On Reducing Maximum Independent Set to Minimum Satis fiabili ty Ale x - - PowerPoint PPT Presentation
A. Ignatiev, A. Morgado, and J. Marques-Silva On Reducing MIS to MinSAT / On Reducing Maximum Independent Set to Minimum Satis fiabili ty Ale x e y Ig n a t ie v , A nton i o Mor gad o , a n d J o a o M a rqu e s - S il v a ,
On Reducing MIS to MinSAT /
Alexey Ignatiev, Antonio Morgado, and Joao Marques-Silva,
INESC-ID/IST, Lisbon, Portugal CASL/CSI, University College Dublin, Ireland
th International Conference on Theory and Applications of Satisfiability Testing Vienna, Austria July ,
On Reducing MIS to MinSAT /
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 all edges of G.
On Reducing MIS to MinSAT /
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 all edges of G. ⇓ Given G, I ⊆ G is an MIS solution ⇔ G \ I is an MVC solution.
On Reducing MIS to MinSAT /
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 all edges of G. ⇓ Given G, I ⊆ G is an MIS solution ⇔ G \ I is an MVC solution. MaxClq = MIS
On Reducing MIS to MinSAT /
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 all edges 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 in F. MaxFalse: compute the largest number of simultaneously falsified clauses in F.
On Reducing MIS to MinSAT /
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 all edges 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 in F. 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.
On Reducing MIS to MinSAT /
MIS ↔ MaxFalse MVC ↔ MinSAT
On Reducing MIS to MinSAT /
v v v v v
F = c = x, c = ¬x, ∨ x, ∨ x, ∨ x, c = ¬x, ∨ x, c = ¬x, c = ¬x, ∨ ¬x,
On Reducing MIS to MinSAT /
v v v v v
F = c = x, c = ¬x, ∨ x, ∨ x, ∨ x, c = ¬x, ∨ x, c = ¬x, c = ¬x, ∨ ¬x,
On Reducing MIS to MinSAT /
v v v v v
F = c = x, c = ¬x, ∨ x, ∨ x, ∨ x, c = ¬x, ∨ x, c = ¬x, c = ¬x, ∨ ¬x,
On Reducing MIS to MinSAT /
v v v v v
F = c = x, c = ¬x, ∨ x, ∨ x, ∨ x, c = ¬x, ∨ x, c = ¬x, c = ¬x, ∨ ¬x,
On Reducing MIS to MinSAT /
v v v v v
F = c = x, c = ¬x, ∨ x, ∨ x, ∨ x, c = ¬x, ∨ x, c = ¬x, c = ¬x, ∨ ¬x,
On Reducing MIS to MinSAT /
v v v v v
F = c = x, c = ¬x, ∨ x, ∨ x, ∨ x, c = ¬x, ∨ x, c = ¬x, c = ¬x, ∨ ¬x,
On Reducing MIS to MinSAT /
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.
On Reducing MIS to MinSAT /
v v v v v
F = c = ¬x c = x c = ¬x ∨ x c = ¬x c = ¬x ∨ ¬x
On Reducing MIS to MinSAT /
v v v v v
F = c = ¬x c = x c = ¬x ∨ x c = ¬x c = ¬x ∨ ¬x
On Reducing MIS to MinSAT /
v v v v v
F = c = ¬x c = x c = ¬x ∨ x c = ¬x c = ¬x ∨ ¬x
On Reducing MIS to MinSAT /
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 and |V| variables.
On Reducing MIS to MinSAT /
On Reducing MIS to MinSAT /
compatible variables can replace each other
On Reducing MIS to MinSAT /
compatible variables can replace each other variable compatibility rules:
no tautology given a clause ¬x ∨ x, variables x and x are not compatible
On Reducing MIS to MinSAT /
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
On Reducing MIS to MinSAT /
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.
On Reducing MIS to MinSAT /
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.
On Reducing MIS to MinSAT /
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.
On Reducing MIS to MinSAT /
basic greedy greedy+vc vars clauses time vars clauses time vars clauses time Instance c-fat-
. .
c-fat-
. .
c-fat-
. .
c-fat-
— .
c-fat-
— .
c-fat-
— .
c-fat-
— .
Running time for MinSatz.
On Reducing MIS to MinSAT /
Benchmarks ( in total):
On Reducing MIS to MinSAT /
Benchmarks ( in total):
crafted MaxClq instances: DIMACS MaxClq, FRB, etc.
On Reducing MIS to MinSAT /
Benchmarks ( in total):
crafted MaxClq instances: DIMACS MaxClq, FRB, etc. MIS instances obtained from Binate Covering Problem benchmarks (BCP)
On Reducing MIS to MinSAT /
Benchmarks ( in total):
crafted MaxClq instances: DIMACS MaxClq, FRB, etc. MIS instances obtained from Binate Covering Problem benchmarks (BCP)
Tools:
MaxCLQ — native MaxClq solver
On Reducing MIS to MinSAT /
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 solver
On Reducing MIS to MinSAT /
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 solver MaxSatz — branch and bound MaxSAT solver
On Reducing MIS to MinSAT /
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 solver MaxSatz — branch and bound MaxSAT solver MiFuMaX — Fu&Malik algorithm for MaxSAT (best for MS industrial in )
On Reducing MIS to MinSAT /
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 solver MaxSatz — branch and bound MaxSAT solver MiFuMaX — Fu&Malik algorithm for MaxSAT (best for MS industrial in )
Machine configuration:
Intel Xeon @GHz with GB RAM
On Reducing MIS to MinSAT /
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 solver MaxSatz — branch and bound MaxSAT solver MiFuMaX — Fu&Malik algorithm for MaxSAT (best for MS industrial in )
Machine configuration:
Intel Xeon @GHz with GB RAM running Fedora Linux
On Reducing MIS to MinSAT /
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 solver MaxSatz — branch and bound MaxSAT solver MiFuMaX — Fu&Malik algorithm for MaxSAT (best for MS industrial in )
Machine configuration:
Intel Xeon @GHz with GB RAM running Fedora Linux GB memout
On Reducing MIS to MinSAT /
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 solver MaxSatz — branch and bound MaxSAT solver MiFuMaX — Fu&Malik algorithm for MaxSAT (best for MS industrial in )
Machine configuration:
Intel Xeon @GHz with GB RAM running Fedora Linux GB memout
Timeout value — seconds
On Reducing MIS to MinSAT /
— CLQ MinSz MaxSz-d MaxSz-mf FM-d FM-mf VBS-d VBS-mf VBS Native MaxFalse-based Direct MaxSAT
On Reducing MIS to MinSAT /
— 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
On Reducing MIS to MinSAT /
20 40 60 80 100 120 140 160 instances 1 10 100 1000 CPU time (s) VBS MaxFalse VBS MaxCLQ MinSatz Direct MaxSAT VBS MaxSatz-dir MaxSatz-mf MiFuMaX-mf MiFuMaX-dir
On Reducing MIS to MinSAT /
proposed a reduction from MIS to MinSAT
On Reducing MIS to MinSAT /
proposed a reduction from MIS to MinSAT
heuristics to reduce obtained MinSAT formulas
On Reducing MIS to MinSAT /
proposed a reduction from MIS to MinSAT
heuristics to reduce obtained MinSAT formulas
experimental results:
comparable to native MaxClq solvers
On Reducing MIS to MinSAT /
proposed a reduction from MIS to MinSAT
heuristics to reduce obtained MinSAT formulas
experimental results:
comparable to native MaxClq solvers
On Reducing MIS to MinSAT /
proposed a reduction from MIS to MinSAT
heuristics to reduce obtained MinSAT formulas
experimental results:
comparable to native MaxClq solvers
portfolios of solvers
On Reducing MIS to MinSAT /
proposed a reduction from MIS to MinSAT
heuristics to reduce obtained MinSAT formulas
experimental results:
comparable to native MaxClq solvers
portfolios of solvers
further improvements to the proposed MinSAT models
On Reducing MIS to MinSAT /
proposed a reduction from MIS to MinSAT
heuristics to reduce obtained MinSAT formulas
experimental results:
comparable to native MaxClq solvers
portfolios of solvers
further improvements to the proposed MinSAT models
portfolios of solvers for NP-hard graph problems
On Reducing MIS to MinSAT /
proposed a reduction from MIS to MinSAT
heuristics to reduce obtained MinSAT formulas
experimental results:
comparable to native MaxClq solvers
portfolios of solvers
further improvements to the proposed MinSAT models
portfolios of solvers for NP-hard graph problems
development of efficient MinSAT solvers
On Reducing MIS to MinSAT /