Archipelagos

Discovering Archipelagos of Tractability
for Constraint Satisfaction and Counting
Robert Ganian, M. S. Ramanujan, Stefan Szeider

Vienna University of Technology
SODA 2016

Satisﬁability
• SATISFIABILITY: Is a given propositional formula
satisﬁable? 
• First NP-Complete problem.

Satisﬁability
• CSP: Is there an assignment to variables from
the domain so that every given constraint is
satisﬁed ? 
• Generalizes SAT.

Backdoor sets
Informally, a set of variables whose instantiation results

in a signiﬁcantly simpliﬁed formula.
Introduced by

Williams, Gomes, Selman (IJCAI 2003)

and

Crama, Ekin, Hammer (D. A. M. 1997)

• (strong) Backdoor to C
Backdoors to SAT/CSP

• (strong) Backdoor to C
All assignments lead to an

instance in the base class C.
X
F1 F2 F3 F4 F2|X|
Backdoors to SAT/CSP

2 Perspectives on backdoor sets

• (a) (Williams, Gomes, Selman) the presence of small
backdoor sets provides a good explanation for the
performance of SAT solvers, the success of random
restarts etc.

restarts etc. 
• (b) (Crama, Ekin, Hammer) backdoor sets provide an
excellent framework to extend tractability results for SAT.

restarts etc. 
• (b) (Crama, Ekin, Hammer) backdoor sets provide an
excellent framework to extend tractability results for SAT.
eg. SAT is in P for 2-cnf formulas—> SAT is in P for

formulas with a strong backdoor of size 10 to 2-cnf.

Islands of Tractability
• Think of the base classes as `Islands of tractability’.

• Think of the base classes as `Islands of tractability’. 
• An instance with a `small’ backdoor to one of these
base classes is `close’ to an island of tractability.

• Think of the base classes as `Islands of tractability’. 
• An instance with a `small’ backdoor to one of these
base classes is `close’ to an island of tractability. 
• Objective: The class of instances close to an island
of tractability, is also tractable.

Research Agenda
Instances with a backdoor

of size c to an island

of size log n to an island

of size log2 n to an island

How to extend tractability results?

• One approach: Find a strong backdoor to ONE FIXED Base
Class, explore all assignments to the backdoor variables.

Class, explore all assignments to the backdoor variables. 
• For each reduced formula, run the algorithm for the base
class.

Class, explore all assignments to the backdoor variables. 
• For each reduced formula, run the algorithm for the base
class. 
• Running time is T1+2|X|
.T2 ; X is the backdoor set, T1 is the
time to detect the backdoor and T2 is the time to solve
SAT for the base class .

Finding Backdoor sets
• For any reasonable island of tractability, detecting if a
formula is `close’ to this island is NP-complete.

Finding Backdoor sets
• For any reasonable island of tractability, detecting if a
formula is `close’ to this island is NP-complete. 
 
• When can we do this detection efﬁciently (for a
relaxed notion of efﬁciency)?

FPT algorithms
• 2-dimensional analysis of algorithms.

FPT algorithms
• 2-dimensional analysis of algorithms. 
• We analyse how the running time depends on the
input AND a second parameter which is also a
function of the input alone.

FPT algorithms
function of the input alone. 
• Aim for running times of the form f(k) |x|c

FPT algorithms
function of the input alone. 
• Aim for running times of the form f(k) |x|c
FPT

FPT + Backdoors + SAT
• Schaefer Classes [Nishimura, Ragde, Szeider SAT ’04]

• Q-Horn [Gaspers,Ordyniak, R., Saurabh, Szeider STACS ’13]

• Acyclic-SAT [Gaspers,Szeider ICALP ’12]

• Heterogenous classes [Gaspers, Misra, Ordyniak, Szeider,
Zivny AAAI 2014]

• Heterogenous classes [Gaspers, Misra, Ordyniak, Szeider,
Zivny AAAI 2014]
• Bounded tw-SAT [Gaspers,Szeider FOCS ’13, Fomin,
Lokshtanov, Misra, R., Saurabh, SODA ’15]

Composite Base Classes
• Natural objective when using backdoors:

• Make the base class as large as possible.

• Make the base class as large as possible.
• Can we `compose’ several base classes to get a bigger
composite class?

Archipelagos of tractability
(x ⋁¬a1 ⋁¬a2…⋁¬an) ⋀ (¬x ⋁¬p1 ⋁¬p2…⋁¬pn) 
⋀ 
(¬x ⋁b1 ⋁ c1)⋀(¬x ⋁b2 ⋁ c2)..⋀(¬x ⋁bn ⋁ cn)  
⋀

(x ⋁q1 ⋁ r1)⋀(x ⋁q2 ⋁ r2)..⋀(x ⋁qn ⋁ rn)

Consider F[x=0]
(x ⋁¬a1 ⋁¬a2…⋁¬an) ⋀ (¬x ⋁¬p1 ⋁¬p2…⋁¬pn) 
⋀ 
(¬x ⋁b1 ⋁ c1)⋀(¬x ⋁b2 ⋁ c2)..⋀(¬x ⋁bn ⋁ cn)  
⋀

(x ⋁q1 ⋁ r1)⋀(x ⋁q2 ⋁ r2)..⋀(x ⋁qn ⋁ rn)
(¬a1 ⋁¬a2…⋁¬an) ⋀

(q1 ⋁ r1)⋀(q2 ⋁ r2).. ⋀ (qn ⋁ rn)

Consider F[x=0]
Consider F[x=1]
(x ⋁¬a1 ⋁¬a2…⋁¬an) ⋀ (¬x ⋁¬p1 ⋁¬p2…⋁¬pn) 
⋀ 
(¬x ⋁b1 ⋁ c1)⋀(¬x ⋁b2 ⋁ c2)..⋀(¬x ⋁bn ⋁ cn)  
⋀

(x ⋁q1 ⋁ r1)⋀(x ⋁q2 ⋁ r2)..⋀(x ⋁qn ⋁ rn)
(¬a1 ⋁¬a2…⋁¬an) ⋀

(q1 ⋁ r1)⋀(q2 ⋁ r2).. ⋀ (qn ⋁ rn)
(¬p1 ⋁¬p2…⋁¬pn) ⋀

(b1 ⋁ c1)⋀(b2 ⋁ c2)..⋀(bn ⋁ cn)

Consider F[x=0]
Consider F[x=1]
(x ⋁¬a1 ⋁¬a2…⋁¬an) ⋀ (¬x ⋁¬p1 ⋁¬p2…⋁¬pn) 
⋀ 
(¬x ⋁b1 ⋁ c1)⋀(¬x ⋁b2 ⋁ c2)..⋀(¬x ⋁bn ⋁ cn)  
⋀

(x ⋁q1 ⋁ r1)⋀(x ⋁q2 ⋁ r2)..⋀(x ⋁qn ⋁ rn)
(¬a1 ⋁¬a2…⋁¬an) ⋀

(q1 ⋁ r1)⋀(q2 ⋁ r2).. ⋀ (qn ⋁ rn)
(¬p1 ⋁¬p2…⋁¬pn) ⋀

(b1 ⋁ c1)⋀(b2 ⋁ c2)..⋀(bn ⋁ cn)
Horn

Consider F[x=0]
Consider F[x=1]
(x ⋁¬a1 ⋁¬a2…⋁¬an) ⋀ (¬x ⋁¬p1 ⋁¬p2…⋁¬pn) 
⋀ 
(¬x ⋁b1 ⋁ c1)⋀(¬x ⋁b2 ⋁ c2)..⋀(¬x ⋁bn ⋁ cn)  
⋀

(x ⋁q1 ⋁ r1)⋀(x ⋁q2 ⋁ r2)..⋀(x ⋁qn ⋁ rn)
(¬a1 ⋁¬a2…⋁¬an) ⋀

(q1 ⋁ r1)⋀(q2 ⋁ r2).. ⋀ (qn ⋁ rn)
(¬p1 ⋁¬p2…⋁¬pn) ⋀

(b1 ⋁ c1)⋀(b2 ⋁ c2)..⋀(bn ⋁ cn)
2-cnf

Consider F[x=0]
Consider F[x=1]
(x ⋁¬a1 ⋁¬a2…⋁¬an) ⋀ (¬x ⋁¬p1 ⋁¬p2…⋁¬pn) 
⋀ 
(¬x ⋁b1 ⋁ c1)⋀(¬x ⋁b2 ⋁ c2)..⋀(¬x ⋁bn ⋁ cn)  
⋀

(x ⋁q1 ⋁ r1)⋀(x ⋁q2 ⋁ r2)..⋀(x ⋁qn ⋁ rn)
(¬a1 ⋁¬a2…⋁¬an) ⋀

(q1 ⋁ r1)⋀(q2 ⋁ r2).. ⋀ (qn ⋁ rn)
(¬p1 ⋁¬p2…⋁¬pn) ⋀

(b1 ⋁ c1)⋀(b2 ⋁ c2)..⋀(bn ⋁ cn) 2-cnf

Choose base classes C1,.. ,Cr
X is a strong backdoor into C1⊕… ⊕Cr if for every
assignment of X, every cluster of the reduced formula

is in some Ci.

Choose base classes C1,.. ,Cr
X is a strong backdoor into C1⊕… ⊕Cr if for every
assignment of X, every cluster of the reduced formula

is in some Ci.
A minimal set of clauses which is variable-disjoint

from the remaining clauses.

Strong backdoor of size c

to composition of

several Islands
Strong backdoor of size c to

a single Island of Tractability

If H is a ﬁnite set of ﬁnite constraint languages, then there
is a Polynomial-time algorithm to check if a given instance
has a strong-backdoor of size O(log log n) into ⊕H.
Our Theorem

Our Theorem

Our Theorem
• Under standard complexity hypotheses, ﬁnite-ness of
the constraint languages is unavoidable.

If H is a ﬁnite set of tractable, ﬁnite constraint languages,
then CSP can be solved in Polynomial time on instances with
a backdoor of size O(log log n) to ⊕H.
Corollary

Proof Sketch
• Work with incidence graphs. 
Variables
Constraints
I/p: CSP F, k
Q: Is there a s.b.d of size k into
comp(H)?

Proof Sketch
Variables
Constraints
w1 w2 w3
Approx. Solution W
I/p: CSP F, k
comp(H)?
,|W|=k+1

Proof Sketch
Variables
Constraints
w1 w2 s1 s2 s3w3
Target Solution SApprox. Solution W
I/p: CSP F, k
comp(H)?
,|W|=k+1

Proof Sketch
Variables
Constraints
w1 w2 s1 s2 s3w3
Target Solution S
• 2 cases depending on the interaction between them
Approx. Solution W
I/p: CSP F, k
comp(H)?
,|W|=k+1

Proof Sketch
Variables
Constraints
w1 w2 s1 s2 s3w3
Case 1: S does not `split’ W
• Work with incidence graphs.

Proof Sketch
Variables
Constraints
w1 w2 s1 s2 s3w3
• 2 cases depending on the interaction between them

Proof Sketch
Variables
Constraints
w1
w2 w3s1 s2 s3
Case 2: S splits W

Proof Sketch
Variables
Constraints
• Case 1 is the `base’ case.
w1
w2 w3s1 s2 s3
Case 2: S splits W

Proof Sketch
w1 w2 s1 s2 s3w3
• Every sub formula Z of G-S
interacts with the rest of the
formula through S, and |S|<=k.

• Also, S is a strong backdoor for
F[S + Z].
SW
Lemma: F has a set of g(k) sub-formulas such that : 
each sub formula Z interacts with the rest of F via at most k variables  
AND

the variables on the boundary of Z already form a strong backdoor set for Z

AND

S intersects the boundary of one of these subformulas.

Proof Sketch
• There is a sub formula Z of G-S
that contains W1 and interacts with
the rest of the formula through S.
• Also, S is a strong backdoor for
F[S + Z].
Case 2: S splits W into W1 and W2
SW
Lemma: F has a set of g(k) sub-formulas such that : 
each sub formula Z interacts with the rest of F via at most k variables  
AND

Z contains W1

AND

S intersects any ‘local optimal solution’ OR the boundary of one of these
subformulas.
w1
w2 w3s1 s2 s3

Summing up
Thank you for your attention!

Summing up
• Backdoors + FPT = extend tractability results for CSP
based on `distance’ to islands of tractability.

Summing up
• Composite base classes allow further generalization.

Summing up
• Also works for #CSP.

Summing up
• Also works for #CSP.
• Better tractability results for speciﬁc composite
classes and special non-ﬁnite cases?

Archipelagos

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