The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems

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The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems

Dániel Marx

Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI)

Budapest, Hungary

(Joint work with Andreas Feldmann)

SIWAG 2016 Polignano a Mare September 26, 2016

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Steiner Tree

Given an edge-weighted graphG and setT ⊆V(G) of terminals, find a minimum-weight tree inG containing every vertex ofT.

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Steiner Tree

Some known results:

NP-hard

Easy 2-approximation: use a minimum spanning tree.

1.386-approximation [Byrka et al. 2013].

3^k ·n^O(1) time algorithm fork terminals using dynamic programming (i.e., fixed-parameter tractable parameterized by the number of terminals)

Can be improved to 2^k ·n^O⁽¹⁾ time using fast subset convolution[Björklund et al. 2006].

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Steiner Forest

Given an edge-weighted graphG and a list(s₁,t₁),. . .,(s_k,t_k)of pairs of terminals, find a minimum-weight forest inG that connects s_i andt_i for every 1≤i ≤k.

s1

s2

t₁t₂t₃ s3

s4

t₄ t₅

s5

s6

t6

Fixed-parameter tractable parameterized byk: Guess a partition of the2k terminals (k^O(k) =2^O(k^log^k)) possibilities) and solve a

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Variants of Steiner Tree

Steiner Tree

Connect all the terminals

Steiner Forest

Create connections satisying every request

r

Directed Steiner Network (DSN) Strongly Connected

Steiner Subgraph (SCSS)

Make all the terminals reachable from each other Make every terminal

reachable from the root Steiner Tree

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Variants of Steiner Tree

Steiner Tree

Connect all the terminals

Steiner Forest

r

Directed Steiner Network (DSN) Strongly Connected

Steiner Subgraph (SCSS)

Make all the terminals Make every terminal

Steiner Tree

Create connections

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Directed Steiner vs. SCSS

The DP forSteiner Tree generalizes to the directed version:

Directed Steiner Treewith k terminals can be solved in time 2^k·n^O(1).

SCSSseems to be much harder: Theorem[Feldman and Ruhl 2006]

Strongly Connected Steiner Subgraphwithk terminals can be solved in timen^O(k).

Theorem[Chitnis, Hajiaghayi, and M. 2014]

Assuming ETH,Strongly Connected Steiner Subgraphis W[1]-hard and has nof(k)n^o(k/^log^k) time algorithm for any functionf.

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Directed Steiner vs. SCSS

The DP forSteiner Tree generalizes to the directed version:

Directed Steiner Treewith k terminals can be solved in time 2^k·n^O(1).

SCSSseems to be much harder:

Theorem[Feldman and Ruhl 2006]

Strongly Connected Steiner Subgraphwithk terminals can be solved in timen^O(k).

Theorem[Chitnis, Hajiaghayi, and M. 2014]

Assuming ETH,Strongly Connected Steiner Subgraphis W[1]-hard and has nof(k)n^o(k/^log^k) time algorithm for any functionf.

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Directed Steiner Network

Theorem[Feldman and Ruhl 2006]

Directed Steiner Networkwithk requests can be solved in timen^O^(k).

Corollary: Strongly Connected Steiner Subgraphwith k terminals can be solved in timen^O(k).

Proof is based on a “pebble game”: O(k) pebbles need to reach their destinations using certain allowed moves, tracing the solution.

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Directed Steiner Network

A new combinatorial result:

Theorem[Feldmann and M. 2016]

[The underlying undireced graph of] every minimum cost solution ofDirected Steiner Networkwith k requests has cutwidth and treewidthO(k).

A new algorithmic result:

If aDirected Steiner Networkinstance with k requests has a minimum cost solution with treewidthw [of the underlying undirected graph], then it can be solved in timef(k,w)·n^O(w). Corollary: A new proof thatDSN andSCSScan be solved in timef(k)n^O(k⁾.

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Directed Steiner Network

A new combinatorial result:

[The underlying undireced graph of] every minimum cost solution ofDirected Steiner Networkwith k requests has cutwidth and treewidthO(k).

A new algorithmic result:

If aDirected Steiner Networkinstance withk requests has a minimum cost solution with treewidthw [of the underlying undirected graph], then it can be solved in timef(k,w)·n^O(w). Corollary: A new proof thatDSN andSCSScan be solved in timef(k)n^O(k⁾.

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Special cases of Directed Steiner Network

Directed Steiner TreeandStrongly Connected

Steiner Subgraphare both restrictions ofDirected Steiner Networkto certain type of patterns:

SCSS Directed Steiner Tree

Goal: characterize the patterns that give rise to FPT/W[1]-hard problems.

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Patterns for Directed Steiner Network

Question:

What is the complexity ofDirected Steiner Networkfor this pattern?

Answer:

Directed Steiner Networkhas ann^O(k) algorithm fork requests, so it is polynomial-time solvable for every fixed pattern.

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Patterns for Directed Steiner Network

Question:

What is the complexity ofDirected Steiner Networkfor this pattern?

Answer:

Directed Steiner Networkhas ann^O(k) algorithm fork

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Patterns for Directed Steiner Network

Goal: For every class ofHof directed patterns, characterize the complexity ofDirected Steiner Networkwhen restricted to demand patterns fromH.

Example:

If His the class of all directed in-stars (or out-stars), then H-DSN is FPT.

If His the class of all directed cycles, thenH-DSN is W[1]-hard.

Main result:

Theorem[Feldmann and M. 2016] For any classHof directed patterns,

if Hhas combinatorial property X, then H-DSNand H-DSN is W[1]-hard otherwise.

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Patterns for Directed Steiner Network

Goal: For every class ofHof directed patterns, characterize the complexity ofDirected Steiner Networkwhen restricted to demand patterns fromH.

Example:

If His the class of all directed in-stars (or out-stars), then H-DSN is FPT.

If His the class of all directed cycles, thenH-DSN is W[1]-hard.

Main result:

For any classHof directed patterns,

ifH has combinatorial property X, then H-DSNand H-DSN is W[1]-hard otherwise.

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FPT special cases

What classesH give FPT cases of H-DSN?

We know that out-stars are FPT.

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FPT special cases

This is also FPT: minimal solutions have bounded treewidth.

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FPT special cases

This is also FPT: minimal solutions have bounded treewidth.

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FPT special cases

C_λ: in- or out-caterpillar of length λ.

Lemma

If the pattern is inC_λ, then every minimal solution has treewidth O(λ²).

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FPT special cases

What about this pattern?

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FPT special cases

Lemma

If the pattern istransitively equivalent to a member ofC_λ, then every minimal solution has treewidthO(λ²).

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FPT special cases

C_λ,δ: in- or out-caterpillar of length λwithδ additional edges.

Lemma

If the pattern istransitively equivalent to a member ofC_λ,δ, then every minimal solution has treewidthO((1+λ)(λ+δ)).

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FPT special cases

Theorem

If everyH∈ H istransitively equivalent to a member ofC_λ,δ for some constantsλ, δ≥0, thenH-DSN is FPT.

Does this cover all the FPT cases?

(Yes)

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FPT special cases

Theorem

If everyH∈ H istransitively equivalent to a member ofC_λ,δ for some constantsλ, δ≥0, thenH-DSN is FPT.

Does this cover all the FPT cases?

(Yes)

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W[1]-hard special cases

We show that the following classesH make H-DSN W[1]-hard:

cycles (SCSS) out-diamonds in-diamonds

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Identifying terminals

IfH⁰ is obtained from H by identifying terminals, then the problem cannot be harder forH⁰ than for H:

0

H H⁰

G G⁰

⇒We can assume that His closed under identifying terminals.

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Combinatorial classification

The following combinatorial result connects the algorithmic and the hardness results:

Theorem

LetHbe a class of patterns closed under identifying terminals and transitive equivalence. Then exactly one of the following holds:

1 There are constants λ, δ such that everyH ∈ H is transitively equivalent to a member of C_λ,δ

2 H contains either all directed cycles, all in-diamonds, all out-diamonds,

all flawed in-diamonds, or all flawed out-diamonds.

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Classification result

Our main result:

LetHbe a class of patterns.

1 If there are constantsλ, δ such that everyH ∈ His transitively equivalent to a member of C_λ,δ, then H-DSNis FPT,

2 and it is W[1]-hard otherwise.