Bidimensionality (Revised) Daniel Lokshtanov Based 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 GraphsPlanar Graphs Exact Algorithm 1.49 n 2 O(n 1/2 ) Parameterized Complexity W[2]-complete 2 O(k 1/2 ) KernelW[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: G Max / 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 g4g4 Γ4Γ4

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(g t ) ≥ ct 2. A problem Π is contraction-bidimensional if: – If H ≤ c G then π(H) ≤ π(G). – There is a constant c such that π(Γ t ) ≥ ct 2.

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 2 O(tw(G)) n time. 2.If H ≤ m G then tw(H) ≤ tw(G). 3.The treewidth of g k 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 g c*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 Γ c*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, g c*tw(G) ≤ m G. Since Π is bidimensional, π(g c*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 2 O(t) n (or t O(t) n) time. Total time taken is 2 π(G) 1/2 n (or π(G) π(G) 1/2 n).

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 g t 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) ≤ dk 1/2. Balanced Separator Lemma applied to G,Z yields decomposition of V(G) into (L, S, R) such that |S|≤ dk 1/2, 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(k 1/2 ) ≤ |Z\R|+ O(k 1/2 ) G\R has a set X L of size T(|Z\R|+ O(k 1/2 ) ) such that tw((G\R)\X L ) ≤ d. G\L has a set X R of size T(|Z\L|+ O(k 1/2 ) ) such that tw((G\L)\X R ) ≤ d.

WNDL recurrence X = X L ∪ X R ∪ S is a set of size T(|X\R|+ O(k 1/2 ) ) + T(|X\L|+ O(k 1/2 ) ) + O(k 1/2 ) 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(k 1/2 )...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 T γ (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 T γ (k) ≤ g(γ)k where g(γ)  0 as γ  ∞. Prove T γ (k) ≤ g(γ)k using balanced separation as in the proof of WNDL.

Recurrence for Scaling Lemma T γ (γ) = 0 T γ (k) ≤ T γ ( ⍺ k + O(k 1/2 )) + T γ ((1- ⍺ )k + O(k 1/2 )) + O(k 1/2 )...where 1/3 ≤ ⍺ ≤ 2/3. See board Thus T γ (k) ≤ g(γ)k but what is lim g(γ) when γ  ∞?

Analyzing g(γ) cheat: set ⍺ = ½ and move lower order terms outside function calls. T γ (γ) = 0 T γ (k) ≤ 2T γ (½k) + O(k ½ )

Analyzing g(γ) T γ (γ) = 0T γ (k) ≤ 2T γ (½k) + O(k ½ ) 2 0 *(½ 0 k) ½ = 2 0/2 k ½ 2 1 *(½ 1 k) ½ = 2 1/2 k ½ 2 2 *(½ 2 k) ½ = 2 2/2 k ½ 2 3 *(½ 3 k) ½ = 2 3/2 k ½

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.

Approximation algorithm for Treewidth-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 X C in G[C]. Add X c and N(C) to X. G  G \ (C ∪ N(C)) Output X

Approximation Ratio We deleted X 1, X 2, X X t ≤ OPT N(C 1 ), N(C 2 )... N(C t ) ≤ ct Each C i 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 C 1, C 2,... C t where t ≤ cπ(X) such that - there are no edges between C i and C j - tw(G[Ci]) ≤ c - tw(G[Cj]) ≤ c

Kernelization Each C i 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.