10 research outputs found
On the Structure and the Number of Prime Implicants of 2-CNFs
Let be the maximum number of prime implicants that any -CNF on n
variables can have. We show that
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)
A Variant of the VC-Dimension with Applications to Depth-3 Circuits
We introduce the following variant of the VC-dimension. Given and a positive integer , we define to be the
size of the largest subset such that the projection of on
every subset of of size is the -dimensional cube. We show that
determining the largest cardinality of a set with a given
dimension is equivalent to a Tur\'an-type problem related to the total number
of cliques in a -uniform hypergraph. This allows us to beat the
Sauer--Shelah lemma for this notion of dimension. We use this to obtain several
results on -circuits, i.e., depth- circuits with top gate OR and
bottom fan-in at most :
* Tight relationship between the number of satisfying assignments of a
-CNF and the dimension of the largest projection accepted by it, thus
improving Paturi, Saks, and Zane (Comput. Complex. '00).
* Improved -circuit lower bounds for affine dispersers for
sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture
under which we get further improvement.
* We make progress towards settling the complexity of the inner
product function and all degree- polynomials over in general.
The question of determining the complexity of IP was recently
posed by Golovnev, Kulikov, and Williams (ITCS'21)
Strong ETH and Resolution via Games and the Multiplicity of Strategies
We consider a restriction of the Resolution proof system in which at most a fixed number of variables can be resolved more than once along each refutation path. This system lies between regular Resolution, in which no variable can be resolved more than once along any path, and general Resolution where there is no restriction on the number of such variables. We show that when the number of re-resolved variables is not too large, this proof system is consistent with the Strong Exponential Time Hypothesis (SETH). More precisely for large n and k we show that there are unsatisfiable k-CNF formulas which require Resolution refutations of size 2^{(1 - epsilon_k)n}, where n is the number of variables and epsilon_k=~O(k^{-1/5}), whenever in each refutation path we only allow at most ~O(k^{-1/5})n variables to be resolved multiple times. However, these re-resolved variables along different paths do not need to be the same. Prior to this work, the strongest proof system shown to be consistent with SETH was regular Resolution [Beck and Impagliazzo, STOC\u2713]. This work strengthens that result and gives a different and conceptually simpler game-theoretic proof for the case of regular Resolution
Cops-Robber games and the resolution of Tseitin Formulas
We characterize several complexity measures for the resolution of Tseitin formulas in terms of a two person cop-robber game. Our game is a slight variation of the one Seymour and Thomas used in order to characterize the tree-width parameter. For any undirected graph, by counting the number of cops needed in our game in order to catch a robber in it, we are able to exactly characterize the width, variable space and depth measures for the resolution of the Tseitin formula corresponding to that graph. We also give an exact game characterization of resolution variable space for any formula. We show that our game can be played in a monotone way. This implies that the corresponding resolution measures on Tseitin formulas correspond exactly to those under the restriction of regular resolution. Using our characterizations we improve the existing complexity bounds for Tseitin formulas showing that resolution width, depth and variable space coincide up to a logarithmic factor, and that variable space is bounded by the clause space times a logarithmic factor
Linear Branching Programs and Directional Affine Extractors
A natural model of read-once linear branching programs is a branching program
where queries are linear forms, and along each path, the queries
are linearly independent. We consider two restrictions of this model, which we
call weakly and strongly read-once, both generalizing standard read-once
branching programs and parity decision trees. Our main results are as follows.
- Average-case complexity. We define a pseudo-random class of functions which
we call directional affine extractors, and show that these functions are hard
on average for the strongly read-once model. We then present an explicit
construction of such function with good parameters. This strengthens the result
of Cohen and Shinkar (ITCS'16) who gave such average-case hardness for parity
decision trees. Directional affine extractors are stronger than the more
familiar class of affine extractors. Given the significance of these functions,
we expect that our new class of functions might be of independent interest.
- Proof complexity. We also consider the proof system
which is an extension of resolution with linear queries. A refutation of a CNF
in this proof system naturally defines a linear branching program solving the
corresponding search problem. Conversely, we show that a weakly read-once
linear BP solving the search problem can be converted to a
refutation with constant blow up