CNF Satisfiability in a Subspace and Related Problems
We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero...
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| Vydáno v: | Algorithmica Ročník 84; číslo 11; s. 3276 - 3299 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.11.2022
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over
F
2
each of which is a product of affine forms. We focus on the case of
k
-CNF formulas (the
k
-
S
U
B
-
S
A
T
problem). Clearly,
k
-
S
U
B
-
S
A
T
is no easier than
k
-SAT, and might be harder. Indeed, via simple reductions we show that
2
-
S
U
B
-
S
A
T
is NP-hard, and
W
[
1
]
-hard when parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-
2
-
S
U
B
-
S
A
T
is NP-hard to approximate better than the trivial 3/4 ratio even on satisfiable instances. On the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with running time
O
∗
(
1.5
)
r
for
2
-
S
U
B
-
S
A
T
, where
r
is the subspace dimension, as well as an
O
∗
(
1.4312
)
n
time algorithm where
n
is the number of variables. Turning to
k
-
S
U
B
-
S
A
T
for
k
⩾
3
, while known algorithms for solving a system of degree
k
polynomial equations already imply a solution with running time
≈
2
r
(
1
-
1
/
2
k
)
, we explore a more combinatorial approach. Based on an analysis of critical variables (a key notion underlying the randomized
k
-SAT algorithm of Paturi, Pudlak, and Zane), we give an algorithm with running time
≈
n
⩽
t
2
n
-
n
/
k
where
n
is the number of variables and
t
is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for
k
-
S
U
B
-
S
A
T
with running time
≈
2
n
-
n
/
2
k
. This algorithm is in fact oblivious to the structure of the subspace, and extends when the subspace-membership constraint is replaced by any constraint for which partial satisfying assignments can be efficiently completed to a full satisfying assignment. Finally, for systems of
O
(
n
) polynomial equations in
n
variables over
F
2
, we give a fast exponential algorithm when each polynomial has bounded degree irreducible factors (but can otherwise have large degree) using a degree reduction trick. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-022-00958-4 |