a) |F|  2
Õ(
p
p
k);
Õ(
p
k) · nc.
Construct F.
For each A 2 F, try to find a k-path in G[A] by DP.
Running time: (2
Õ(
p
k)
· nc
) · (2
Õ(
p
k)
· n) = 2
Õ(
p
k)
· nc+1
79

Applications
Applies to a range of pattern-searching problems:
80

Applications
(directed) k-path of largest weight;
80

Applications
cycle of length exactly k;
80

Applications
k-local search for Planar Vertex Cover.
80

Applications
Needed: 2
˜O(w)
· nc
algorithm on graphs of treewidth w.
80

Applications
Needed: 2
˜O(w)
· nc
Subgraph Isomorphism:
Given G and H with |V (H)|  k, is H a subgraph of G?
80

Applications
Needed: 2
˜O(w)
· nc
When H is connected and has maximum degree bounded by a
constant, we obtain an algorithm with running time 2
˜O(
p
k)
·nc
.
80

Applications
Needed: 2
˜O(w)
· nc
˜O(
p
k)
·nc
.
Without the maxdeg bound, we get running time
2O(k/ log k)
· nc
using (Bodlaender, Nederlof, van der Zanden;
ICALP’16).
80

Applications
Needed: 2
˜O(w)
· nc
˜O(
p
k)
·nc
.
Without the maxdeg bound, we get running time
2O(k/ log k)
· nc
using (Bodlaender, Nederlof, van der Zanden;
ICALP’16).
This running time is tight under ETH! (Bodlaender et al.)
80

Example: grid
Take a k ⇥ k grid.
81

Example: grid
Baker: Guess an index modulo
p
k s.t. there are 
p
k
vertices of the pattern in the corresponding rows.
81

Example: grid
p
p
k
Guess columns in which these vertices lie, kp
k options.
81

Example: grid
p
p
k
k options.
Remove the rows apart from intersections with columns.
What remains has treewidth O(
p
k).
81

Example: grid
p
p
k
k options.
Remove the rows apart from intersections with columns.
What remains has treewidth O(
p
k).
In total,
p
k · kp
k = 2O(
p
k log k) possible guesses.
81

Glimpse into the proof
Randomized statement:
There is a polytime algorithm that samples A with treewidth
˜O(
p
k) s.t. every k-pattern is covered with probability
(2
˜O(
p
k) · nc) 1.
82

˜O(
p
(2
˜O(
p
k) · nc) 1.
Idea: Emulate a recursive approximation algorithm for
treewidth, trying to construct a decomposition of width
p
k.
82

Õ(
p
(2
Õ(
p
k) · nc) 1.
p
k.
When the algorithm gets stuck, apply Baker-like sparsification.
82

Õ(
p
(2
Õ(
p
k) · nc) 1.
p
k.
State of the algorithm:
Graph G with Õ(
p
k) terminals on the outer face.
82

Õ(
p
(2
Õ(
p
k) · nc) 1.
p
k.
State of the algorithm:
Graph G with Õ(
p
k) terminals on the outer face.
Pieces of the (unkown) pattern “hang” from terminals.
82

Proof strategy
Observation: Everything would be trivial if the pattern would
be within an ˜O(
p
k) margin from the outer face.
˜O(
p
k)
83

Proof strategy
be within an ˜O(
p
Otherwise, one of three possibilities holds:
˜O(
p
k)
83

Proof strategy
be within an ˜O(
p
1 Balanced separator that avoids the pattern outside the margin.
83

Proof strategy
be within an ˜O(
p
Otherwise we “locate” a vertex z of the pattern outside the
margin.
z
83

Proof strategy
be within an ˜O(
p
margin.
2 Many concentric short separators separating z from the outer
face.
z
83

Proof strategy
be within an ˜O(
p
margin.
2 Many concentric short separators separating z from the outer
face.
3 Many “almost” disjoint paths connecting z with the outer face.
83

Achieving progress
Balanced separator: Remove outside margin and split.
84

Achieving progress
Progress: The graph size and the pattern size are halved.
84

Achieving progress
Concentric short separators:
z
84

Achieving progress
Progress: Guess a separator that splits the pattern in a
balanced way, and has a small intersection with it.
z
84

Achieving progress
Nearly disjoint paths:
84

Achieving progress
Progress: Guess a path where private are not in the pattern.
84

Achieving progress
Contract along it, thus dragging more of the pattern into the
margin.
84

Achieving progress
Contract along it, thus dragging more of the pattern into the
margin.
Keep track of 3 potentials measuring the progress to estimate
the success probability.
84

Generalizations
We rely on two properties of the graph class C:
85

Generalizations
Minor-closed: If H G and G 2 C, then H 2 C.
85

Generalizations
Locally bounded treewidth: tw(G)  f (rad(G)) for some f .
85

Generalizations
Classes C satisfying the above are exactly apex-minor-free.
85

Generalizations
Corollary: All the results hold on any apex-minor-free class.
85

Generalizations
We can generalize to H-minor-free graphs for any H, but we
need to assume that the pattern has bounded maximum
degree.
85

Generalizations
degree.
Issue: Apices can “teleport” the pattern almost arbitrarily.
85

Generalizations
degree.
Open:
85

Generalizations
degree.
Open:
Handling disconnected patterns.
85

Generalizations
degree.
Open:
Handling disconnected patterns.
Extending the technique to general H-minor-free graphs.
85

Contribution: New tool for developing subexp. parameterized
algorithms on topologically-constrained graph classes.
86

Usage: Searching for small patterns with prescribed properties.
86

Seems more robust than bidimensionality.
86

Outlook:
86

Outlook:
A similar tool can be given for graph classes with polynomial
growth (Marx, Pilipczuk; 2016+).
86

Outlook:
A similar tool can be given for graph classes with polynomial
growth (Marx, Pilipczuk; 2016+).
Questions:
Derandomization, disconnected patterns, H-minor-free graphs.
86

Conclusions
A robust understanding of why certain problems can be solved
in time 2O(
p
n) etc. on planar graphs and why the square root
is best possible.
87

Conclusions
in time 2O(
p
is best possible.
Going beyond the basic toolbox requires new problem-speciﬁc
algorithmic techniques and hardness proofs with tricky gadget
constructions.
87

Conclusions
in time 2O(
p
is best possible.
Going beyond the basic toolbox requires new problem-speciﬁc
algorithmic techniques and hardness proofs with tricky gadget
constructions.
The lower bound technology on planar graphs cannot give a
lower bound without a square root factor. Does this mean that
there are matching algorithms for other problems as well?
2O(
p
k)
· nO(1)
time algorithm for Steiner Tree with k
terminals in a planar graph?
. . .
87

Conclusions
Chapter 1: Subexponential algorithms using treewidth.
Algorithms: standard treewidth algorithms.
Lower bounds: textbook NP-completeness proofs + ETH.
Chapter 2: Grid minors and bidimensionality.
Algorithms: standard treewidth algorithms + excluded grid
theorem.
Chapter 3: Finding bounded-treewidth solutions.
Algorithms: the solution can be represented by a graph of
treewidth O(
p
k).
Lower bounds: grid-like W[1]-hardness proofs to rule out
f (k) · no(
p
k)
algorithms.
Chapter 4: Pattern Coverage.
New Tool: A kind of graphic hash functions covering patterns
of treewidth O(
p
k). Much more to be explored here.
88

Conclusions
Chapter 1: Subexponential algorithms using treewidth.
Algorithms: standard treewidth algorithms.
Chapter 2: Grid minors and bidimensionality.
Algorithms: standard treewidth algorithms + excluded grid
theorem.
Chapter 3: Finding bounded-treewidth solutions.
Algorithms: the solution can be represented by a graph of
treewidth O(
p
k).
Lower bounds: grid-like W[1]-hardness proofs to rule out
f (k) · no(
p
k)
algorithms.
Chapter 4: Pattern Coverage.
New Tool: A kind of graphic hash functions covering patterns
of treewidth O(
p
k). Much more to be explored here.
Thanks for listening!
88

A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: From Undirected to Directed

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A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: From Undirected to Directed