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

A Quest for Subexponential Time ParameterizedAlgorithms for Planar-k-Path: From Undirected to

Directed Graphs

Saket Saurabh1

Institute for Mathematical Sciences, Chennai, India,

and University of Bergen, Norway.

NMI Workshop on Complexity Theory, IITGN, Saturday 5thNovember, 2016

(Slides by D. Marx, M. Pilipzuck and S. Saurabh)

Main message

NP-hard problems become easier on planar graphsand geometric objects, and usually exactly by asquare root factor.

Planar graphs Geometric objects

Better exponential algorithms

Most NP-hard problems (e.g., 3-Coloring, Independent Set,Hamiltonian Cycle, Steiner Tree, etc.) remain NP-hard onplanar graphs,1 so what do we mean by “easier”?

The running time is still exponential, but significantly smaller:

2O(n) ) 2O(pn)

nO(k) ) nO(pk)

2O(k) · nO(1) ) 2O(pk) · nO(1)

1Notable 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 onplanar graphs,1 so what do we mean by “easier”?

The running time is still exponential, but significantly smaller:

2O(n) ) 2O(pn)

nO(k) ) nO(pk)

2O(k) · nO(1) ) 2O(pk) · nO(1)

1Notable 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.

Chapter 1: Subexponential algorithms using treewidth

Treewidth is a measure of “how treelike the graph is.”

We need only the following basic facts:

Treewidth1 If a graph G has treewidth k , then many classical NP-hard

problems can be solved in time 2O(k) · nO(1) or2O(k log k) · nO(1) on G .

2 A planar graph on n vertices has treewidth O(pn).

Treewidth — a measure of “tree-likeness”Tree decomposition: Vertices are arranged in a tree structuresatisfying the following properties:

1 If u and v are neighbors, then there is a bag containing bothof 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.

e f g h

g , hb, e, fa, b, c

d , f , gb, c, f

c, d , f

A subtree communicates with the outside worldonly via the root of the subtree.

Treewidth — a measure of “tree-likeness”Tree decomposition: Vertices are arranged in a tree structuresatisfying the following properties:

1 If u and v are neighbors, then there is a bag containing bothof 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.

g , hb, e, fa, b, c

d , f , gb, c, f

c, d , f

A subtree communicates with the outside worldonly via the root of the subtree.

Finding tree decompositions

Various algorithms for finding optimal or approximate treedecompositions 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(

plogw)-approximation in polynomial time

[Feige, Hajiaghayi, Lee 2008].As we are mostly interested in algorithms with running time2O(w) · nO(1), we may assume that we have a decomposition.

3-Coloring and tree decompositions

TheoremGiven a tree decomposition of width w , 3-Coloring can besolved 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 valueE [x , c], which is true if and only if c canbe extended to a proper 3-coloring of Vx .

Claim:We can determine E [x , c] if all the values areknown 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=Fbcf=T bcf=F. . . . . .

Subexponential algorithm for 3-Coloring

Theorem [textbook dynamic programming]

3-Coloring can be solved in time 2O(w) · nO(1) on graphs oftreewidth w .

Theorem [Robertson and Seymour]

A planar graph on n vertices has treewidth O(pn).

+Corollary

3-Coloring can be solved in time 2O(pn) on planar graphs.

textbook algorithm + combinatorial bound+

subexponential algorithm

Theorem [textbook dynamic programming]

3-Coloring can be solved in time 2O(w) · nO(1) on graphs oftreewidth w .

+Corollary

subexponential algorithm9

Theorem3-Coloring can be solved in time 2O(w) · nO(1) on graphs oftreewidth w .

+Corollary

subexponential algorithm10

Lower bounds

Corollary

Two natural questions:Can we achieve this running time on general graphs?Can we achieve even better running time (e.g., 2O( 3

pn)) on

planar graphs?

P 6= NP is not a sufficiently strong hypothesis: it is compatible with3SAT being solvable in time 2O(n1/1000) or even in time nO(log n).

We need a stronger hypothesis!

Lower bounds

Corollary

Two natural questions:Can we achieve this running time on general graphs?Can we achieve even better running time (e.g., 2O( 3

pn)) on

planar graphs?

P 6= NP is not a sufficiently strong hypothesis: it is compatible with3SAT being solvable in time 2O(n1/1000) or even in time nO(log n).

We need a stronger hypothesis!

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.

Are there algorithms that are subexponential in the size n +m ofthe 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.

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 variablesm clauses

)Graph G

O(n +m) verticesO(n +m) edges

(we can assume n = O(m))

Corollary

Assuming ETH, there is no 2o(n) algorithm for 3-Coloring on ann-vertex graph G .

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:

)Graph G

O(m) verticesO(m) edges

(we can assume n = O(m))

Corollary

Assuming ETH, there is no 2o(n) algorithm for 3-Coloring on ann-vertex graph G .

Transfering boundsThere are polynomial-time reductions from, say, 3-Coloring tomany other problems such that the reduction increases the numberof vertices by at most a constant factor.

Consequence: Assuming ETH, there is no 2o(n) time algorithm onn-vertex graphs for

Independent Set

Clique

Dominating Set

Vertex Cover

Hamiltonian Path

Feedback Vertex Set

Lower bounds based on ETHWhat about 3-Coloring on planar graphs?

The textbook reduction from 3-Coloring to Planar3-Coloring uses a “crossover gadget” with 4 external connectors:

In every 3-coloring of the gadget, opposite external connectorshave the same color.Every coloring of the external connectors where the oppositevertices have the same color can be extended to the wholegadgets.If two edges cross, replace them with a crossover gadget.

The reduction from 3-Coloring to Planar 3-Coloringintroduces O(1) new edge/vertices for each crossing.A graph with m edges can be drawn with O(m2) crossings.

)Graph G

O(m) verticesO(m) edges

)Planar graph G 0

O(m2) verticesO(m2) edges

Corollary

Assuming ETH, there is no 2o(pn) algorithm for 3-Coloring on

an n-vertex planar graph G .

(Essentially observed by [Cai and Juedes 2001])

Summary of Chapter 1

Streamlined way of obtaining tight upper and lower bounds forplanar problems.

Upper bound:Standard bounded-treewidth algorithm + treewidth bound onplanar graphs give 2O(

pn) time subexponential algorithms.

Lower bound:Textbook NP-hardness proof with quadratic blow up + ETHrule out 2o(

pn) algorithms.

Works for Hamiltonian Cycle, Vertex Cover,Independent Set, Feedback Vertex Set, DominatingSet, Steiner Tree, . . .

Chapter 2: Grid minors and bidimensionalityMore refined analysis of the running time: we express the runningtime as a function of input size n and a parameter k .

DefinitionA problem is fixed-parameter tractable (FPT) parameterized byk if it can be solved in time f (k) · nO(1) for some computablefunction 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, whichis best possible on general graphs.

W[1]-hardness

Negative evidence similar to NP-completeness. If a problem isW[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)).

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) timealgorithm on general graphs.(As this would imply a 2o(n) algorithm.)

For most problems, we cannot expect a 2o(pk) · nO(1) time

algorithm on planar graphs.(As this would imply a 2o(

pn) algorithm.)

However, 2O(pk) · 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.

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) timealgorithm on general graphs.(As this would imply a 2o(n) algorithm.)

For most problems, we cannot expect a 2o(pk) · nO(1) time

algorithm on planar graphs.(As this would imply a 2o(

pn) algorithm.)

However, 2O(pk) · 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.

Minors

DefinitionGraph H is a minor of G (H G ) if H can be obtained from G bydeleting edges, deleting vertices, and contracting edges.

deleting uv

u vcontracting uv

Note: length of the longest path in H is at most the length of thelongest path in G .

Planar Excluded Grid Theorem

Theorem [Robertson, Seymour, Thomas 1994]

Every planar graph with treewidth at least 5k has a k ⇥ k gridminor.

Note: for general graphs, treewidth at least k100 (now even k19) orso guarantees a k ⇥ k grid minor [Chekuri and Chuzhoy 2013] and[Chuzhoy 2016]!

Bidimensionality for k-Path

Observation: If the treewidth of a planar graph G is at least 5pk

) It has apk ⇥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 .

We use this observation to find a path of length at least k onplanar graphs:

) It has apk ⇥p

If treewidth w of G is at least 5pk :

we answer “there is a path of length atleast k .”If treewidth w of G is less than 5

then we can solve the problem in time2O(w) · nO(1) = 2O(

pk) · nO(1).

) It has apk ⇥p

Set w := 5pk .

Find an O(1)-approximate treedecomposition.

If treewidth is at least w : we cananswer “there is a path of length atleast k .”If we get a tree decomposition ofwidth O(w), then we can solve theproblem in time2O(w) · nO(1) = 2O(

pk) · nO(1).

BidimensionalityDefinitionA graph invariant x(G ) is minor-bidimensional if

x(G 0) x(G ) for every minor G 0 of G , andIf 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.

Tight bounds for minor-bidimensional planar problems.

Upper bound:Standard bounded-treewidth algorithm + planar excluded gridtheorem give 2O(

pk) · nO(1) time FPT algorithms.

Lower bound:Textbook NP-hardness proof with quadratic blow up + ETHrule out 2o(

pn) time algorithms ) no 2o(

pk) · nO(1) time

algorithm.

Variant of theory works for contraction-bidimensional problems,e.g., Independent Set, Dominating Set.

Limits of bidimensionality?Perhaps the most basic problem:

Subgraph IsomorphismGiven graphs H and G , decide if G has a subgraph isomorphic toH.

Theorem [Eppstein 1999]

Subgraph Isomorphism for planar graphs can be solved in time2O(k log k) · n for k := |V (H)|.

Question already asked several time over the last decades:

Does the square root phenomenon appear for SubgraphIsomorphism on planar graphs?

Assuming ETH, there is no 2o(k/ log k)nO(1) time algorithm forgeneral patterns.

[Bodlaender, Nederlof, van der Zanden; ICALP’16)]

Assuming ETH, there is no 2o(k/ log k)nO(1) time algorithm forgeneral patterns.

[Bodlaender, Nederlof, van der Zanden; ICALP’16)]

There is a 2O(k/ log k)nO(1) time (randomized) algorithm forconnected 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.

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.

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.

Bidimensionality and counting

Does not work for counting k-paths in a planar graph:

If treewidth w is O(pk): can be solved in time

2O(w)nO(1) = 2O(pk)nO(1) using dynamic programming.

If treewidth w is larger than cpk : answer is positive, but how

much exactly?

What about counting vertex covers of size k in a planar graph?Works for counting vertex covers of size k in a planar graph:

If treewidth w is larger than cpk : answer is 0.

much exactly?

What about counting vertex covers of size k in a planar graph?

Works for counting vertex covers of size k in a planar graph:

much exactly?

What about counting vertex covers of size k in a planar graph?

Works for counting vertex covers of size k in a planar graph:

Counting k-matching

Counting matchings can be significantly harder than finding amatching!

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(pk) · nO(1) algorithm for counting k

matchings in planar graphs?

Counting k-matching

Counting matchings can be significantly harder than finding amatching!

Counting perfect matchings in planar graphs ispolynomial-time solvable.[Kasteleyn 1961], [Temperley and Fischer 1961].Corollary: we can count matchings covering n � k vertices intime nO(k)

. . . but (assuming ETH) there is no f (k)no(k/ log k) timealgorithm [Curticapean and Xia 2015].

Chapter 3: Finding bounded-treewidth solutions

So far, we have exploited that the input has bounded treewidthand used standard algorithms.

Change of viewpoint:

In many cases, we have to exploit instead that the solution hasbounded treewidth.

So far, we have exploited that the input has bounded treewidthand used standard algorithms.

Change of viewpoint:

In many cases, we have to exploit instead that the solution hasbounded treewidth.

Chapter 3: Finding bounded-treewidth solutionsSo far the way we have used treewidth is to find something (e.g.,Hamiltonian cycle) in a large bounded-treewidth graph:

If the problem can be formulated as finding a graph of treewidthO(

pk), then we get an nO(

pk) time algorithm.

Chapter 3: Finding bounded-treewidth solutionsSo far the way we have used treewidth is to find something (e.g.,Hamiltonian cycle) in a large bounded-treewidth graph:

pk) time algorithm.

Chapter 3: Finding bounded-treewidth solutionsBut we can also find small bounded-treewidth objects in an arbitrarylarge graph.

Theorem [Alon, Yuster, Zwick 1994]

Given a graph H and weighted graph G , we can find a minimumweight subgraph of G isomorphic to H in time 2O(|V (H)|) · nO(tw(H)).

pk) time algorithm.

Chapter 3: Finding bounded-treewidth solutionsBut we can also find small bounded-treewidth objects in an arbitrarylarge graph.

Theorem [Alon, Yuster, Zwick 1994]

Given a graph H and weighted graph G , we can find a minimumweight subgraph of G isomorphic to H in time 2O(|V (H)|) · nO(tw(H)).

pk) time algorithm.

Examples

Several examples:Planar k-Terminal CutImprovement from nO(k) to 2O(k) · nO(

Planar Strongly Connected SubgraphImprovement from nO(k) to 2O(k log k) · nO(

Subset TSP with k cities in a planar graphImprovement from 2O(k) · nO(1) to 2O(

pk log k) · nO(1).

Rectilinear Steiner Tree with n points in a plane.Improvement from 2O(n) · nO(1) to 2O(

pn log n) · nO(1).

TSPTSP

Input: A set T of cities and a distance function d on TOutput: 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 ,T 0) be the minimum length of path from v

to vvisiting all the cities T 0 ✓ T .

Subset TSP on planar graphsAssume that the cities correspond to a subset T of a planar graphand distance is measured in this planar graph.

Can be solved in time 2O(pn).

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

pk) · nO(1)?

Subset TSP on planar graphsAssume that the cities correspond to a subset T of a planar graphand distance is measured in this planar graph.

Theorem [Klein and Marx]Subset TSP for k cities in a (unit-weight) planar graph can besolved in time 2O(

pk log k) · nO(1).

TSP and treewidthWe wanted to formulate the problem as finding a lowtreewidth 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).

c-change TSP

c-change operation: removing c steps of the tour andconnecting 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 isthe maximum distance (if distances are integers).

c-change TSP

The treewidth boundConsider the union of an optimum solution and a 4-OPT solutionas a graph on k vertices:

LemmaThe union of an optimum solution and a 4-OPT solution hastreewidth O(

pk) [some techincal details omitted].

The treewidth bound

The union has separators of size O(pk).

In each component, the set of cities visited by the optimumsolution is nice: it is the same as what O(

pk) segments of the

4-OPT tour visited (kO(pk) possibilities).

Use board for few intuition!

The treewidth bound

The union has separators of size O(pk).

In each component, the set of cities visited by the optimumsolution is nice: it is the same as what O(

pk) segments of the

4-OPT tour visited (kO(pk) possibilities).

Use board for few intuition!38

Steiner Tree

Steiner TreeInput: A graph G , a set T of n terminal vertices, a weight functionw : E (G ) ! Q�0

Parameter: |T |Question: A minimum weight Steiner tree for T in G .

Steiner Tree

Dreyfus-Wagner gave an algorithm with running time3n · logW · |V (G )|O(1). W is the maximum weight on an edge.

In 2013, Nederlof et. al. gave an alternative algorithm withrunning time 2n · |V (G )|O(1) ·W .Approximation algorithm: Byrka et. al. gave an algorithm withapproximation factor ln(4) + ✏.

Steiner Tree

Dreyfus-Wagner gave an algorithm with running time3n · logW · |V (G )|O(1). W is the maximum weight on an edge.In 2013, Nederlof et. al. gave an alternative algorithm withrunning time 2n · |V (G )|O(1) ·W .

Approximation algorithm: Byrka et. al. gave an algorithm withapproximation factor ln(4) + ✏.

Steiner Tree

Dreyfus-Wagner gave an algorithm with running time3n · logW · |V (G )|O(1). W is the maximum weight on an edge.In 2013, Nederlof et. al. gave an alternative algorithm withrunning time 2n · |V (G )|O(1) ·W .Approximation algorithm: Byrka et. al. gave an algorithm withapproximation factor ln(4) + ✏.

Rectilinear Steiner Tree

Rectilinear Steiner TreeInput: A set T of n terminal points, recdistG or the taxi-cab/rectilinear metric.Parameter: |T | = nQuestion: A minimum length network connecting any two pointsin T .

Figure: The set T

Figure: A network connecting T

Result

Theorem

Rectilinear Steiner Tree can be solved in time 2O(pn log n)nO(1).

Outline of the algorithm

Step 1: Interpret the problem as a Steiner Tree problem on agrid graph.

Step 2: Find a 2-approximate solution bS for Rectilinear SteinerTree.Step 3: Show that there exists an optimal solution S

suchthat the subgraph S = bS [ S

has a “useful” structure.Step 4: Use a “ghost” structure to do dynamic programmingto solve the problem.

Step 1: Interpret the problem as a Steiner Tree problem on agrid graph.Step 2: Find a 2-approximate solution bS for Rectilinear SteinerTree.

Step 3: Show that there exists an optimal solution Sopt

Step 1: Interpret the problem as a Steiner Tree problem on agrid graph.Step 2: Find a 2-approximate solution bS for Rectilinear SteinerTree.Step 3: Show that there exists an optimal solution S

has a “useful” structure.

Step 4: Use a “ghost” structure to do dynamic programmingto solve the problem.

Step 1

Figure: The Hanan grid.

Step 1: Graphic Interpretation

Rectilinear Steiner TreeInput: A set T of n terminal points, the Hanan grid G of T andrecdistG .Parameter: |T | = nQuestion: A minimum Steiner tree for T in G .

A Hanan grid with n terminals has at most n2 vertices in total.

Step 2: Shortest path RST

Shortest path RST

Can ensure that at each iterative step, at most one bendvertex is created. Total of n bend vertices.

The tree is constructed in polynomial time.

Shortest path RST

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

Step 1: Interpret the problem as a Steiner Tree problem on aHanan grid.Step 2: Find a 2-approximate solution bS for Rectilinear SteinerTree.Step 3: Show that there exists an optimal solution S

Step 1: Interpret the problem as a Steiner Tree problem on aHanan grid.Step 2: Find a Shortest Path RST bS .Step 3: Show that there exists an optimal solution S

has bounded treewidth.Step 4: Use a “ghost” structure to do dynamic programmingto solve the problem.

has bounded treewidth.Step 4: Use a “ghost” tree decomposition to do dynamicprogramming to solve the problem.

Step 3: Bounded treewidth subgrid

Let Sopt

be the optimal solution that has minimum number ofbends.

has 3n bend vertices.

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

Let Sopt

has 3n bend vertices.bS has n bend vertices.

There are n terminal vertices.

Let Sopt

has 3n bend vertices.bS has n bend vertices.There are n terminal vertices.

Attempt 1:S = bS [ S

is a planar graph.

Has treewidth at mostp|V (S)|.

In this way, bound on treewidth as bad as O(n).

is a planar graph.Has treewidth at most

p|V (S)|.

In this way, bound on treewidth as bad as O(n).

is a planar graph.Has treewidth at most

p|V (S)|.In this way, bound on treewidth as bad as O(n).

Suppose S has treewidth at most ↵pn. Then there is a

9pn ⇥ 9

pn grid as a minor.

Suppose S has treewidth at most ↵pn. Then there is a

9pn ⇥ 9

pn grid as a minor.

Bounded treewidth subgrid

The subgraph S = bS [ Sopt

has treewidth 41pn.

Potential tree decomposition with bounded treewidth.

The subgraph S = bS [ Sopt

has treewidth 41pn.

Potential tree decomposition with bounded treewidth.

Step 4: Treewidth Dynamic programming

If we knew the tree decomposition, we can do boundedtreewidth DP.

We don’t know S .

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

Upper graph

Lower graph

• • • ••• ••

• • • •

Steiner Tree Treewidth DP

The important step in the algorithm for Steiner Tree is to computethe following information:

for each bag Xt , X ✓ Xt and a partitionP = (P

, . . . ,Pq) of X , the value c[t,X ,P] is the minimum weightof a subgraph F of St with the following properties:

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 toconnected components of F .

2 Xt \ V (F ) = X . That is, the vertices of Xt \ X are untouchedby F .

3 T \ Vt ✓ V (F ). That is, all the terminal vertices in St belongto F .

The important step in the algorithm for Steiner Tree is to computethe following information: for each bag Xt , X ✓ Xt and a partitionP = (P

In our case, we define types. Bags of a tree decomposition 'types.A type is a tuple (Y ,Y 0 ✓ Y ,P,T 0) such that the followingholds.

(i) Y is a subset of V (G ) of size at most q = 41pn + 2.

(ii) P is a partition of Y 0.(iii) There exists a set of components C

, . . . ,Cq in bS � Y suchthat T 0 = T \ �

Y 0 [Sqi=1

V (Ci )�.

In our case, we define types. Bags of a tree decomposition 'types.A type is a tuple (Y ,Y 0 ✓ Y ,P,T 0) such that the followingholds.(i) Y is a subset of V (G ) of size at most q = 41

pn + 2.

(ii) P is a partition of Y 0.

(iii) There exists a set of components C1

Y 0 [Sqi=1

V (Ci )�.

In our case, we define types. Bags of a tree decomposition 'types.A type is a tuple (Y ,Y 0 ✓ Y ,P,T 0) such that the followingholds.(i) Y is a subset of V (G ) of size at most q = 41

pn + 2.

(ii) P is a partition of Y 0.(iii) There exists a set of components C

Y 0 [Sqi=1

V (Ci )�.

There are at most 2pn log n types. This helps to design a dynamic

programming for the final step of the algorithm.

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: solutionconsists of O(k) paths/strips.Subset TSP on planar graphs: the union of an optimumsolution and a 4-OPT solution has treewidth O(

Rectilinear Steiner Tree on planar graphs: the union ofan optimum solution and an approximation solution hastreewidth O(

Lower bounds:To rule out f (k) · no(

pk) time algorithms, we have to prove

W[1]-hardness by reduction from Grid Tiling.

Moving towards directed graphs.

Grid minors and treewidth

Grid Minor Theorem Robertson,

Seymour; GM V

There is a function f such that tw(G ) � f (k) implies the existenceof a k ⇥ k grid minor in G .

Upper bound: f (k) 2 O(k19) (Chuzhoy ’15)Lower bound: f (k) 2 ⌦(k2 log k)Minor-free: f (k) 2 ⇥(k) for graphs excluding a fixed minor.

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.

Seymour; GM V

Upper bound: f (k) 2 O(k19) (Chuzhoy ’15)

Lower bound: f (k) 2 ⌦(k2 log k)Minor-free: f (k) 2 ⇥(k) for graphs excluding a fixed minor.

Seymour; GM V

Upper bound: f (k) 2 O(k19) (Chuzhoy ’15)Lower bound: f (k) 2 ⌦(k2 log k)

Minor-free: f (k) 2 ⇥(k) for graphs excluding a fixed minor.

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.

Seymour; GM V

Bidimensionality

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

Goal: 2O(pk) · n algorithm for planar graphs.

Fact 1: k-Path can be solved in time 2O(w) · n on a treedecomposition of width w .Fact 2: If there is

pk ⇥p

k grid minor, then there is a k-path.

Bidimensionality

pk ⇥p

Bidimensionality

Fact 1: k-Path can be solved in time 2O(w) · n on a treedecomposition of width w .

Fact 2: If there ispk ⇥p

Bidimensionality

pk ⇥p

Algorithm

Approximate treewidth.

If tw � 100pk then there is a

pk ⇥p

k grid minor, so also ak-path.Otherwise we have a tree decomposition of width O(

Apply dynamic programming.

Algorithm

Approximate treewidth.If tw � 100

pk then there is a

pk ⇥p

k grid minor, so also ak-path.

Otherwise we have a tree decomposition of width O(pk).

Algorithm

Approximate treewidth.If tw � 100

pk then there is a

pk ⇥p

k grid minor, so also ak-path.Otherwise we have a tree decomposition of width O(

Bidimensionality: drawbacksThe argument is elegant, but very delicate.

Assumption “large grid minor ) certain answer” is very strong.Directed k-Path: simple path on k vertices in adirected graph.

A large grid minor in the underlying undirected graph tellslittle about directed k-paths.Similar problem for searching for k-path of maximum weight,or a cycle of length exactly k , ...

Can such problems be solved in time 2O(pk) · nc on planar

graphs?

Until Recently, no better algorithms were known than2O(k) · nc inherited from the general setting.

General: Let a k-pattern be a vertex subset P such that|P | k and G [P] is connected.

We are looking with k-patterns with some prescribed property.Idea: Find a small family of subgraphs of small treewidth thatcover every pattern.

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

Directed k-Path: simple path on k vertices in adirected graph.

graphs?

A large grid minor in the underlying undirected graph tellslittle about directed k-paths.

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

graphs?

graphs?Until Recently, no better algorithms were known than2O(k) · nc inherited from the general setting.

We are looking with k-patterns with some prescribed property.

Idea: Find a small family of subgraphs of small treewidth thatcover every pattern.

Pattern coverage

Main resultSuppose G is a planar graph on n vertices, and suppose k is apositive integer. Then there exists a family F of subsets of V (G )such that:(a) |F| 2O(

pk) · nc for some constant c ;

(b) For each A 2 F , we have tw(G [A]) O(pk);

(c) For every k-pattern P , there is some A 2 F such that P ✓ A.

Moreover, F can be constructed in randomized time 2O(pk) · nc .

Directed k-Path algorithm:

Construct F .For each A 2 F , try to find a k-path in G [A] by DP.Running time: (2O(

pk) · nc) · (2O(

pk) · n) = 2O(

pk) · nc+1

Pattern coverage

Directed k-Path algorithm:

Construct F .For each A 2 F , try to find a k-path in G [A] by DP.Running time: (2O(

pk) · nc) · (2O(

pk) · n) = 2O(

pk) · nc+1

Pattern coverage

Directed k-Path algorithm:Construct F .

For each A 2 F , try to find a k-path in G [A] by DP.Running time: (2O(

pk) · nc) · (2O(

pk) · n) = 2O(

pk) · nc+1

Pattern coverage

Directed k-Path algorithm:Construct F .For each A 2 F , try to find a k-path in G [A] by DP.

Running time: (2O(pk) · nc) · (2O(

pk) · n) = 2O(

pk) · nc+1

Pattern coverage

Directed k-Path algorithm:Construct F .For each A 2 F , try to find a k-path in G [A] by DP.Running time: (2O(

pk) · nc) · (2O(

pk) · n) = 2O(

pk) · nc+1

Applications

Applies to a range of pattern-searching problems:

(directed) k-path of largest weight;cycle of length exactly k ;k-local search for Planar Vertex Cover.Needed: 2O(w) · nc algorithm on graphs of treewidth w .

Subgraph Isomorphism:Given G and H with |V (H)| k , is H a subgraph of G?

When H is connected and has maximum degree bounded by aconstant, we obtain an algorithm with running time 2O(

pk) ·nc .

Without the maxdeg bound, we get running time2O(k/ log k) · nc using (Bodlaender, Nederlof, van der Zanden;ICALP’16).This running time is tight under ETH! (Bodlaender et al.)

Applications

Applies to a range of pattern-searching problems:(directed) k-path of largest weight;

cycle of length exactly k ;k-local search for Planar Vertex Cover.Needed: 2O(w) · nc algorithm on graphs of treewidth w .

pk) ·nc .

Applications

Applies to a range of pattern-searching problems:(directed) k-path of largest weight;cycle of length exactly k ;

k-local search for Planar Vertex Cover.Needed: 2O(w) · nc algorithm on graphs of treewidth w .

pk) ·nc .

Applications

Applies to a range of pattern-searching problems:(directed) k-path of largest weight;cycle of length exactly k ;k-local search for Planar Vertex Cover.

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

pk) ·nc .

Applications

Applies to a range of pattern-searching problems:(directed) k-path of largest weight;cycle of length exactly k ;k-local search for Planar Vertex Cover.Needed: 2O(w) · nc algorithm on graphs of treewidth w .

pk) ·nc .

Applications

pk) ·nc .

Applications

pk) ·nc .

Applications

pk) ·nc .

Without the maxdeg bound, we get running time2O(k/ log k) · nc using (Bodlaender, Nederlof, van der Zanden;ICALP’16).

This running time is tight under ETH! (Bodlaender et al.)

Applications

pk) ·nc .

Example: grid

Take a k ⇥ k grid.

Baker: Guess an index modulopk s.t. there are p

kvertices of the pattern in the corresponding rows.Guess columns in which these vertices lie,

� kpk

�options.

Remove the rows apart from intersections with columns.What remains has treewidth O(

In total,pk · � kp

�= 2O(

pk log k) possible guesses.

Example: grid

Take a k ⇥ k grid.Baker: Guess an index modulo

pk s.t. there are p

kvertices of the pattern in the corresponding rows.

Guess columns in which these vertices lie,� kp

�options.

�= 2O(

Example: grid

pk s.t. there are p

� kpk

�options.

�= 2O(

Example: grid

pk s.t. there are p

� kpk

�options.

�= 2O(

Example: grid

pk s.t. there are p

� kpk

�options.

�= 2O(

Glimpse into the proofRandomized statement:There is a polytime algorithm that samples A with treewidthO(

pk) s.t. every k-pattern is covered with probability

(2O(pk) · nc)�1.

Idea: Emulate a recursive approximation algorithm fortreewidth, trying to construct a decomposition of width

When the algorithm gets stuck, apply Baker-like sparsification.

State of the algorithm:Graph G with O(

pk) terminals on the outer face.

Pieces of the (unkown) pattern “hang” from terminals.

(2O(pk) · nc)�1.

Proof strategy

Observation: Everything would be trivial if the pattern wouldbe within an O(

pk) margin from the outer face.

Otherwise, one of three possibilities holds:

1 Balanced separator that avoids the pattern outside the margin.

Otherwise we “locate” a vertex z of the pattern outside themargin.

2 Many concentric short separators separating z from the outerface.

3 Many “almost” disjoint paths connecting z with the outer face.

Proof strategy

Otherwise, one of three possibilities holds:

1 Balanced separator that avoids the pattern outside the margin.

Proof strategy

Otherwise, one of three possibilities holds:1 Balanced separator that avoids the pattern outside the margin.

Proof strategy

Achieving progressBalanced separator: Remove outside margin and split.

Progress: The graph size and the pattern size are halved.Concentric short separators:

Progress: Guess a separator that splits the pattern in abalanced way, and has a small intersection with it.

Nearly disjoint paths:

Progress: Guess a path where private are not in the pattern.Contract along it, thus dragging more of the pattern into themargin.

Keep track of 3 potentials measuring the progress to estimatethe success probability.

Progress: The graph size and the pattern size are halved.

Concentric short separators:

Nearly disjoint paths:Progress: Guess a path where private are not in the pattern.

Contract along it, thus dragging more of the pattern into themargin.

Nearly disjoint paths:Progress: Guess a path where private are not in the pattern.Contract along it, thus dragging more of the pattern into themargin.

Generalizations

We rely on two properties of the graph class C:

Minor-closed: If H � G and G 2 C, then H 2 C.Locally bounded treewidth: tw(G ) f (rad(G )) for some f .Classes C satisfying the above are exactly apex-minor-free.

Corollary: All the results hold on any apex-minor-free class.We can generalize to H-minor-free graphs for any H, but weneed to assume that the pattern has bounded maximumdegree.

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

Handling disconnected patterns.Extending the technique to general H-minor-free graphs.

Generalizations

We rely on two properties of the graph class C:Minor-closed: If H � G and G 2 C, then H 2 C.

Locally bounded treewidth: tw(G ) f (rad(G )) for some f .Classes C satisfying the above are exactly apex-minor-free.

Generalizations

We rely on two properties of the graph class C:Minor-closed: If H � G and G 2 C, then H 2 C.Locally bounded treewidth: tw(G ) f (rad(G )) for some f .

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

Generalizations

We rely on two properties of the graph class C:Minor-closed: If H � G and G 2 C, then H 2 C.Locally bounded treewidth: tw(G ) f (rad(G )) for some f .Classes C satisfying the above are exactly apex-minor-free.

Generalizations

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

We can generalize to H-minor-free graphs for any H, but weneed to assume that the pattern has bounded maximumdegree.

Generalizations

Open:Handling disconnected patterns.

Extending the technique to general H-minor-free graphs.

Generalizations

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

Contribution: New tool for developing subexp. parameterizedalgorithms on topologically-constrained graph classes.

Usage: Searching for small patterns with prescribed properties.Seems more robust than bidimensionality.

Outlook:

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

Usage: Searching for small patterns with prescribed properties.

Seems more robust than bidimensionality.

Outlook:

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

Questions:Derandomization, disconnected patterns, H-minor-free graphs.

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

Conclusions

A robust understanding of why certain problems can be solvedin time 2O(

pn) etc. on planar graphs and why the square root

is best possible.

Going beyond the basic toolbox requires new problem-specificalgorithmic techniques and hardness proofs with tricky gadgetconstructions.The lower bound technology on planar graphs cannot give alower bound without a square root factor. Does this mean thatthere are matching algorithms for other problems as well?

2O(pk) · nO(1) time algorithm for Steiner Tree with k

terminals in a planar graph?. . .

Conclusions

is best possible.Going beyond the basic toolbox requires new problem-specificalgorithmic techniques and hardness proofs with tricky gadgetconstructions.

The lower bound technology on planar graphs cannot give alower bound without a square root factor. Does this mean thatthere are matching algorithms for other problems as well?

Conclusions

is best possible.Going beyond the basic toolbox requires new problem-specificalgorithmic techniques and hardness proofs with tricky gadgetconstructions.The lower bound technology on planar graphs cannot give alower bound without a square root factor. Does this mean thatthere are matching algorithms for other problems as well?

ConclusionsChapter 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 gridtheorem.Lower bounds: textbook NP-completeness proofs + ETH.

Chapter 3: Finding bounded-treewidth solutions.Algorithms: the solution can be represented by a graph oftreewidth O(

Lower bounds: grid-like W[1]-hardness proofs to rule outf (k) · no(

pk) algorithms.

Chapter 4: Pattern Coverage.New Tool: A kind of graphic hash functions covering patternsof treewidth O(

pk). Much more to be explored here.

Thanks for listening!

ConclusionsChapter 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 gridtheorem.Lower bounds: textbook NP-completeness proofs + ETH.

Chapter 3: Finding bounded-treewidth solutions.Algorithms: the solution can be represented by a graph oftreewidth O(

Lower bounds: grid-like W[1]-hardness proofs to rule outf (k) · no(

pk) algorithms.

Chapter 4: Pattern Coverage.New Tool: A kind of graphic hash functions covering patternsof treewidth O(

pk). Much more to be explored here.

Thanks for listening!

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