Node Multiway Cut and Subset Feedback Vertex Set on Graphs of - - PowerPoint PPT Presentation

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-width

Benjamin Bergougnoux⋆, Charis Papadopoulos† and Jan Arne Telle⋆. WG 2020, June 26, 2020

⋆ University of Bergen, Norway † University of Ioannina, Greece

The parameters

Treewidth

◮ Unbounded in any dense graph class. ◮ Tractability results on dense graph classes can be explained with rank-width and mim-width.

Divide a graph

Recursively decompose your graph...

G

Divide a graph

Recursively decompose your graph... into simple cuts. → We describe the simplicity of a cut with a function f: cut → Q.

Different notions of simplicity = different width measures.

Divide a graph

Width of a decomposition D := max f(cut) among the cuts of D.

G

Divide a graph

Width of a graph G := min width of a decomposition of G.

G G G G

Rank-width Defined from the function rw(A, B) := the rank of adjacency matrix between A and B over GF(2).

B A

  1 1 1 1 1 1  =2

rw

Maximum Induced Matching width (mim-width) Defined from the function mim(A, B) := the size of a maximum induced matching in the bipartite graph between A and B.

B A

Hierarchy

Cographs, hereditary distance Interval, permutation, k-polygon, Dilworth-k, convex graphs,...

MSO1

tree-width clique-width rank-width mim-width

(σ, ρ)-Dominating Set problems

MSO2

Modeling Power Meta-algorithmic applications

Tree, partial k-tree

Computation complexity

◮ NP-hard for all these widths measures. ◮ Efficient algorithms for tree-width and rank-width. → Running time: 2O(k) · nO(1). ◮ Tough open question for mim-width.

The problem

Feedback Vertex Set

Feedback Vertex Set Find a set of vertices X of minimum weight hitting all cycles (G − X is a forest).

X

Subset Feedback Vertex Set

Subset Feedback Vertex Set Find a set of vertices X of minimum weight hitting all cycles going through a prescribed set of vertices S.

S X

Subset Feedback Vertex Set

Subset Feedback Vertex Set Find a set of vertices X of minimum weight hitting all cycles going through a prescribed set of vertices S.

S

Subset FVS is harder

Corneil and Fonlupt 1988 FVS is solvable in polynomial time on split graphs. Fomin et al. 2014 Subset FVS is NP-hard on split graphs. Bodalender, Cygan, Kratsch and Nederlof 2015 FVS is solvable in time 2O(tw) · n. B., Bonnet, Brettell and Kwon [ArxiV] Subset FVS can not be solved in time 2o(tw log tw) · nO(1) unless ETH fails.

Recent results

Papadopoulos and Tzimas 2019 Subset FVS is solvable in polynomial time on interval graphs and permutation graphs. Jaffke, Kwon and Telle 2019 | B. and Kanté 2019 FVS is solvable in time nO(mim) given a decomposition. Our result Subset FVS is solvable in time nO(mim2) given a decomposition.

Algorithmic consequences

Corollary Subset FVS is solvable in polynomial time on interval, permutation, bi-interval graphs, circular arc and circular permutation graphs, convex graphs, k-polygon, Dilworth-k and co-k-degenerate graphs for fixed k. Theorem Subset FVS is solvable in time 2O(rw3) · nO(1). Same results holds for Node Multiway Cut.

Our Approach

Neighbor equivalence

Definition X, Y ⊆ A are 1-neighbor equivalent over A if N(X) ∩ B = N(Y ) ∩ B. nec1(A) := number of equivalence classes over A.

B X Y A

Lemma For every A ⊆ V (G), have nec2(A) ≤ nO(mim).

Reduce

Lemma It is sufficient to keep for each cut (A, B) (nec2(A) + nec2(B))O(mim) · mimO(mim) ≤ nO(mim2) partial solutions. Set of partial solutions C ⊆ 2A Reduce D ⊆ C small and equivalent to C

∀Y ⊆ B, best(C, Y ) = best(D, Y )

|D| ≤ nO(mim2)

Corollary Subset FVS is solvable in time nO(mim2) given a decomposition.

S-forest Minimum Subset FVS Complement: S-forest

S-forest Minimum Subset FVS Complement↓: forest

S-forest − → forest

Lemma We can contract the components of G[X] − S and unions of the components of G[Y ] − S such that the resulting graph is a forest.

A B X Y

S-forest − → forest

Lemma We can contract the components of G[X] − S and unions of the components of G[Y ] − S such that the resulting graph is a forest.

A B Y↓ X↓

S-forest − → forest

Lemma We can contract the components of G[X] − S and unions of the components of G[Y ] − S such that the resulting graph is a forest.

A B Y↓ X↓

Small vertex cover

Lemma [B. and Kanté 2019] After contractions, every induced forest between A and B have a vertex cover of size 4mim whose vertices have different neighborhoods.

A B Y↓ X↓

Indices

Definition An index i is a collection of at most 4mim + 1 representatives for the 2-neighbor equivalence over A and B. Number of indices ≤ (nec2(A) + nec2(B))O(mim) ≤ nO(mim2). A B X↓ Y↓ i

X \ V C

Index association

We associate each index with some partial solutions and some subsets of B.

A B X↓ i

X \ V C

Index association

We associate each index with some partial solutions and some subsets of B.

Y \ V C

A B Y↓ i

Index association

Lemma For every solution F, there exists an index i such that F ∩ A and F ∩ B are associated with i.

A B F ∩ A↓ F ∩ B↓ i

Index association

Definition Two partial solutions X, W associated with i are i-equivalent if they connect the vertices of the vertex cover in the same way. Number of i-equivalence classes ≤ mimO(mim) Lemma If X and W are i-equivalent, then for every Y ⊆ B associated with i, G[X ∪ Y ] is a solution iff G[W ∪ Y ] is also a solution.

Result

Lemma It is sufficient to keep, for every index i, a partial solutions of maximum weight for each i-equivalence class. The number of partial solutions we keep is at most (nec2(A) + nec2(B))O(mim2)mimO(mim) ≤ nO(mim2) Theorem Subset FVS is solvable in time nO(mim2) given a decomposition.

Algorithmic consequences

We have nec2(A) ≤ 2O(rw2) and mim ≤ rw Lemma It is sufficient to keep, for every index i, a partial solutions of maximum weight for each i-equivalence class. The number of partial solutions we keep is at most (nec2(A) + nec2(B))O(mim2)mimO(mim) ≤ 2O(rw3) Theorem Subset FVS is solvable in time 2O(rw3) · nO(1).

Node Multiway Cut

X

T

Theorem Node Multiway Cut is solvable in time 2O(rw3) and nO(mim2) given a decomposition.

Node Multiway Cut

T

Theorem Node Multiway Cut is solvable in time 2O(rw3) and nO(mim2) given a decomposition.

Node Multiway Cut

X

T S

Theorem Node Multiway Cut is solvable in time 2O(rw3) and nO(mim2) given a decomposition.

Neighbor equivalence relations

These relations have been used for many problems: ◮ (σ, ρ)-Dominating Set problems [Bui-Xuan, Telle and Vatshelle 2013] ◮ Their connected and acyclic variants and Max Cut [B. and Kanté 2019] ◮ Subset FVS and Node Multiway Cut [B., Papadapoulos and Telle] for many parameters: tree-width, clique-width, rank-width, Q-rank-width, mim-width

Open questions

◮ Can we characterize which problems are in XP parameterized by the mim-width of a given decomposition?

• B. and Kanté 2019

The connected and acyclic variants of (σ, ρ)-Dominating Set problems are solvable in time nO(mim) given a decomposition. ◮ Can we approximate/compute the mim-width of a graph in XP-time?

◮ Can we recognize the graphs of mim-width one?

Casual dynamic programming

G

Combine and Reduce

S1 S2 S

Combination Reduction

{X1 ∪ X2 | X1 ∈ S1, X2 ∈ S2}

Reduce: tough and key step

In general If it is enough to keep N partial solutions at each step, then we can solve the problem in time N O(1) · nO(1).