Pruned Dynamic Programming for Steiner Tree Yoichi Iwata (NII) - PowerPoint PPT Presentation

Separator-based Pruned Dynamic Programming for Steiner Tree Yoichi Iwata (NII) Takuto Shigemura (U-Tokyo) appeared in AAAI 2019 https://www.aaai.org/Papers/AAAI/2019/AAAI-IwataY.4578.pdf 1

Steiner Tree Problem Steiner tree problem Input: graph 𝐻 , terminals 𝐵 ⊆ 𝑊(𝐻) Output: minimum-weight tree connecting 𝐵 𝐵 Various applications:  VLSI design 𝑈  fiber-optic network design  team formulation in social networks 2

PACE Challenge 2018 https://pacechallenge.wordpress.com/pace-2018/ Track 1: small number of terminals ( 𝐵 ≤ 136 )  𝑃 3 |𝐵| 𝑜 + 2 |𝐵| 𝑛 + 𝑜 log 𝑜 by DP [Erickson-Monma-Veinott 87]  EMV + 𝐵 ∗ -search [Hougardy-Silvanus-Vygen 17] Track 2: small tree-width ( 𝑥 ≤ 47 )  𝑥 𝑃 𝑥 𝑛 by DP  2 𝑃 𝑥 𝑛 by a rank-based DP [Bodlaender-Cygan-Kratsch- Nederlof 15] Track 3: heuristic 3

Results https://pacechallenge.org/files/PACE18-report.pdf Track 1: small number of terminals 1. Our team solved 95/100 by pruned DP 2. Maziarz and Polak solved 94/100 by HSV 3. Koch and Rehfeldt solved 93/100 by ILP Track 2: small tree-width 1. Koch and Rehfeldt solved 92/100 by ILP Our team solved 77/100 by pruned DP + 𝑥 𝑥 -DP 2. 3. Tom van der Zanden solved 58/100 by rank-based DP for public instances: pruned DP alone solved 81/100 𝑥 𝑥 -DP alone solved 44/100 combined solved 84/100 4

Outline 1. EMV Algorithm 2. Separator-based Pruning 3. Other techniques 4. Experiments 5

Dynamic Programming [Dreyfus-Wagner 71] For 𝑇 ⊆ 𝑊 , let opt 𝑇 ≔ weight of min Steiner tree for terminals 𝑇 . opt 𝑇 ∪ {𝑣} = min ቐopt 𝑇 ′ ∪ {𝑣} + opt 𝑇 ∖ 𝑇 ′ ∪ {𝑣} 𝑇 ′ ⊆ 𝑇 opt 𝑇 ∪ {𝑤} + 𝑥 𝑤𝑣 𝑤𝑣 ∈ 𝐹 𝐻 ′ 𝑇 𝑇 ′ 𝑣 𝑇 𝑤 𝑣 ∖ 𝑇 6

EMV Algorithm [Erickson-Monma-Veinott 87] 𝑒 𝑇, 𝑣 = ∞ for ∀𝑇 ⊆ 𝐵 and ∀𝑣 ∈ 𝑊 𝑒 𝑏 , 𝑏 = 0 for ∀𝑏 ∈ 𝐵 for 𝑇 ⊆ 𝐵 in ascending order of |𝑇| update 𝑒(𝑇, 𝑣) for ∀𝑣 by Dijkstra for 𝑇 ′ ⊆ 𝐵 ∖ 𝑇 update 𝑒(𝑇 ∪ 𝑇 ′ , 𝑣) for ∀𝑣 𝑃 3 |𝐵| 𝑜 + 2 |𝐵| 𝑛 + 𝑜 log 𝑜 time Can we avoid computing all 𝑒(𝑇, 𝑣) ? → Yes, for special instances. [EMV 87] 7

Special Case [Erickson-Monma-Veinott 87] If the graph if planar and all the terminals are on a single face, the running time is improved to 𝑃( 𝐵 3 𝑜 + 𝐵 2 𝑜 log 𝑜) . 𝐵 1 2 4 3 8

Special Case [Erickson-Monma-Veinott 87] If the graph if planar and all the terminals are on a single face, the running time is improved to 𝑃( 𝐵 3 𝑜 + 𝐵 2 𝑜 log 𝑜) . But, of course, this is too special to apply in practice… 𝐵 1 Steiner tree for {1, 3} always separates {2, 4} . So we can skip the computation 2 of 𝑒( 1,3 ,∗) . In general, we only need to compute 𝑒( 𝑗, 𝑗 + 1, … , 𝑘 ,∗) 4 3 9

Outline 1. EMV Algorithm 2. Separator-based Pruning 3. Other techniques 4. Experiments 10

Important Partial Solution A Steiner tree 𝑈 for 𝑇 ∪ {𝑣} is called important if there is a Steiner tree 𝑈 ′ for 𝐵 ∖ 𝑇 ∪ {𝑣} s.t. 𝑈 + 𝑈 ′ is a minimum Steiner tree for 𝐵 . 𝐵 𝐵 𝑈 𝑈 𝑣 𝑣 unimportant important 11

Important Partial Solution A Steiner tree 𝑈 for 𝑇 ∪ {𝑣} is called important if there is a Steiner tree 𝑈 ′ for 𝐵 ∖ 𝑇 ∪ {𝑣} s.t. 𝑈 + 𝑈 ′ is a minimum Steiner tree for 𝐵 . In EMV algorithm, we can safely skip computations for unimportant 𝑇, 𝑣 . But how can we check the importance? 12

Necessary Condition of Importance Key Lemma: A Steiner tree 𝑈 for 𝑇 ∪ {𝑣} is not important if there is an (𝐵 ∖ 𝑇) -separator 𝐷 such that for every 𝑤 ∈ 𝐷 , there is a Steiner tree for 𝑇 ∪ {𝑤} of weight strictly less than the weight of 𝑈 . 𝐷 𝐷 𝐵 𝐵 𝑥 𝑈 ′′ < 𝑥(𝑈) 𝑈 𝑈′′ 𝑣 𝑣 𝑤 𝑤 𝑈′ 𝑈′ 13

Separator Construction After computing 𝑒(𝑇,∗) , we compute the minimum 𝑦 s.t. 𝐷 𝑦 ≔ 𝑣 ∈ 𝑊 𝑒 𝑇, 𝑣 ≤ 𝑦 separates 𝐵 ∖ 𝑇 . Then, 𝐷 𝑦 satisfies the condition for every 𝑇, 𝑣 with 𝑒 𝑇, 𝑣 > 𝑦 . 𝑇 4 3 4 5 5 3 5 5 4 3 3 4 5 5 4 5 14

Pruned DP Algorithm 𝑄 ← ∅ # set of processed 𝑇 while ∃ unprocessed 𝑇 s.t. 𝑒 𝑇, 𝑣 ≠ ∞ for some 𝑣 pick smallest such 𝑇 update 𝑒(𝑇, 𝑣) for ∀𝑣 by Dijkstra compute minimum 𝑦 s.t. 𝐷 𝑦 separates 𝐵 ∖ 𝑇 drop (𝑇, 𝑣) from 𝑒 for ∀ 𝑇, 𝑣 s.t. 𝑒 𝑇, 𝑣 > 𝑦 for 𝑇 ′ ∈ 𝑄 s.t. 𝑇 ∩ 𝑇 ′ ≠ ∅ update 𝑒(𝑇 ∪ 𝑇 ′ , 𝑣) for ∀𝑣 push 𝑇 into 𝑄 15

Restore dropped information (1) For unimportant (𝑇, 𝑣) , we may have 𝑒 𝑇, 𝑣 > opt 𝑇 ∪ 𝑣 . This can interfere with the pruning… 𝑇 𝑇 3 4 3 4 5 Dijkstra 3 5 3 5 6 3 4 3 4 5 5 5 5 6 𝐷 4 does not separate 𝐵 ∖ 𝑇  16

Restore dropped information (2) Before running Dijkstra, we partially restore 𝑒(𝑇, 𝑣) as follows. For each 𝑣 with 𝑒 𝑇, 𝑣 ≠ ∞ , we construct the corresponding Steiner tree 𝑈 𝑣 for 𝑇 ∪ {𝑣} , and update 𝑒 𝑇, 𝑤 with 𝑒(𝑇, 𝑣) for ∀𝑤 ∈ 𝑊(𝑈 𝑣 ) . 𝑇 𝑇 3 4 3 4 5 Dijkstra 3 5 3 5 5 3 3 4 3 3 4 5 5 4 5 𝐷 4 separates 𝐵 ∖ 𝑇  17

Restore dropped information (3) If a Steiner tree 𝑈 for 𝑇 ∪ {𝑣} is unimportant, 𝑈 + 𝑄 𝑣𝑤 is also unimportant. So we can safely drop such (𝑇, 𝑤) . 𝑇 𝑇 3 4 3 4 5 Dijkstra 3 5 3 5 5 3 3 4 3 3 4 5 5 4 5 𝐷 4 separates 𝐵 ∖ 𝑇  18

Restore dropped information (3) If a Steiner tree 𝑈 for 𝑇 ∪ {𝑣} is unimportant, 𝑈 + 𝑄 𝑣𝑤 is also unimportant. So we can safely drop such (𝑇, 𝑤) . 𝑇 𝑇 3 4 3 4 Dijkstra 3 3 3 3 4 3 𝐷 4 separates 𝐵 ∖ 𝑇  19

Outline 1. EMV Algorithm 2. Separator-based Pruning 3. Other techniques 4. Experiments 20

Data Structure (1) 𝑄 ← ∅ # set of processed 𝑇 while ∃ unprocessed 𝑇 s.t. 𝑒 𝑇, 𝑣 ≠ ∞ for some 𝑣 … for 𝑇 ′ ∈ 𝑄 s.t. 𝑇 ∩ 𝑇 ′ ≠ ∅ update 𝑒(𝑇 ∪ 𝑇 ′ , 𝑣) for ∀𝑣 push 𝑇 into 𝑄 How can we apply the merge operation efficiently? Let valid 𝑇 ≔ 𝑣 ∈ 𝑊 𝑒 𝑇, 𝑣 ≠ ∞ . We want to find all 𝑇 ′ such that 𝑇 ∩ 𝑇 ′ = ∅ and 1. valid 𝑇 ∩ valid 𝑇 ′ ≠ ∅ 2. 21

Data Structure (2) We maintain the set of processed 𝑇 using the following binary trie. Each leaf 𝑢 keeps 𝑇 𝑢 ⊆ 𝐵 . Each internal node 𝑢 with leaves 𝑀 𝑢 keeps 𝐽 𝑗 ≔ځ 𝑢∈𝑀 𝑗 𝑇 𝑢 1. 𝑉 𝑗 ≔ڂ 𝑢∈𝑀 𝑗 valid(𝑇 𝑢 ) 2. When searching 𝑇′ , we can stop if 𝑇 ∩ 𝐽 𝑗 ≠ ∅ or valid 𝑇 ∩ 𝑉 𝑗 = ∅ . 22

Meet in the Middle Any Steiner tree for 𝐵 can be written as a sum of three Steiner trees for 𝑇 1 , 𝑇 2 , 𝑇 3 with 1 ≤ 𝑇 1 ≤ 𝑇 2 ≤ 𝑇 3 ≤ |𝐵|/2 . 1. We can stop after processing all 𝑇 of size ≤ 𝐵 /2 . 2. When merging, we can iterate only over 𝑇 ′ of size at most 2 𝐵 − 2|𝑇| . 𝑄 ← ∅ # set of processed 𝑇 while ∃ unprocessed 𝑇 s.t. 𝑒 𝑇, 𝑣 ≠ ∞ for some 𝑣 … for 𝑇 ′ ∈ 𝑄 s.t. 𝑇 ∩ 𝑇 ′ ≠ ∅ update 𝑒(𝑇 ∪ 𝑇 ′ , 𝑣) for ∀𝑣 push 𝑇 into 𝑄 23

Outline 1. EMV Algorithm 2. Separator-based Pruning 3. Other techniques 4. Experiments 24

Environment Data set: from the DIMACS and PACE List of Solvers: • Pruned: the proposed pruned DP algorithm • EMV: the classical DP algorithm HSV: EMV + 𝐵 ∗ -search • • SCIP-Jack [Gamrath,Koch,Maher,Rehfeldt,Shinano 17] : ILP solver (PACE version) Setting: Intel Xeon E5-2670 (2.6 GHz), single thread, time limit = 30 minutes, memory limit = 6GB (same as PACE) 25

Comparison with EMV  No reductions  200 public instances from PACE 26

Power of other techniques DS: Data Structure MM: Meet in the Middle 27

Comparison with HSV  No reductions 𝐵 is ‘ better ’ than 𝐶 on an instance 𝑗 if 𝐵 could solve 𝑗 but B couldn’t or (run-time of 𝐶 ) > (run-time of 𝐵 ) × 10 + 1s. 28

Comparison with HSV 29

Comparison with SCIP-Jack average of 𝑙/𝑜  Pruned used the same reductions as SCIP-Jack  Omit too-easy instances solved by the reductions alone 30

Comparison with SCIP-Jack 31

Conclusion and Open Problems  Pruned DP is quite effective for Steiner Tree.  Further speedup?  Pruning interferes with future pruning…  Pruning for tree-decomposition DP? 32