Unsatisfiable Linear CNF Formulas Are Large and Complex

We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear k-CNF formulas with at most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic means, and we have no explicit construction coming even close to it. One reason for this is that unsatisfiable linear formulas exhibit a more complex structure than general (non-linear) formulas: First, any treelike resolution refutation of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$. This implies that small unsatisfiable linear k-CNF formulas are hard instances for Davis-Putnam style splitting algorithms. Second, if we require that the formula F have a strict resolution tree, i.e. every clause of F is used only once in the resolution tree, then we need at least a^a^...^a clauses, where a is approximately 2 and the height of this tower is roughly k.Comment: 12 pages plus a two-page appendix; corrected an inconsistency between title of the paper and title of the arxiv submissio

Satisfiability of Almost Disjoint CNF Formulas

We call a CNF formula linear if any two clauses have at most one variable in common. Let m(k) be the largest integer m such that any linear k-CNF formula with <= m clauses is satisfiable. We show that 4^k / (4e^2k^3) <= m(k) < ln(2) k^4 4^k. More generally, a (k,d)-CSP is a constraint satisfaction problem in conjunctive normal form where each variable can take on one of d values, and each constraint contains k variables and forbids exacty one of the d^k possible assignments to these variables. Call a (k,d)-CSP l-disjoint if no two distinct constraints have l or more variables in common. Let m_l(k,d) denote the largest integer m such that any l-disjoint (k,d)-CSP with at most m constraints is satisfiable. We show that 1/k (d^k/(ed^(l-1)k))^(1+1/(l-1))<= m_l(k,d) < c (k^2/l ln(d) d^k)^(1+1/(l-1)). for some constant c. This means for constant l, upper and lower bound differ only in a polynomial factor in d and k

PPSZ is better than you think

PPSZ, for long time the fastest known algorithm for $k$-SAT, works by going through the variables of the input formula in random order; each variable is then set randomly to $0$ or $1$, unless the correct value can be inferred by an efficiently implementable rule (like small-width resolution; or being implied by a small set of clauses). We show that PPSZ performs exponentially better than previously known, for all $k \geq 3$. For Unique-$3$-SAT we bound its running time by $O(1.306973^{n})$, which is somewhat better than the algorithm of Hansen, Kaplan, Zamir, and Zwick, which runs in time $O(1.306995^n)$. Before that, the best known upper bound for Unique-$3$-SAT was $O(1.3070319^n)$. All improvements are achieved without changing the original PPSZ. The core idea is to pretend that PPSZ does not process the variables in uniformly random order, but according to a carefully designed distribution. We write "pretend" since this can be done without any actual change to the algorithm

Impatient PPSZ - A Faster Algorithm for CSP

PPSZ is the fastest known algorithm for (d,k)-CSP problems, for most values of d and k. It goes through the variables in random order and sets each variable randomly to one of the d colors, excluding those colors that can be ruled out by looking at few constraints at a time. We propose and analyze a modification of PPSZ: whenever all but 2 colors can be ruled out for some variable, immediately set that variable randomly to one of the remaining colors. We show that our new "impatient PPSZ" outperforms PPSZ exponentially for all k and all d ? 3 on formulas with a unique satisfying assignment

Tighter Hard Instances for PPSZ

We construct uniquely satisfiable k-CNF formulas that are hard for the PPSZ algorithm, the currently best known algorithm solving k-SAT. This algorithm tries to generate a satisfying assignment by picking a random variable at a time and attempting to derive its value using some inference heuristic and otherwise assigning a random value. The "weak PPSZ" checks all subformulas of a given size to derive a value and the "strong PPSZ" runs resolution with width bounded by some given function. Firstly, we construct graph-instances on which "weak PPSZ" has savings of at most (2 + epsilon)/k; the saving of an algorithm on an input formula with n variables is the largest gamma such that the algorithm succeeds (i.e. finds a satisfying assignment) with probability at least 2^{- (1 - gamma) n}. Since PPSZ (both weak and strong) is known to have savings of at least (pi^2 + o(1))/6k, this is optimal up to the constant factor. In particular, for k=3, our upper bound is 2^{0.333... n}, which is fairly close to the lower bound 2^{0.386... n} of Hertli [SIAM J. Comput.\u2714]. We also construct instances based on linear systems over F_2 for which strong PPSZ has savings of at most O(log(k)/k). This is only a log(k) factor away from the optimal bound. Our constructions improve previous savings upper bound of O((log^2(k))/k) due to Chen et al. [SODA\u2713]

Super Strong ETH Is True for PPSZ with Small Resolution Width

We construct k-CNFs with m variables on which the strong version of PPSZ k-SAT algorithm, which uses resolution of width bounded by O(√{log log m}), has success probability at most 2^{-(1-(1 + ε)2/k)m} for every ε > 0. Previously such a bound was known only for the weak PPSZ algorithm which exhaustively searches through small subformulas of the CNF to see if any of them forces the value of a given variable, and for strong PPSZ the best known previous upper bound was 2^{-(1-O(log(k)/k))m} (Pudlák et al., ICALP 2017)