Bidimensionality (Revised)

Daniel LokshtanovBased on joint work with Hans Bodlaender ,Fedor

Fomin,Eelko Penninkx, Venkatesh Raman, Saket Saurabh and Dimitrios Thilikos

Background

Most interesting graph problems are NP-hard on general graphs.

Often input graphs are planar or almost planar. Can this be used to give efficient algorithms?

Most interesting graph problems remain NP-hard on planar graphs.

Are planar graphs as hard as general graphs?

On planar graphs many problems admit:- Faster exact algorithms.- Faster parameterized algorithms.- Good preprocessing rules (kernels).- Better approximation algorithms.

Case Study: Dominating Set

General Graphs Planar Graphs

Exact Algorithm 1.49n 2O(n1/2)

Parameterized Complexity W[2]-complete 2O(k1/2)

Kernel W[2]-complete O(k)

Approximation log(n) 1+ε

Bidimensionality [DFHT]

A framework that gives fast exact algorithms, paramterized algorithms, kernels and approximation schemes for problems on planar graphs.

Main tool: Graph Minors theory of Robertson and Seymour.

Extends to larger classes of graphs.

Preliminaries

Problems considered

Input: GMax / Min: κ(G,S) (S V(G) / S E(G))⊆ ⊆Subject to: φ(G,S)

Technical note: we demand that κ(G,S) ≤ |S| and that κ(G,OPT) = |OPT|.

Value of optimal solution on G = π(G).

Minors and Contractions

H is a minor of G (H ≤m G)if H can be obtained from G by a sequence of edge contractions, edge deletions and vertex deletions.

H is a contraction of G (H ≤c G) if H can be obtained from G by a sequence of edge contractions.

grids and Γammas

g4 Γ4

Bidimensionality

A problem Π is (minor)-bidimensional if:– If H ≤m G then π(H) ≤ π(G).

– There is a constant c such that π(gt) ≥ ct2.

A problem Π is contraction-bidimensional if:– If H ≤c G then π(H) ≤ π(G).

– There is a constant c such that π(Γt) ≥ ct2.

Examples of Bidimensional problems

• Vertex Cover, Feedback Vertex Set, Longest Path and Cycle Packing are minor-bidimensional.

• Dominating Set, Connected Vertex Cover and Independent Set are contraction-bidimensional.

Facts about Treewidth

1. Many graph probems can be solved in 2O(tw(G))n time.2. If H ≤m G then tw(H) ≤ tw(G).3. The treewidth of gk is k.4. Every graph G has a balanced separator of size tw(G).5. On H-minor free graphs, treewidth is constant factor

approximable.

Excluded Grid Theorem

Theorem [RS]: For every fixed graph H there is a constant c such that any graph G which excludes H as a minor contains gc*tw(G) as a minor.

Excluded Γamma Theorem

Theorem [FGT]: For every fixed apex graph H there is a constant c such that any graph G which excludes H as a minor contains Γc*tw(G) as a contraction.

Subexponential Parameterized Algorithms

Parameter-treewidth bound

Lemma [Parameter-treewidth bound]: For every bidimensional problem Π there is a constant c such that for any planar graph G, tw(G) ≤ cπ(G)1/2

Proof: By excluded grid theorem, gc*tw(G) ≤m G. Since Π is bidimensional, π(gc*tw(G)) ≥ c’tw(G)2. Since Π is minor closed, π(G) ≥ c’tw(G)2.

Algorithm on planar graphs

Constant-factor approximate treewidth. Output a decomposition of width t = O(π(G)1/2).

Solve problem in 2O(t)n (or tO(t)n) time. Total time taken is 2π(G)1/2n (or π(G)π(G)1/2n).

More general graph classes

Note: The only place we used planarity was for the excluded grid theorem. So results hold on H-minor-free graphs for minor-bidimensional problems and apex-minor-free graphs for contraction-bidimensional problems.

Exercise 1:

Prove: For any fixed H, d, if G is H-minor-free and has a set X such that tw(G \ X) ≤ d then tw(G) ≤ d + O(|X|1/2).

Soln: Vertex deletion into treewidth d graphs is minor closed and at least (t/(d+1))2 on gt grids.

Approximation

Separability

Want: EPTASes for all bidimensional problems on (apex)-minor-free graphs.

Can’t handle Longest Path. Parameter-treeewidth bound is not enough, but ”almost enough”.

(1+ε)-approximation in f(ε)poly(n) time.

Separability

A problem Π is separable* if for any partition of V(G) into L, S, R such that there is no edge from L to R, and optimal solution OPT V(G)⊆ :

- π(G \ R) ≤ κ(G \ R, OPT \ R) + O(|S|)- π(G \ L) ≤ κ(G \ L, OPT \ L) + O(|S|)

*For contraction-bidimensional problems a slightly different definition is used.

Excercise 2

Show that Vertex Cover is separable.

Solution: OPT \ R is a feasible solution for G[L ∪S]. Hence π(G \ R) ≤ |OPT \ R|.

Exercise 3:

Show that Independent Set is separable.

Solution: Let OPT be a maximum independent set of G. Suppose π(G \ R) > |OPT \ R| + |S|. Then π(G[L]) > |OPT \ R| Then G has an independent set of size: π(G[L]) + |OPT ∩ R| > |OPT \ R| + |OPT ∩ R| =|OPT|.

Decomposition Lemma

Lemma: For any minor-bidimensional, separable problem Π on H-minor-free graphs, there is a function f : N N and polynomial time algorithm that given G and ε > 0 outputs a set X such that

- |X| ≤ επ(G) - tw(G \ X) ≤ f(ε).

Exercise 4:

Assume Feedback Vertex Set (FVS) is minor-bidimensional,and separable. Give an EPTAS for FVS on H-minor-free graphs using the decomposition lemma.

Solution: For a fixed ε and given G find X. Solve FVS optimally on G \ X in g(ε)n time. Add X to the solution. Solution size ≤ (1+ε)π(G).

Decomposition’ Lemma

Lemma: For any contraction-bidimensional, separable problem Π on apex-minor-free graphs, there is a function f : N N and polynomial time algorithm that given G and ε > 0 outputs a set X such that

- |X| ≤ επ(G) - tw(G \ X) ≤ f(ε).

Exercise 5:

Assume Dominating Set (DS) is minor-bidimensional,and separable. Give an EPTAS for DS on apex-minor-free graphs using the decomposition’ lemma.

Solution: For a fixed ε and given G find X. Mark N(X). Find a smallest set S in G\X that dominates all unmarked vertices of G\X. Now S

X ∪ is a DS of G of size ≤ (1+ε)π(G).

Remainder of talk:Proof Sketch of Decomposition Lemma

Balanced Separator Lemma

For any graph G of treewidth t and vertex set X there is a partition of V(G) into L, S, R such that:

- There is no edge between L and R- The separator S is small; |S| ≤ t.- The separator is balanced;

|X ∩ L| ≤ 2|X|/3 and |X ∩ R| ≤ 2|X|/3

Weak, Non-constructive, Decomposition Lemma

WNDL: For any minor-bidimensional, separable problem Π on H-minor-free graphs, there is a constant c such that any instance G has a vertex set X such that

- |X| ≤ cπ(G) - tw(G \ X) ≤ c.

WNDL Proof

1. By parameter-treewidth bound, there is a constant d such that tw(G) ≤ dπ(G)1/2.

2. Let T(k) be the smallest number t such that any H-minor free graph G with π(G) = k contains a set X of size t such that tw(G \ X) ≤ d.

3. Need to prove T(k) = O(k).4. Base Case: T(1) = 0 since tw(G) ≤ dπ(G)1/2 ≤ d.

WNDL recurrence

Let Z be an optimal solution in G, then k=|Z|=π(G).

Now, tw(G) ≤ dk1/2.

Balanced Separator Lemma applied to G,Z yields decomposition of V(G) into (L, S, R) such that |S|≤ dk1/2 , L ∩ Z ≤ 2|Z|/3, R ∩ Z ≤ 2|Z|/3.

WNDL recurrence

Since Π is separable: π(G \ R) ≤ κ(G \ R, Z \ R) + O(k1/2) ≤ |Z\R|+ O(k1/2)

G\R has a set XL of size T(|Z\R|+ O(k1/2) ) such that tw((G\R)\XL) ≤ d.

G\L has a set XR of size T(|Z\L|+ O(k1/2) ) such that tw((G\L)\XR) ≤ d.

WNDL recurrence

X = XL X∪ R S∪ is a set of size T(|X\R|+ O(k1/2) ) + T(|X\L|+ O(k1/2) ) + O(k1/2) such that tw(G \ X) ≤ d.

Observe: |X\R| + |X\L| ≤ |X| + |S|.

WNDL recurrence

T(k) ≤ T( k + O(k⍺ 1/2)) + T((1- )k + O(k⍺ 1/2)) + O(k1/2)...where 1/3 ≤ ≤ 2/3⍺ .

This solves to T(k) = O(k).

Breathe Break

Questions?

Scaling Lemma

For any H and c there is a polynomial time algorithm and a function f : N N that given a H-minor free graph G, a set X such that tw(G\X) ≤ c, and ε > 0 outputs a set X’ of size ε|X| such that for any component C of G \ X’

- |C ∩ X| ≤ f(ε) - |N(C)| ≤ f(ε) Implies tw(G[C]) ≤ f’(ε)

Proof Idea for Scaling Lemma

For a fixed γ let Tγ(k) be the smallest integer t such that any G with X such that |X|≤ k and tw(G\X) ≤ d contains a set X’ of size ≤ t such that for any component C of G \ X’

- |C ∩ X| ≤ γ - |N(C)| ≤ γ

Proof Idea for Scaling Lemma

For every γ > d prove that Tγ(k) ≤ g(γ)k where g(γ) 0 as γ ∞.

Prove Tγ(k) ≤ g(γ)k using balanced separation as in the proof of WNDL.

Recurrence for Scaling Lemma

Tγ(γ) = 0

Tγ(k) ≤ Tγ( k + O(k⍺ 1/2)) + Tγ((1- )k + O(k⍺ 1/2)) + O(k1/2)

...where 1/3 ≤ ≤ 2/3⍺ .

See board

Thus Tγ(k) ≤ g(γ)kbut what is lim g(γ) when γ ∞?

Analyzing g(γ)

cheat: set = ½ ⍺ and move lower order terms outside function calls.

Tγ(γ) = 0Tγ(k) ≤ 2Tγ(½k) + O(k½)

Analyzing g(γ)

Tγ(γ) = 0 Tγ(k) ≤ 2Tγ(½k) + O(k½)

20 *(½0k)½ = 20/2k½

21 *(½1k)½ = 21/2k½

22 *(½2k)½ = 22/2k½

23 *(½3k)½ = 23/2k½

Making Proof of Scaling Lemma constructive

Proof naturally makes a divide and conquer algorithm for constructing X’ from G, X and ε.

Only computationally hard step is computing treewidth. Can be constant-factor approximated instead since G is H-minor-free.

What we have, what we want

Have: Weak Nonconstructive Decomposition Lemma and Scaling Lemma

If we could make WNDL constructive, we would be done!

Want: Constant factor approximation of ”treewidth-d deletion” on H-minor free graphs.

Protrusion Lemma

For every H, d, there are constants c such that if G is H-minor-free and tw(G)>d then there is a vertex set C such that:– d < tw(G[C]) ≤ c– N(C) ≤ c

Proof: Let X be smallest set such that tw(G)<d. Apply Scaling Lemma on X with ε=½. Set c=f(½). Since X’ < X some component C of G\X’has tw(G[C]) > d.

Approximation algorithm forTreewidth-d deletion

Let c be as in Protrusion Lemma. While tw(G) > d:

Find a vertex set C such that d < tw(G[C]) ≤ c and N(C) ≤ c. Find best treewidth-d-deletion XC in G[C]. Add Xc and N(C) to X.G G \ (C N(C))∪

Output X

Approximation Ratio

We deletedX1, X2, X3.... Xt ≤ OPTN(C1), N(C2) ... N(Ct) ≤ ct

Each Ci contains a vertex from OPT so t ≤ |OPT|.Hence |X| ≤ (c+1)|OPT|

Proof of Decomposition Lemma

By WNDL there exists a treewidth d-deletion of size O(π(G)).

By approximation we can find a treewidth treewidth d-deletion X of size O(π(G)).

By Scaling Lemma we can turn X into a treewidth- f(ε) deletion set X’ of size ε|X|. Choosing ε small enough we get |X’| ≤ επ(G).

Approximation - recap

Saw a decomposition lemma for bidiemsional, separable problems on H-minor-free graphs and how it can be used to give EPTAS’es for many problems on H-minor free graphs

Kernelization

The decomposition lemma can be modified as follows:

Lemma: For any minor-bidimensional, separable problem Π on H-minor-free graphs, there is a constant c and polynomial time algorithm that given G outputs a set X such that |X| ≤ cπ(X) and G\X can be partitioned into C1, C2, ... Ct where t ≤ cπ(X) such that- there are no edges between Ci and Cj

- tw(G[Ci]) ≤ c - tw(G[Cj]) ≤ c

Kernelization

Each Ci can be replaced with a constant size graph using techniques from [BFLPST09].

Kernels of size O(π(G)).

Very Short Summary

Bidimensionality is a framework for giving subexponential time algorithms, EPTAS’es and kernels, based on excluded grid theorems and balanced separation techniques.

Thank You!