21,670 research outputs found
A Randomized Algorithm for 3-SAT
In this work we propose and analyze a simple randomized algorithm to find a
satisfiable assignment for a Boolean formula in conjunctive normal form (CNF)
having at most 3 literals in every clause. Given a k-CNF formula phi on n
variables, and alpha in{0,1}^n that satisfies phi, a clause of phi is critical
if exactly one literal of that clause is satisfied under assignment alpha.
Paturi et. al. (Chicago Journal of Theoretical Computer Science 1999) proposed
a simple randomized algorithm (PPZ) for k-SAT for which success probability
increases with the number of critical clauses (with respect to a fixed
satisfiable solution of the input formula). Here, we first describe another
simple randomized algorithm DEL which performs better if the number of critical
clauses are less (with respect to a fixed satisfiable solution of the input
formula). Subsequently, we combine these two simple algorithms such that the
success probability of the combined algorithm is maximum of the success
probabilities of PPZ and DEL on every input instance. We show that when the
average number of clauses per variable that appear as unique true literal in
one or more critical clauses in phi is between 1 and 1.9317, combined algorithm
performs better than the PPZ algorithm
Improved Exact Algorithms for Mildly Sparse Instances of Max SAT
We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2^{(1-mu(c))n}) for instances with n variables and m=cn clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with mu(c)=1/O(c * log(c)) due to Dantsin and Wolpert [SAT 2006], a randomized polynomial space algorithm with mu(c)=1/O(c * log^3(c)) and a deterministic polynomial space algorithm with mu(c)=1/O(c^2 * log^2(c)) due to Sakai, Seto and Tamaki [Theory Comput. Syst., 2015]. Our first result is a deterministic polynomial space algorithm with mu(c)=1/O(c * log(c)) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2^n only if m=O(n^2).
Our second results are deterministic exponential space algorithms for Max SAT with mu(c)=1/O((c * log(c))^{2/3}) and for Max 3-SAT with mu(c)=1/O(c^{1/2}) that run super-polynomially faster than 2^n when m=o(n^{5/2}/log^{5/2}(n)) and m=o(n^3/log^2(n)) respectively
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers
A line of work initiated by Fortnow in 1997 has proven model-independent
time-space lower bounds for the problem and related problems
within the polynomial-time hierarchy. For example, for the
problem, the state-of-the-art is that the problem cannot be solved by
random-access machines in time and space simultaneously for .
We extend this lower bound approach to the quantum and randomized domains.
Combining Grover's algorithm with components from time-space
lower bounds, we show that there are problems verifiable in time with
quantum Merlin-Arthur protocols that cannot be solved in time and
space simultaneously for , a
super-quadratic time lower bound. This result and the prior work on
can both be viewed as consequences of a more general formula for
time lower bounds against small-space algorithms, whose asymptotics we study in
full.
We also show lower bounds against randomized algorithms: there are problems
verifiable in time with (classical) Merlin-Arthur protocols that cannot
be solved in randomized time and space simultaneously for , improving a result of Diehl. For quantum Merlin-Arthur protocols, the
lower bound in this setting can be improved to .Comment: 38 pages, 5 figures. To appear in ITCS 202
Hamming Approximation of NP Witnesses
Given a satisfiable 3-SAT formula, how hard is it to find an assignment to
the variables that has Hamming distance at most n/2 to a satisfying assignment?
More generally, consider any polynomial-time verifier for any NP-complete
language. A d(n)-Hamming-approximation algorithm for the verifier is one that,
given any member x of the language, outputs in polynomial time a string a with
Hamming distance at most d(n) to some witness w, where (x,w) is accepted by the
verifier. Previous results have shown that, if P != NP, then every NP-complete
language has a verifier for which there is no
(n/2-n^(2/3+d))-Hamming-approximation algorithm, for various constants d > 0.
Our main result is that, if P != NP, then every paddable NP-complete language
has a verifier that admits no (n/2+O(sqrt(n log n)))-Hamming-approximation
algorithm. That is, one cannot get even half the bits right. We also consider
natural verifiers for various well-known NP-complete problems. They do have
n/2-Hamming-approximation algorithms, but, if P != NP, have no
(n/2-n^epsilon)-Hamming-approximation algorithms for any constant epsilon > 0.
We show similar results for randomized algorithms
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 of a system of polynomials over ?? each of which is a product of affine forms.
We focus on the case of k-CNF formulas (the k-Sub-Sat problem). Clearly, k-Sub-Sat is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that 2-Sub-Sat 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-Sub-Sat 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 (1.5)^r for 2-Sub-Sat, where r is the subspace dimension, as well as an O^*(1.4312)? time algorithm where n is the number of variables.
Turning to k-Sub-Sat for k ? 3, while known algorithms for solving a system of degree k polynomial equations already imply a solution with running time ? 2^{r(1-1/2k)}, 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 choose {?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-Sub-Sat with running time ? 2^{n-n/2k}. 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 ??, 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
Adding cardinality constraints to integer programs with applications to maximum satisfiability
Max-SAT-CC is the following optimization problem: Given a formula in CNF and a bound k, find an assignment with at most k variables being set to true that maximizes the number of satisfied clauses among all such assignments. If each clause is restricted to have at most ℓ literals, we obtain the problem Max-ℓSAT-CC. Sviridenko [Algorithmica 30 (3) (2001) 398–405] designed a (1−e−1)-approximation algorithm for Max-SAT-CC. This result is tight unless P=NP [U. Feige, J. ACM 45 (4) (1998) 634–652]. Sviridenko asked if it is possible to achieve a better approximation ratio in the case of Max-ℓSAT-CC. We answer this question in the affirmative by presenting a randomized approximation algorithm whose approximation ratio is 1-(1-1/ℓ)ℓ-ε. To do this, we develop a general technique for adding a cardinality constraint to certain integer programs. Our algorithm can be derandomized using pairwise independent random variables with small probability space
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