1. Nearly ETH-tight algorithms for Planar Steiner Tree with Terminals on Few Faces
Sándor Kisfaludi-Bak Jesper Nederlof Erik Jan van Leeuwen
TU Eindhoven TU Eindhoven Utrecht University

2. Steiner Tree (ST)
Given : 𝑛-vertex graph 𝐺 and 𝑇⊆𝑉(𝐺) (called terminals), Find : smallest (-weight) tree in 𝐺 that contains 𝑇
Well-studied in Parameterized Complexity (PC):
[DW’71, L’71] : 𝑂 ∗ (3 |𝑇| ) time ( 𝑂 ∗ omits factors poly in input)
[BHKK’07] : 𝑂 ∗ ( 2 |𝑇| 𝑊) time (W denotes max input weight)
[FKMSPW’07]: 𝑂 ∗ ( 2+𝜀 |𝑇| ) time (for any 𝜖>0)
[N’13] : 𝑂 ∗ ( 2 |𝑇| 𝑊) W time and poly space
[FKLPS’15] : 𝑂 ∗ ( 7.97 |𝑇| ) W time and poly space

3. Steiner Tree (ST)
Use Dynamic Programming (DP) based on following observation:
If 𝑆 is an optimal ST for T and 𝑣𝑤∈𝑆, then S∖ 𝑣,𝑤 = 𝑆 1 ∪ 𝑆 2 where
𝑆 1 is an optimal ST for 𝑇 1 ∪𝑣, and
𝑆 2 is an optimal ST for 𝑇 2 ∪𝑤,
For some partition 𝑇 1 , 𝑇 2 of 𝑇.
Dynamic Programming (DP): Only 2 |𝑇| 𝑛 subproblems
𝑂 ∗ (3 |𝑇| ) time, 2 |𝑇| space
Recursion: Guess {𝑣,𝑤} and 𝑇 1 , 𝑇 2 s.t | 𝑇 1 |≈ 𝑇 2 , recurse
𝑛 log |𝑇| 2 𝑂( 𝑇 ) time, polynomial space 𝑣 𝑤
𝑇 2
𝑇 1

4. Planar Steiner Tree (PST)
Well-studied in e.g. approximation algorithms
In PC, planarity often allows for sub-exponential time algorithms:
find simple paths on 𝑘 vertices: 𝑂 ∗ (2 𝑂( 𝑘 ) ) [DPBF’06]
find shortest tour visiting 𝑘 given vertices: 𝑂 ∗ ( 2 𝑂( 𝑘 log 𝑘 ) ) [MPP’18]
Steiner tree: 𝑂 ∗ ( 2 |𝑆| log |𝑆| ) [PPSvL’13]
Natural question: PST in 2 𝑜(|𝑇|) time?
[MPP’18]: Not in 2 𝑜(|𝑇|) time assuming ETH
[MPP’18]: 𝑛 𝑂( |𝑇| ) W time

5. Planar Steiner Tree with Terminals on Few Faces
Assume terminals are on only 𝑘 faces (called terminals faces)
[EMV’87]: In 𝑛 𝑂(𝑘) time
Even for 𝑘=1 interesting:
Used as subroutine in
Approximation algo’s [P’88,BKM’07]
Kernelization algo’s [PPSvL’14]
[B’90] improved constant in 𝑂() to 2
Natural question: Can we improve? Maybe to 𝑂 ∗ (𝑓(𝑘)) time?
Our results:
An 2 𝑂(𝑘) 𝑛 𝑂( 𝑘 ) time, polynomial space algorithm
An 𝑓(𝑘) 𝑛 𝑜( 𝑘 ) time lower bound, assuming ETH

6. Steiner Tree (ST) with terminals on outer face
Use Dynamic Programming (DP) based on following observation:
If 𝑆 is an optimal ST for T and 𝑣𝑤∈𝑆, then S∖ 𝑣,𝑤 = 𝑆 1 ∪ 𝑆 2 where
𝑆 1 is an optimal ST for 𝑇 1 ∪𝑣, and
𝑆 2 is an optimal ST for 𝑇 2 ∪𝑤,
For some partition 𝑇 1 , 𝑇 2 of 𝑇 in intervals
𝑇 1 𝑣 𝑤
𝑇 2
If 𝑇 1 , 𝑇 2 are not intervals, 𝑆 1 and 𝑆 2 cross -> 𝑆 not min weight

7. Steiner Tree (ST) with terminals on outer face
Use Dynamic Programming (DP) based on following observation:
If 𝑆 is an optimal ST for T and 𝑣𝑤∈𝑆, then S∖ 𝑣,𝑤 = 𝑆 1 ∪ 𝑆 2 where
𝑆 1 is an optimal ST for 𝑇 1 ∪𝑣, and
𝑆 2 is an optimal ST for 𝑇 2 ∪𝑤,
For some partition 𝑇 1 , 𝑇 2 of 𝑇 in intervals X
𝑇 1 𝑣 𝑤
𝑇 2
If 𝑇 1 , 𝑇 2 are not intervals, 𝑆 1 and 𝑆 2 cross -> 𝑆 not min weight
At most 𝑛 3 subproblems for DP
[EMV’87]: for 𝑘 faces, use subproblems for each 𝑘-tuple of intervals
DP gives 𝑛 𝑂(𝑘) time, 𝑛 𝑂(𝑘) space algorithm

8. Planar Steiner Tree with Terminals on Few Faces
Set of edges of all terminals faces
Planar Steiner Tree with Terminals on Few Faces
Our Idea: Guess a separator 𝑋 in the graph 𝐻=𝑆∪𝐹

9. Planar Steiner Tree with Terminals on Few Faces
Set of edges of all terminals faces
Planar Steiner Tree with Terminals on Few Faces
Our Idea: Guess a separator 𝑋 in the graph 𝐻=𝑆∪𝐹
Splits solution in two sub solutions 𝑆 1 and 𝑆 2
Separates terminal faces untouched by 𝑋

10. Planar Steiner Tree with Terminals on Few Faces
Set of edges of all terminals faces
Planar Steiner Tree with Terminals on Few Faces
Our Idea: Guess a separator 𝑋 in the graph 𝐻=𝑆∪𝐹
Splits solution in two sub solutions 𝑆 1 and 𝑆 2
Separates terminal faces untouched by 𝑋
Obs: 𝑡𝑤 𝐻 =𝑂( 𝑘 )
(planar dual of 𝐻 has dom set of size 𝑘)
 𝐻 has separator 𝑋 of size 𝑂( 𝑘 ), such that 𝑆 1 and 𝑆 2 avoid ≥𝑘/3 terminals faces each
Algorithm: Guess
The separator 𝑋 ( 𝑛 |𝑋| )
Which terminal faces untouched by 𝑋 appear in 𝑆 1 and 𝑆 2 ( 2 𝑘 )
Which of the terminal faces segment cut by 𝑋 are in 𝑆 1 and in 𝑆 2 ( 2 |𝑋| )
Who of 𝑋 is connected to whom via 𝑆 1 and 𝑆 ( 𝑋 |𝑋| )
And search for 𝑆 1 and 𝑆 2 recursively.
Runtime: 𝑇 𝑘 ≤ 2 𝑘 𝑛 𝑂 𝑘 𝑇(2𝑘/3) = 2 𝑂(𝑘) 𝑛 𝑂 𝑘

11. Lower Bound
We show no 𝑓(𝑘) 𝑛 𝑜( 𝑘 ) time algorithm exists, assuming ETH
Builds heavily on 2 𝑂(|𝑇|) lower bound from [MPP’18]
Roughly same template. Major differences:
Reduces from Grid Tiling instead of CNF-SAT,
Flower gadget that communicates a lot of information with only one terminal face

12. Grid Tiling
Given sets 𝑆 𝑎,𝑏 ⊆ 𝑛 ×[𝑛] for 𝑎,𝑏∈[𝑘] Find 𝑥𝑎 𝑦 𝑎 for 𝑎∈[𝑘] such that: 𝑥𝑎, 𝑦𝑏 ∊ 𝑆 𝑎,𝑏 for all a,b
Theorem [M’12]:
Cannot solve Grid Tiling in 𝑓 𝑘 𝑛 𝑜(𝑘) time under ETH

13. The reduction ∨ Picture and basic construction idea by [MPP’18]
Gadget from [MPP’18]
Picture and basic construction idea by [MPP’18]

14. Flower Gadget 𝑃 𝑇 There exists a planar graph 𝐺 𝑡 with two faces
𝑇 of 𝑡 terminals
𝑃 of 𝑡 portals
Such that if 𝑆⊆𝐸(𝐺) connects every terminal to some portal, 𝑆 ≥10𝑡 and
If 𝑆 uses one portal, 𝑆 >10𝑡
If 𝑆 uses two portals and 𝑆 =10𝑡, the two used portals are opposite
Moreover, an 𝑆 with two portals of cost 10𝑡 exists 𝑃 𝑇

15. Flower Gadget 𝑃 𝑇 There exists a planar graph 𝐺 𝑡 with two faces
𝑇 of 𝑡 terminals
𝑃 of 𝑡 portals
Such that if 𝑆⊆𝐸(𝐺) connects every terminal to some portal, 𝑆 ≥10𝑡 and
If 𝑆 uses one portal, 𝑆 >10𝑡
If 𝑆 uses two portals and 𝑆 =10𝑡, the two used portals are opposite
Moreover, an 𝑆 with two portals of cost 10𝑡 exists
Proved using the displayed graph with carefully chosen weights on edges Can get rid of weights by replacing edges with paths 𝑃 𝑇

16. The reduction

17. The reduction
Given grid tiling instance with k rows/columns, PST instance has 𝑂 𝑘 2 terminal faces
Thus an 𝑓(𝑘)𝑛 𝑜( 𝑘 ) time algo violates ETH by [M’12].

18. Concluding Remarks Our results:
An 2 𝑂(𝑘) 𝑛 𝑂( 𝑘 ) time, polynomial space algorithm
An 𝑓(𝑘) 𝑛 𝑜( 𝑘 ) time lower bound, assuming ETH
Embedding with 𝑘 terminal faces can be found in 𝑐 𝑘 time if exists [BM’88]
Open: Is an 𝑛 𝑂( 𝑘 ) time algorithm possible?
Open: How fast can we solve the problem for constant 𝑘?