Parameterized complexity and kernelizability of
Max Ones and Exact Ones problems
Stefan Kratsch1
Dániel Marx2
Magnus Wahlström1
November 15, 2010
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones1 problem
/ 19
Introduction
Dichotomy theorems for generalized satisfiability problems.
For every problem in a family of problems, we want to know if it is
I
I
I
polynomial-time solvable,
fixed-parameter tractable,
has a polynomial kernel.
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones2 problem
/ 19
Generalized satisfiability
Let Γ be a set Boolean of relations. An Γ-formula is a conjunction of
relations in Γ:
R1 (x1 , x4 , x5 ) ∧ R2 (x2 , x1 ) ∧ R1 (x3 , x3 , x3 ) ∧ R3 (x5 , x1 , x4 , x1 )
Γ-SAT
Input: A Γ-formula ϕ
Find: A variable assignment satisfying ϕ
Γ = {a 6= b} ⇒ Γ-SAT = 2-coloring of a graph
Γ = {a ∨ b, a ∨ b̄, ā ∨ b̄} ⇒ Γ-SAT = 2SAT
Γ = {a ∨ b ∨ c, a ∨ b ∨ c̄, a ∨ b̄ ∨ c̄, ā ∨ b̄ ∨ c̄} ⇒ Γ-SAT = 3SAT
Question: For which Γ is Γ-SAT polynomial time solvable?
For which Γ is it NP-complete?
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones3 problem
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Schaefer’s Dichotomy Theorem (1978)
For every Γ, the Γ-SAT problem is polynomial time solvable if one of the
following holds, and NP-complete otherwise:
Every relation is satisfied by the all 0 assignment
Every relation is satisfied by the all 1 assignment
Every relation can be expressed by a 2SAT formula (bijunctive)
Every relation can be expressed by a Horn formula
Every relation can be expressed by an dual-Horn formula
Every relation is an affine subspace over GF (2)
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones4 problem
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Other dichotomy results
Approximability of Max SAT, Min UnSAT [Khanna et al., 2001]
Approximability of Max Ones, Min Ones [Khanna et al., 2001]
Generalization to 3 valued variables [Bulatov, 2002]
Inverse satisfiability [Kavvadias and Sideri, 1999]
etc.
Our contribution: Parameterized complexity analysis of Max Ones and
Exact Ones.
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones5 problem
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Problems with prescribed cardinalities
Min Ones SAT(Γ): Given a Γ-formula and integer k, is there a satisfying
assignment with at most k 1’s?
For Γ = {x ∨ y }, this is equivalent to Minimum Vertex Cover.
Max Ones SAT(Γ): Given a Γ-formula and integer k, is there a satisfying
assignment with at least k 1’s?
For Γ = {x̄ ∨ ȳ }, this is equivalent to Maximum Independent Set.
Exact Ones SAT(Γ): Given a Γ-formula and integer k, is there a satisfying
assignment with exactly k 1’s?
Can express both Minimum Vertex Cover and Maximum
Independent Set.
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones6 problem
/ 19
First question
Which sets Γ make these problems polynomial-time solvable?
Previous results [Creignou et al. 2001, Khanna et al. 2001]:
Min Ones SAT(Γ): In P if Γ is 0-valid, Horn, or width-2 affine,
otherwise NP-hard.
Max Ones SAT(Γ): In P if Γ is 1-valid, dual-Horn, or width-2 affine,
otherwise NP-hard.
Exact Ones SAT(Γ): In P if Γ is width-2 affine, otherwise NP-hard.
The problems are well understood with respect to polynomial-time
solutions.
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones7 problem
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Second question:
Which sets Γ make these problems fixed-parameter tractable?
Easy: Min Ones SAT(Γ) is FPT for every Γ.
Theorem: [Marx 2004] Exact Ones SAT(Γ) is FPT if every relation R ∈ Γ
is weakly separable and W[1]-hard otherwise.
What about Max Ones?
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones8 problem
/ 19
Partial polymorphisms
Weak separability (last slide) defined by partial polymorphism:
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1
1
1
FPT(1)
0 0
1 0
1 1
0 1
0
0
0
0
1
1
1
1
FPT(2)
1 0
0 0
0 1
1 1
0
0
0
0
Applying a partial polymorphism, say FPT(1):
pick 3 tuples from R and write them as rows of a matrix
apply FPT(1) on each column; if all are defined, this gives a new tuple
R is closed under FPT(1) if it contains all tuples created this way
A relation is weakly separable if it is closed under FPT(1) and FPT(2).
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of Max
November
Ones and
15, Exact
2010 Ones9 problem
/ 19
Third question
For which sets Γ do these problems have a polynomial kernel?
Theorem (Kratsch, W. 2010)
Min Ones SAT(Γ) has a polynomial kernel if it is in P, or if Γ is mergeable;
otherwise not (unless the polynomial hierarchy collapses).
Theorem (This work)
Exact Ones SAT(Γ) has a polynomial kernel if Γ is both mergeable and
semi-separable (slight strengthening of weak separability); otherwise not
(unless the polynomial hierarchy collapses).
Note: Exact Ones no easier than Min Ones (by adding k dummy variables)
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
10 problem
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Max Ones result
Max Ones is very different: not even clear if it can be solved in
polynomial-time for fixed k.
Theorem (This work)
Every finite Γ can be classified into one of the following 5 types:
1
Max Ones SAT(Γ) is polynomial-time solvable.
2
Max Ones SAT(Γ) is NP-hard, but FPT and admits a polynomial
kernel.
3
Max Ones SAT(Γ) is FPT, but has no polynomial kernel (unless the
polynomial hierarchy collapses).
4
Max Ones SAT(Γ) is W[1]-hard, but can be solved in polynomial time
for fixed k.
5
Max Ones SAT(Γ) is NP-hard even for k = 1.
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
11 problem
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Proof sketch (Max Ones)
Observation: If SAT(Γ) is NP-hard, then Max Ones SAT(Γ) is NP-hard as
well already for k = 0, hence not FPT.
Thus to classify Max Ones SAT(Γ), we need to go through the 6 classes of
Schaefer’s Theorem (0-valid, 1-valid, bijunctive, horn, dual-horn, affine).
The 1-valid and dual-Horn cases are in P.
Affine case:
1
Remove "frozen" variables that have the same value in every satisfying
assignment.
2
Observation: If there are no frozen variables, then at least half of the
variables can be set to 1.
3
If there are at least 2k variables, the answer is yes. Otherwise, we
have a kernel of size less than 2k.
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
12 problem
/ 19
Proof sketch (Max Ones)
The bijunctive (2SAT) cases fall into several classes:
1
Polynomial kernel for e.g. Γ1 = {(x ∨ y ), (x 6= y ), (x = y )}
2
FPT, but no poly kernel for e.g. Γ2 = {(x ∨ y ), (x 6= y ), (x → y )}
3
W[1]-hard if Γ can express (¬x ∨ ¬y )
Further details:
Kernel for Γ1 by counting (x 6= y )-groups
Algorithm for Γ2 by branching over (x 6= y )-groups
Kernelization lower bound also for slightly weaker
relation (x 6= y ) ∧ (z → x); will sketch next.
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
13 problem
/ 19
Lower Bound Problem
Multiple Compatible Patterns (MCP)
Input: Patterns in {0, 1, F}r , number k
Parameter: r + k
Question: Is there a string compatible with at least k patterns?
Patterns are compatible if they match the same string; F matches 0 or 1.
FPT: search space 2r
NP-complete: reduction from Clique
No polynomial kernel unless PH collapses by compositionality
Will show Max Ones SAT(R) hardness by parameter-preserving reduction
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
14 problem
/ 19
Lower Bound Problem
Multiple Compatible Patterns (MCP)
Input: Patterns in {0, 1, F}r , number k
Parameter: r + k
Question: Is there a string compatible with at least k patterns?
Patterns are compatible if they match the same string; F matches 0 or 1.
FPT: search space 2r
NP-complete: reduction from Clique
No polynomial kernel unless PH collapses by compositionality
Will show Max Ones SAT(R) hardness by parameter-preserving reduction
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
14 problem
/ 19
Reduction MCP → Max Ones SAT((x 6= y ) ∧ (z → x))
P1 = 00 . . . 0
z1
x1 6= y1
P2 = 1F . . . 0
z2
x2 6= y2
P3 = F1 . . . 1
z3
...
xr 6= yr
2r variables xi 6= yi for the output string, zj for the patterns Pj
(xi 6= yi ) ∧ (zj → xi ): bit i of pattern j is 0
(yi 6= xi ) ∧ (zj → yi ): bit i of pattern j is 1
Weight k + r solution iff k compatible patterns
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
15 problem
/ 19
Reduction MCP → Max Ones SAT((x 6= y ) ∧ (z → x))
P1 = 00 . . . 0
z1
x1 6= y1
P2 = 1F . . . 0
z2
x2 6= y2
P3 = F1 . . . 1
z3
...
xr 6= yr
2r variables xi 6= yi for the output string, zj for the patterns Pj
(xi 6= yi ) ∧ (zj → xi ): bit i of pattern j is 0
(yi 6= xi ) ∧ (zj → yi ): bit i of pattern j is 1
Weight k + r solution iff k compatible patterns
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
15 problem
/ 19
Reduction MCP → Max Ones SAT((x 6= y ) ∧ (z → x))
P1 = 00 . . . 0
z1
x1 6= y1
P2 = 1F . . . 0
z2
x2 6= y2
P3 = F1 . . . 1
z3
...
xr 6= yr
2r variables xi 6= yi for the output string, zj for the patterns Pj
(xi 6= yi ) ∧ (zj → xi ): bit i of pattern j is 0
(yi 6= xi ) ∧ (zj → yi ): bit i of pattern j is 1
Weight k + r solution iff k compatible patterns
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
15 problem
/ 19
Max Ones overview
all relations
0-valid
1-valid
Horn
dual-Horn
bijunctive
width-2 affine
P
PK
affine
FPT
W[1]-h
NP-h for k = O(1)
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
16 problem
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Summary
We looked at the parameterized complexity of generalized satisfiability
problems from a modern viewpoint: we characterized not only
fixed-parameter tractability, but kernelizability as well.
Exact Ones SAT(Γ):
Known: FPT characterization.
New result: kernelizability characterization.
Max Ones SAT(Γ):
New result: FPT characterization.
New result: kernelizability characterization.
Min Ones SAT(Γ):
Known: FPT for every Γ.
Known: kernelizability characterization.
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
17 problem
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Bonus round: Exact Ones kernelizability
1
Zero-valid relations, kernelizable for Exact Ones, are really easy:
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Exact Ones SAT(Γ) W[1]-hard for (¬x ∨ ¬y ), (x → y )
Min Ones SAT(Γ) is hard to kernelize for most zero-valid relations,
except for (¬x1 ∨ · · · ∨ ¬xd ), (x → y ))
Remains: (x = y ) and (x = 0).
2
Exact Ones SAT(R) with R(x, y , z) = (x 6= y ) ∧ (z → y ) covers MCP
problem: FPT, but hard to kernelize
3
Otherwise polynomial kernel (sunflower rules)
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
18 problem
/ 19
Bonus round: Exact Ones kernelizability
1
Zero-valid relations, kernelizable for Exact Ones, are really easy:
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Exact Ones SAT(Γ) W[1]-hard for (¬x ∨ ¬y ), (x → y )
Min Ones SAT(Γ) is hard to kernelize for most zero-valid relations,
except for (¬x1 ∨ · · · ∨ ¬xd ), (x → y ))
Remains: (x = y ) and (x = 0).
2
Exact Ones SAT(R) with R(x, y , z) = (x 6= y ) ∧ (z → y ) covers MCP
problem: FPT, but hard to kernelize
3
Otherwise polynomial kernel (sunflower rules)
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
18 problem
/ 19
Bonus round: Exact Ones kernelizability
1
Zero-valid relations, kernelizable for Exact Ones, are really easy:
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I
Exact Ones SAT(Γ) W[1]-hard for (¬x ∨ ¬y ), (x → y )
Min Ones SAT(Γ) is hard to kernelize for most zero-valid relations,
except for (¬x1 ∨ · · · ∨ ¬xd ), (x → y ))
Remains: (x = y ) and (x = 0).
2
Exact Ones SAT(R) with R(x, y , z) = (x 6= y ) ∧ (z → y ) covers MCP
problem: FPT, but hard to kernelize
3
Otherwise polynomial kernel (sunflower rules)
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
18 problem
/ 19
Bonus round: Exact Ones kernelizability
1
Zero-valid relations, kernelizable for Exact Ones, are really easy:
I
I
Exact Ones SAT(Γ) W[1]-hard for (¬x ∨ ¬y ), (x → y )
Min Ones SAT(Γ) is hard to kernelize for most zero-valid relations,
except for (¬x1 ∨ · · · ∨ ¬xd ), (x → y ))
Remains: (x = y ) and (x = 0).
2
Exact Ones SAT(R) with R(x, y , z) = (x 6= y ) ∧ (z → y ) covers MCP
problem: FPT, but hard to kernelize
3
Otherwise polynomial kernel (sunflower rules)
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
18 problem
/ 19
Bonus round: Exact Ones kernelizability
1
Zero-valid relations, kernelizable for Exact Ones, are really easy:
I
I
Exact Ones SAT(Γ) W[1]-hard for (¬x ∨ ¬y ), (x → y )
Min Ones SAT(Γ) is hard to kernelize for most zero-valid relations,
except for (¬x1 ∨ · · · ∨ ¬xd ), (x → y ))
Remains: (x = y ) and (x = 0).
2
Exact Ones SAT(R) with R(x, y , z) = (x 6= y ) ∧ (z → y ) covers MCP
problem: FPT, but hard to kernelize
3
Otherwise polynomial kernel (sunflower rules)
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
18 problem
/ 19
Γ
Min Ones
Exact Ones
Max Ones
width-2 affine
{ODD3 , (¬x)}
{EVEN3 , (x)}
{ODD4 }, general affine
P
PK
FPT
FPT
P
PK
FPT
FPT
P
PK
PK
PK
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
19 problem
/ 19
Γ
Min Ones
Exact Ones
Max Ones
width-2 affine
{ODD3 , (¬x)}
{EVEN3 , (x)}
{ODD4 }, general affine
P
PK
FPT
FPT
P
PK
FPT
FPT
P
PK
PK
PK
{(x ∨ y ), (x 6= y )}
{((x → y ) ∧ (y 6= z))}
{(x ∨ y ), (x 6= y ), (x → y )}
bijunctive
PK
PK
PK
PK
PK
FPT
W[1]
W[1]
PK
FPT
FPT
W[1]-hard, XP
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
19 problem
/ 19
Γ
Min Ones
Exact Ones
Max Ones
width-2 affine
{ODD3 , (¬x)}
{EVEN3 , (x)}
{ODD4 }, general affine
P
PK
FPT
FPT
P
PK
FPT
FPT
P
PK
PK
PK
{(x ∨ y ), (x 6= y )}
{((x → y ) ∧ (y 6= z))}
{(x ∨ y ), (x 6= y ), (x → y )}
bijunctive
PK
PK
PK
PK
PK
FPT
W[1]
W[1]
PK
FPT
FPT
W[1]-hard, XP
P {R1-in-3 }
{ i xi = p (mod q)}
general
PK
FPT
FPT
PK
FPT
W[1]
not in XP
not in XP
not in XP
S Kratsch, D Marx, M Wahlström () Parameterized complexity and kernelizability of November
Max Ones 15,
and2010
Exact Ones
19 problem
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