14 research outputs found
Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs
We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2tw(G)Ω(1/d) in depth-d Frege systems for d < (Formula presented.) where tw(G) is the treewidth of G. This extends HÄstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)O(1/d)poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution
Bounded-depth Frege complexity of Tseitin formulas for all graphs
We prove that there is a constant K such that Tseitin formulas for a connected graph G requires proofs of size 2tw(G)javax.xml.bind.JAXBElement@531a834b in depth-d Frege systems for [Formula presented], where tw(G) is the treewidth of G. This extends HĂ„stad's recent lower bound from grid graphs to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)javax.xml.bind.JAXBElement@25a4b51fpoly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution
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Proof Complexity and Beyond
Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form âhow difficult is it to prove certain mathematical facts?â The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples
Subspace-Invariant AC Formulas
We consider the action of a linear subspace of on the set of
AC formulas with inputs labeled by literals in the set , where an element acts on formulas by
transposing the th pair of literals for all such that . A
formula is {\em -invariant} if it is fixed by this action. For example,
there is a well-known recursive construction of depth formulas of size
computing the -variable PARITY function; these
formulas are easily seen to be -invariant where is the subspace of
even-weight elements of . In this paper we establish a nearly
matching lower bound on the -invariant depth
formula size of PARITY. Quantitatively this improves the best known
lower bound for {\em unrestricted} depth
formulas, while avoiding the use of the switching lemma. More generally,
for any linear subspaces , we show that if a Boolean function is
-invariant and non-constant over , then its -invariant depth
formula size is at least where is the minimum Hamming
weight of a vector in
A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm
We show that for every there exists such that any
-uniform hypergraph with edges and maximum vertex degree
contains a set of at most edges the removal of
which breaks the hypergraph into connected components with at most edges.
We use this to give an algorithm running in time that
decides satisfiability of -variable -CSPs in which every variable
appears in at most constraints, where depends only on and
. Furthermore our algorithm solves the corresponding #CSP-SAT
and Max-CSP-SAT of these CSPs. We also show that CNF representations of
unsatisfiable -CSPs with variable frequency can be refuted in
tree-like resolution in size . Furthermore for Tseitin
formulas on graphs with degree at most (which are -CSPs) we give a
deterministic algorithm finding such a refutation
A separator theorem for hypergraphs and a CSP-SAT algorithm
We show that for every râ„2 there exists Ï”r>0 such that any r-uniform hypergraph with m edges and maximum vertex degree o(mâââ) contains a set of at most (12âÏ”r)m edges the removal of which breaks the hypergraph into connected components with at most m/2 edges. We use this to give an algorithm running in time d(1âÏ”r)m that decides satisfiability of m-variable (d,k)-CSPs in which every variable appears in at most r constraints, where Ï”r depends only on r and kâo(mâââ). Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable (2,k)-CSPs with variable frequency r can be refuted in tree-like resolution in size 2(1âÏ”r)m. Furthermore for Tseitin formulas on graphs with degree at most k (which are (2,k)-CSPs) we give a deterministic algorithm finding such a refutation
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
Efficient local search for Pseudo Boolean Optimization
Algorithms and the Foundations of Software technolog