on reducing maximum independent set to minimum satisfiability

A . Ignatiev, A . Morgado, and J. Marques-Silva On Reducing MIS to MinSAT /

On Reducing Maximum Independent Setto Minimum Satisfiability

Alexey Ignatiev , Antonio Morgado , and Joao Marques-Silva,

INESC-ID/IST, Lisbon, Portugal CASL/CSI, University College Dublin, Ireland

th International Conference onTheory and Applications of Satisfiability Testing

Vienna, AustriaJuly ,


Problems statements

MIS: compute the largest number of pairwise non-connected vertices in G.MVC: compute the smallest number of vertices in G that are incident to alledges of G.

⇓

Given G, I ⊆ G is an MIS solution ⇔ G \ I is an MVC solution.

MaxClq = MIS

MinSAT: compute the smallest number of simultaneously satisfied clauses inF.MaxFalse: compute the largest number of simultaneously falsified clauses in F.

⇓

Given F, M ⊆ F is a MaxFalse solution ⇔ F \M is a MinSAT solution.


MIS and MaxFalse

MIS↔ MaxFalse

MVC↔ MinSAT


Basic reduction

v

v

v

v

v

F =

c = x,

c = ¬x, ∨ x, ∨ x, ∨ x,

c = ¬x, ∨ x,

c = ¬x,

c = ¬x, ∨ ¬x,

Given a graph G = (V ,E), basic reduction constructs a formula F with exactly|V | clauses and |E| variables.


Basic reduction

v

v

v

v

v

F =

c = x,

c = ¬x, ∨ x, ∨ x, ∨ x,

c = ¬x, ∨ x,

c = ¬x,

c = ¬x, ∨ ¬x,

Given a graph G = (V ,E), basic reduction constructs a formula F with exactly

|V | clauses and |E| variables.


Greedy reduction

v

v

v

v

v

F =

c = ¬x

c = x

c = ¬x ∨ x

c = ¬x

c = ¬x ∨ ¬x

Given a graph G = (V ,E), greedy reduction constructs a formula F withexactly |V | clauses and6 |V | variables.


Greedy reduction

v

v

v

v

v

F =

c = ¬x

c = x

c = ¬x ∨ x

c = ¬x

c = ¬x ∨ ¬x

Given a graph G = (V ,E), greedy reduction constructs a formula F with

exactly |V | clauses and6 |V | variables.


Variable compatibility

original idea — compatible states in finite-state machines simplification

compatible variables can replace each other

variable compatibility rules: no tautology

given a clause ¬x ∨ x , variables x and x are not compatible

no new connection

given two clauses ¬x ∨ x and ¬x , variables x and x are not compatible


Variable compatibility

v

v

v

v

v

(a) Graph G

c = x

c = ¬x ∨ x ∨ x ∨ x

c = ¬x ∨ x

c = ¬x

c = ¬x ∨ ¬x

(b) Set of clauses for G

x x x x x

x –x **, –x **, –x **, –x *, * *, –

Literal compatibility is similar.


Weighting clauses and removing duplicates

basic greedy greedy+vc

vars clauses time vars clauses time vars clauses time

Instan

ce

c-fat- . . c-fat- . . c-fat- . . c-fat- — . c-fat- — . c-fat- — . c-fat- — .

Running time for MinSatz.


Experimental evaluation

Benchmarks ( in total):

crafted MaxClq instances: DIMACS MaxClq, FRB, etc.MIS instances obtained from Binate Covering Problem benchmarks (BCP)

Tools:MaxCLQ — native MaxClq solverMinSatz—branch and bound MinSAT solverMaxSatz—branch and bound MaxSAT solverMiFuMaX— Fu&Malik algorithm for MaxSAT (best for MS industrial in )

Machine configuration:Intel Xeon @GHz with GB RAMrunning Fedora LinuxGB memout

Timeout value — seconds



Benchmarks ( in total):crafted MaxClq instances: DIMACS MaxClq, FRB, etc.

MIS instances obtained from Binate Covering Problem benchmarks (BCP) Tools:

MaxCLQ — native MaxClq solverMinSatz—branch and bound MinSAT solverMaxSatz—branch and bound MaxSAT solverMiFuMaX— Fu&Malik algorithm for MaxSAT (best for MS industrial in )





Benchmarks ( in total):crafted MaxClq instances: DIMACS MaxClq, FRB, etc.MIS instances obtained from Binate Covering Problem benchmarks (BCP)







Tools:MaxCLQ — native MaxClq solver

MinSatz—branch and bound MinSAT solverMaxSatz—branch and bound MaxSAT solverMiFuMaX— Fu&Malik algorithm for MaxSAT (best for MS industrial in )






Tools:MaxCLQ — native MaxClq solverMinSatz—branch and bound MinSAT solver

MaxSatz—branch and bound MaxSAT solverMiFuMaX— Fu&Malik algorithm for MaxSAT (best for MS industrial in )






Tools:MaxCLQ — native MaxClq solverMinSatz—branch and bound MinSAT solverMaxSatz—branch and bound MaxSAT solver

MiFuMaX— Fu&Malik algorithm for MaxSAT (best for MS industrial in ) Machine configuration:

Intel Xeon @GHz with GB RAMrunning Fedora LinuxGB memout






Machine configuration:Intel Xeon @GHz with GB RAM

running Fedora LinuxGB memout






Machine configuration:Intel Xeon @GHz with GB RAMrunning Fedora Linux

GB memout Timeout value — seconds


Experimental results

— CLQ MinSz MaxSz-d MaxSz-mf FM-d FM-mf VBS-d VBS-mf VBS

Native

MaxFalse-based

Direct MaxSAT

— CLQ MinSz MaxSz-d MaxSz-mf FM-d FM-mf VBS-d VBS-mf VBS

Crafted Clq BCP Total


Performance of the considered approaches

0 20 40 60 80 100 120 140 160

instances

0

1

10

100

1000

CPU

time(s)

VBSMaxFalse VBSMaxCLQMinSatzDirect MaxSAT VBSMaxSatz-dirMaxSatz-mfMiFuMaX-mfMiFuMaX-dir


Summary and future work

proposed a reduction from MIS to MinSAT

heuristics to reduce obtained MinSAT formulas experimental results:

comparable to native MaxClq solversoutperforms MaxSAT-based approachesportfolios of solvers

further improvements to the proposed MinSAT models portfolios of solvers for NP-hard graph problems development of efficient MinSAT solvers



proposed a reduction from MIS to MinSAT heuristics to reduce obtained MinSAT formulas

experimental results:comparable to native MaxClq solversoutperforms MaxSAT-based approachesportfolios of solvers




proposed a reduction from MIS to MinSAT heuristics to reduce obtained MinSAT formulas experimental results:

comparable to native MaxClq solvers

outperforms MaxSAT-based approachesportfolios of solvers





comparable to native MaxClq solversoutperforms MaxSAT-based approaches

portfolios of solvers






further improvements to the proposed MinSAT models

portfolios of solvers for NP-hard graph problems development of efficient MinSAT solvers





further improvements to the proposed MinSAT models portfolios of solvers for NP-hard graph problems

development of efficient MinSAT solvers


¸ank you for your attention!

on reducing maximum independent set to minimum satisfiability

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