via via exa xact and nd he heuri ristic lists of f min inim - PowerPoint PPT Presentation

Computing tr treewidth via via exa xact and nd he heuri ristic lists of f min inim imal l se separa rators rs Hisao Tamaki Meiji University 2019/3/7 Shonan Meeting 144

Con Contents • Overview of the approach • Experimental results • Technical details • Dynamic programming • Listing minimal separators 2019/3/7 Shonan Meeting 144

Ov Overv rview of of the he appro approach: thr hree com omponents msDP( 𝐻, 𝑙, Δ ) 𝐻 : graph 𝑙 : positive integer, Δ : a set of minimal separators of 𝐻 Decides if 𝐻 has a tree-decomposition of width ≤ 𝑙 that uses minimal separators only from Δ listExact( 𝐻 , 𝑙 ) Generates Δ 𝑙 (𝐻) , the set of all minimal separators of cardinality ≤ 𝑙 listHeuristic( 𝐻 , 𝑈, 𝑙 ) 𝑈 : a tree decomposition of 𝐻 Iteratively generates expanding subsets Δ 0 ⊂ Δ 1 ⊂ Δ 2 ⊂ Δ 3 … ⊆ Δ 𝑙 𝐻 2019/3/7 Shonan Meeting 144

Thre Three algo algori rithms for or com omputing the he treewidth of of 𝐻 Ascend for 𝑙 ascending from a trivial lower bound: decide if 𝑢𝑥 𝐻 ≤ 𝑙 by calling msDP( 𝐻 , 𝑙 , listExact( 𝐻 , 𝑙 )) if YES then stop Descend 𝑈 := heuristic tree-decomposition of 𝐻 by a greedy heuristic heuristically improve 𝑈 as log as possible to improve 𝑈 of width 𝑥 , use listHeuristic( 𝐻 , 𝑈 , 𝑥 - 1) to generate Δ 0 ⊂ Δ 1 ⊂ Δ 2 ⊂ Δ 3 … ⊆ Δ 𝑥−1 𝐻 and try msDP( 𝐻 , 𝑙 , Δ 𝑗 ) for 𝑗 = 0, 1 , 2, … try to show 𝑈 of width 𝑥 is optimal by msDP( 𝐻 , 𝑥 – 1, listExact( 𝐻 , 𝑥 – 1)) Alternate Alternate between Descend and Ascend, with some resource balancing 2019/3/7 Shonan Meeting 144

Ra Rand ndom in inst stances 2019/3/7 Shonan Meeting 144

Som Some DI DIMACS gr grap aph col olori ring in inst stances 2019/3/7 Shonan Meeting 144

Com Computing envi nviro ronment CPU: Intel Core i7-6700 (4 cores), 3.40GHz, 8192KB cache RAM:32GB Operating system: Ubuntu 18.04.1 LTS Programming language: Java 1.8 JVM: jre1.8.0_111 The maximum heap size: 28GB Implementations are single threaded, except that multiple threads may be invoked for garbage collection by JVM. The time measured is the elapsed time. To minimize the influence of system processes, the computer is detached from the network and the graphic user interface is disabled. 2019/3/7 Shonan Meeting 144

Per Perform rman ances of of min min sep sep li list sting algo algori rithms 2019/3/7 Shonan Meeting 144

Pe Performances of of the the tr treewidth al algorith thms on on ran andom inst stances (1) (1) 2019/3/7 Shonan Meeting 144

Pe Performances of of the the tr treewidth al algorith thms on on ran andom inst stances (2) (2) 6-hour time-out time (seconds) is that of the last improvement 2019/3/7 Shonan Meeting 144

Pe Performances of of the the tr treewidth al algorith thms on on DIMACS inst stances 6-hour time-out time (seconds) is that of the last improvement 2019/3/7 Shonan Meeting 144

Min inimal l se sepa para rators rs 𝑇 ⊆ 𝑊 𝐻 is a separator of 𝐻 full components if 𝐻 ∖ 𝑇 ≔ 𝐻 𝑊 𝐻 ∖ 𝑇 is disconnected Each connected component of 𝐻 ∖ 𝑇 is called 𝑇 a component associated with separator 𝑇 A component 𝐷 associated with 𝑇 is a full component if 𝑂 𝐷 = 𝑇 𝑇 is a minimal separator if it has at least two full components associated with it or, equivalently, if 𝑇 separates a pair of vertices but no proper subset of 𝑇 does not separator this components pair. associated with 𝑇 2019/3/7 Shonan Meeting 144

Feas Fe asibili ility of of a a con onnected se set: sub subpro roblem for or DP DP 𝑂(𝐷) Fix 𝐻 and 𝑙. A connected 𝐷 ⊆ 𝑊 𝐻 is feasible C with respect to Δ ⊆ Δ 𝑙 𝐻 if there is a tree-decomposition of 𝐻 𝑂 𝐷 ? of width ≤ 𝑙 that • has a bag containing 𝑂(𝐷) and • uses separators only from Δ ∈ Δ 2019/3/7 Shonan Meeting 144

Dynamic pr progra ramming for or treewidth Dynamic programming of Bouchitte and Todinca 2001: 1. List minimal separators and potential maximal cliques 2. Decide the feasibility of components associated with minimal separators, through a recurrence involving potential maximal cliques Positive instance driven (PID) variant (Tamaki 2017) Does not list minimal separators or potential maximal cliques in advance Generates “on the fly” • feasible components associated with minimal separators • potential maximal cliques needed to show their feasibility New approach 1. List minimal separators, but not potential maximal cliques 2. Decide the feasibility of components associated with minimal separators through a direct recurrence, in which potential maximal cliques are implicit 2019/3/7 Shonan Meeting 144

Well ll-form rmed ed tree ee-decompositi itions (1) (1) 𝑉 ⊆ 𝑊 𝐻 is baggy if • there is no connected set 𝐷 such that 𝑂(𝐷) = 𝑉 and U • for every non-empty 𝑌 ⊆ 𝑉 , • there is a connected set 𝐷 containing 𝑌 such that 𝑂 𝐷 = 𝑉 ∖ 𝑌 . Note: X A potential maximal clique is baggy, but not vice versa 2019/3/7 Shonan Meeting 144

Well ll-form rmed ed tree ee-decompositi itions (1) (1) 𝑉 ⊆ 𝑊 𝐻 is baggy if • there is no connected set 𝐷 such that 𝑂(𝐷) = 𝑉 and U • for every non-empty 𝑌 ⊆ 𝑉 , 𝑂(𝐷) • there is a connected set 𝐷 containing 𝑌 such that 𝑂 𝐷 = 𝑉 ∖ 𝑌 . Note: X A potential maximal clique is baggy, C but not vice versa 2019/3/7 Shonan Meeting 144

Well ll-form rmed ed tree ee-decompositi itions (2) (2) Tree-decomposition T of G is well-formed if • every bag of T is baggy and, • for every connected vertex set C of G such that C is X a component of 𝐻 ∖ 𝑌 for some bag 𝑌 of 𝑈 , • there is a subtree 𝑈′ of 𝑈 and a bag 𝑍 in 𝑈′ such that • 𝑈′ is a tree-decomposition of 𝐻[𝑂[𝐷]] , 𝑂(𝐷) • 𝑍 is adjacent to a bag of 𝑈 , say 𝑎 , not in 𝑈′ with 𝑍 ∩ 𝑎 = 𝑂 𝐷 . C Proposition Every graph 𝐻 has a well-formed tree-decomposition of width tw( 𝐻 ). Reason: The tree-decomposition corresponding to a minimal triangulation of 𝐻 is well-formed 2019/3/7 Shonan Meeting 144

Well ll-feas asibility y of of con onnected se sets A connected 𝐷 ⊆ 𝑊 𝐻 is well-feasible with respect to Δ ⊆ Δ 𝑙 𝐻 if there is a well-formed tree-decomposition of 𝐻[𝑂[𝐷]] that • has a bag containing 𝑂(𝐷) and • uses separators only from Δ Note: For Δ = Δ 𝑙 (𝐻) , 𝐷 is well-feasible if and only if 𝐷 is feasible. 2019/3/7 Shonan Meeting 144

Or Orie ienting mini minimal l se sepa para rators rs Assume a total order on the vertices: outbound 𝑊 𝐻 = 1, 2, … , 𝑜 Induced partial order on vertex sets: 𝑇 𝑉 < 𝑊 if min 𝑉 < min 𝑊 2 4 A connected set 𝐷 is inbound if there is a full component 𝐸 associated 5 8 with 𝑂(𝐷) such that 𝐸 < 𝐷 otherwise 𝐷 is outbound 6 7 inbound 2019/3/7 Shonan Meeting 144

Dyn Dynamic pr progra ramming algo algori rithm (1) (1) Main iteration of msDP: decides the feasibility of each inbound connected set with respect to Δ ⊆ Δ 𝑙 (𝐻) 2019/3/7 Shonan Meeting 144

is isFe Feas asible(C) (C) 𝑂(𝐷) 𝑂(𝐷) X ? C C 2019/3/7 Shonan Meeting 144

is isFe Feas asible(C) (C) 𝑂(𝐷) 𝑂(𝐷) X ? C feasible inbounds 2019/3/7 Shonan Meeting 144

is isFe Feas asible(C) (C) 𝑂(𝐷) X min 𝐷 ? C X try all feasible inbound 𝐸 with D 𝑛𝑗𝑜 𝐷 ∈ 𝐸 , 𝐸 ⊆ 𝐷 , and |𝑂(𝐸) ∪ 𝑂(𝐷)| ≤ 𝑙 + 1 2019/3/7 Shonan Meeting 144

isFe is Feas asible(C) (C) 𝑇 = 𝑂 𝐷 ∪ {min(𝐷)} 𝑂(𝐷) X 𝑇 = 𝑂 𝐷 ∪ 𝑂 𝐸 min 𝐷 ? C X try all feasible inbound 𝐸 with D 𝑛𝑗𝑜 𝐷 ∈ 𝐸 , 𝐸 ⊆ 𝐷 , and |𝑂(𝐸) ∪ 𝑂(𝐷)| ≤ 𝑙 + 1 2019/3/7 Shonan Meeting 144

is isFe Feas asible(C) (C) Case 1: 𝑇 does not have a full component 𝑇 inbounds A right bag 𝑌 = 𝑇 is found, if all these inbounds are feasible 2019/3/7 Shonan Meeting 144

is isFe Feas asible(C) (C) Case 2: 𝑇 has a full component 𝐸 𝑌 𝑇 𝐸 inbounds full component 2019/3/7 Shonan Meeting 144

is isFe Feas asible(C) (C) Case 2: 𝑇 has a full component 𝐸 𝑌 𝑇 If any of the inbounds are not marked feasible, then fail Otherwise, call isFeasible( 𝐸 ) 𝐸 full component inbounds 2019/3/7 Shonan Meeting 144

Dyn Dynamic pr progra ramming algo algori rithm (2) (2) 2019/3/7 Shonan Meeting 144

The Theore rem: corr orrectness of of the he algo algori rithm Let 𝐷 ⊆ 𝑊 𝐻 be connected with 𝑂 𝐷 ≤ 𝑙 . • If call isFeasible ( 𝐷 ) is made during the execution of our dynamic programming algorithm, and returns true then 𝐷 is feasible with respect to Δ . • if 𝐷 is inbound with 𝑂 𝐷 ∈ Δ and moreover is well-feasible with respect to Δ , then the algorithm marks 𝐷 as feasible. 2019/3/7 Shonan Meeting 144