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Exploring the Use of GPUs in Constraint Solving A Preliminary Investigation Federico Campeotto 1 , 2 Alessandro Dal Pal` u 3 Agostino Dovier 1 Ferdinando Fioretto 1 , 2 Enrico Pontelli 2 1. Universit` a di Udine 2. New Mexico State University

SLIDE 1

Exploring the Use of GPUs in Constraint Solving

A Preliminary Investigation Federico Campeotto1,2 Alessandro Dal Pal` u3 Agostino Dovier1 Ferdinando Fioretto1,2 Enrico Pontelli2

1. Universit`

a di Udine

2. New Mexico State University
3. Universit`

a di Parma

San Diego CA, January 2014

SLIDE 2

Introduction

Every new desktop/laptop comes equipped with a powerful graphic processor unit (GPU) These GPUs are general purpose (i.e., we can program them) For most of their life, however, they are absolutely idle (unless some kid is continuously playing with your PC) The question is: can we exploit this computation power for constraint solving? We present a preliminary investigation, focusing on constraint solving

FD-ADP-AD-FF-EP (UD-NMSU-PR) Exploring the Use of GPUs in Constraint Solving 2 / 1

SLIDE 3

Introduction

Constraint Satisfaction Problems

A Constraint Satisfaction Problem (CSP) is defined by: X = {x1, . . . , xn} is a n-tuple of variables D = {Dx1, . . . , Dxn} set of variable’s domains C finite set of constraints over X: c(xi1, . . . , xim) is a relation c(xi1, . . . , xim) ⊆ Dxi1 × . . . × Dxim. A solution of a CSP is a tuple s1, . . . , sn ∈ ×n

i=1Dxi such that for each

c(xi1, . . . , xim) ∈ C, we have si1, . . . , sim ∈ c. CSP solvers alternate 2 steps:

Labeling: select a variable and (non-deterministically) assign a value from its domain

Constraint propagation: propagate the assignment through the constraints, and possibly detect inconsistencies

FD-ADP-AD-FF-EP (UD-NMSU-PR) Exploring the Use of GPUs in Constraint Solving 3 / 1

SLIDE 4

Introduction

Consistency techniques

Idea: replace the current CSP by a “simpler” one, yet equivalent Definition (Arc Consistency) The most common notion of local consistency is arc consistency (AC). Let us consider a binary constraint c ∈ C, where scp(c) = {xi, xj} and xi, xj ∈ X. We say that c is arc consistent if:

∀a ∈ Dxi∃b ∈ Dxj(a, b) ∈ c;
∀b ∈ Dxj∃a ∈ Dxi(a, b) ∈ c;

It is possible to ensure AC by iteratively removing all the values of the variables involved in the constraint that are not consistent with the constraint until a fixpoint is reached

FD-ADP-AD-FF-EP (UD-NMSU-PR) Exploring the Use of GPUs in Constraint Solving 4 / 1

SLIDE 5

Introduction

Consistency techniques

Idea: replace the current CSP by a “simpler” one, yet equivalent Definition (Arc Consistency) The most common notion of local consistency is arc consistency (AC). Let us consider a binary constraint c ∈ C, where scp(c) = {xi, xj} and xi, xj ∈ X. We say that c is arc consistent if:

∀a ∈ Dxi∃b ∈ Dxj(a, b) ∈ c;
∀b ∈ Dxj∃a ∈ Dxi(a, b) ∈ c;

It is possible to ensure AC by iteratively removing all the values of the variables involved in the constraint that are not consistent with the constraint until a fixpoint is reached The propagation engine computes a mutual fixpoint of all the constraint

FD-ADP-AD-FF-EP (UD-NMSU-PR) Exploring the Use of GPUs in Constraint Solving 4 / 1

SLIDE 6

Introduction

Consistency techniques

Idea: replace the current CSP by a “simpler” one, yet equivalent Definition (Arc Consistency) The most common notion of local consistency is arc consistency (AC). Let us consider a binary constraint c ∈ C, where scp(c) = {xi, xj} and xi, xj ∈ X. We say that c is arc consistent if:

∀a ∈ Dxi∃b ∈ Dxj(a, b) ∈ c;
∀b ∈ Dxj∃a ∈ Dxi(a, b) ∈ c;

It is possible to ensure AC by iteratively removing all the values of the variables involved in the constraint that are not consistent with the constraint until a fixpoint is reached The propagation engine computes a mutual fixpoint of all the constraint Several algorithms based on fixpoint loop iteration to achieve (Arc) Consistency: AC3, AC4, AC6, etc.

FD-ADP-AD-FF-EP (UD-NMSU-PR) Exploring the Use of GPUs in Constraint Solving 4 / 1

SLIDE 7

GPUs, in few minutes Why GPUs?