Improving the Performance of Consistency Algorithms by Localizing - - PowerPoint PPT Presentation

▶

Oct 22, 2022 103 likes •261 views

Improving the Performance of Consistency Algorithms by Localizing and Bolstering Propagation in a Tree Decomposition Shant Karakashian, Robert Woodward & Berthe Y. Choueiry Constraint Systems Laboratory University of Nebraska-Lincoln

SLIDE 1

Improving the Performance of Consistency Algorithms by Localizing and Bolstering Propagation in a Tree Decomposition

Shant Karakashian, Robert Woodward & Berthe Y. Choueiry

Constraint Systems Laboratory

University of Nebraska-Lincoln

Acknowledgments:

Experiments conducted at UNL’s Holland Computing Center
NSF Award RI-111795

SLIDE 2

Outline

Introduction
Background

– Tree decomposition – Relational consistency property R(∗,m)C

Key ideas

– Localize consistency to clusters of a tree decomposition – Bolstering propagation at separators

Evaluation

– Theoretical: Comparing resulting consistency properties – Empirical: Solving CSPs in a backtrack-free manner

Conclusions & Future Work

SLIDE 3

Introduction

Constraint Satisfaction Problems (CSPs)

– NP-complete in general – Islands of tractability are classes of CSPs solvable in polynomial time

One tractability condition links

[Freuder 82]

– Consistency level to – Width of the constraint network, a structural parameter

Our approach: exploit a tree decomposition

– Localize application of the consistency algorithm – Add redundant constraints at separators to enhance propagation – Practical tractability aims to solve CSP instances in a backtrack-free manner

SLIDE 4

Tree Decomposition

A tree decomposition:〈T, , 〉

– T: a tree of clusters – : maps variables to clusters – : maps constraints to clusters

Conditions

– Each constraint appears in at least one cluster with all the variables in its scope – For every variable, the clusters where the variable appears induce a connected subtree

{A,B,C,E} , {R2,R3} {A,B,D},{R3,R5} {A,E,F},{R1} {A,D,G},{R4} C1 C2 C3 C4

Hypergraph Tree decomposition

A B C D E F G R4 R5 R2 R1 R3 (C1) (C1)

SLIDE 5

Tree Decomposition: Separators

A separator of two adjacent clusters is the set of

variables associated to both clusters

Width of a decomposition/network

– Treewidth = maximum number of variables in clusters - 1

A B C D E F G C1 C2 C3 C4

{A,B,C,E},{R2,R3} {A,B,D},{R3,R5} {A,E,F},{R1} {A,D,G},{R4} C1 C2 C3 C4

SLIDE 6

Relational Consistency Property R(∗,m)C

A CSP is R(∗,m)C iff

– Every tuple in a relation can be extended – to the variables in the scope of any (m-1) other relations – in an assignment satisfying all m relations simultaneously

R(∗,m)C ≡ Every set of m relations is minimal

..…

∀ m-1 relations

∀ tuple ∀ relation

[Karakashian+ AAAI 10]

SLIDE 7

Localize Consistency

Restricting R(∗,m)C to clusters: cl-R(∗,m)C
Two clusters communicate via their separator

– Constraints common to the two clusters – Domains of variables common to the two clusters

E R6 R5 R7 R4 R2 R1 R3 B A D C F R4 A D B A D C

SLIDE 8

Bolstering Propagation at Separators

Localization cl-R(*,m)C

– Fewer combinations of m relations – Reduces the enforced consistency level

Ideally: add unique constraint

– Space overhead, major bottleneck

Enhance propagation by bolstering

– Projection of existing constraints – Adding binary constraints – Adding clique constraints

E F R6 R5 R7 R2 R1 R3 B A D C Rsep

SLIDE 9

Bolstering Schemas: Approximate Unique Separator Constraint

E R6 R5 R7 R4 R2 R1 R3 B A D C F R3

’

E F R’3 B A D C Ra E

B A D C F

R’3

Ry Rx

B A D C B A D C

Projection cl+proj-R(,m)C Binary constraints cl+bin-R(,m)C Clique constraints cl+clq-R(*,m)C

B A D C

Ry Rx

Ra E R3 D C E R6 R2 B A F E R5 R1 B C F

SLIDE 10

Resulting Consistency Properties

[Bessiere+ 2008]

SLIDE 11

Empirical Evaluations

+ maxRPWC, m=3,4 wR(∗,2)C R(∗,| (cli)|)C #inst. GAC global local Proj. binary clique local Proj. binary clique Completed UNSAT 479 167 34.9% 170 35.5% 167 34.9% 172 35.9% 169 35.3% 162 33.8% 285 59.5% 286 59.7% 282 58.9% 271 56.6% SAT 200 174 87.0% 179 89.5% 178 89.0% 176 88.0% 169 84.5% 104 52.0% 152 76.0% 138 69.0% 124 62.0% 113 56.5% BT-Free UNSAT 479 0.0% 70 14.6% 39 8.1% 70 14.6% 70 14.6% 74 15.4% 187 39.0% 223 46.6% 223 46.6% 213 44.5% SAT 200 44 22.0% 55 27.5% 37 18.5% 53 26.5% 52 26.0% 38 19.0% 39 19.5% 77 38.5% 71 35.5% 58 29.0% Min(#NV) UNSAT 479 17 3.5% 73 15.2% 43 9.0% 72 15.0% 72 15.0% 77 16.1% 220 45.9% 249 52.0% 248 51.8% 239 49.9% SAT 200 47 23.5% 64 32.0% 37 18.5% 62 31.0% 61 30.5% 39 19.5% 83 41.5% 111 55.5% 100 50.0% 79 39.5% Fastest UNSAT 479 72 15.0% 13 2.7% 35 7.3% 5 1.0% 1 0.2% 1 0.2% 176 36.7% 108 22.5% 42 8.8% 37 7.7% SAT 200 121 60.5% 45 22.5% 47 23.5% 23 11.5% 14 7.0% 12 6.0% 34 17.0% 18 9.0% 13 6.5% 12 6.0%

SLIDE 12

Cumulative Count of Instances Solved w/o Backtracking

50 100 150 200 250 100 200 300

Treewidth

cl+proj-R(∗,|ψ(cli)|)C GAC cl-proj-wR(∗,3)C cl-proj-wR(∗,2)C cl-R(∗,|ψ(cli)|)C 10 20 30 40 50 60 70 80 50 100 150 200