12 research outputs found
Characterization and Lower Bounds for Branching Program Size using Projective Dimension
We study projective dimension, a graph parameter (denoted by pd for a
graph ), introduced by (Pudl\'ak, R\"odl 1992), who showed that proving
lower bounds for pd for bipartite graphs associated with a Boolean
function imply size lower bounds for branching programs computing .
Despite several attempts (Pudl\'ak, R\"odl 1992 ; Babai, R\'{o}nyai, Ganapathy
2000), proving super-linear lower bounds for projective dimension of explicit
families of graphs has remained elusive.
We show that there exist a Boolean function (on bits) for which the
gap between the projective dimension and size of the optimal branching program
computing (denoted by bpsize), is . Motivated by the
argument in (Pudl\'ak, R\"odl 1992), we define two variants of projective
dimension - projective dimension with intersection dimension 1 (denoted by
upd) and bitwise decomposable projective dimension (denoted by
bitpdim).
As our main result, we show that there is an explicit family of graphs on vertices such that the projective dimension is , the
projective dimension with intersection dimension is and the
bitwise decomposable projective dimension is .
We also show that there exist a Boolean function (on bits) for which
the gap between upd and bpsize is . In contrast, we
also show that the bitwise decomposable projective dimension characterizes size
of the branching program up to a polynomial factor. That is, there exists a
constant and for any function , . We also study two other
variants of projective dimension and show that they are exactly equal to
well-studied graph parameters - bipartite clique cover number and bipartite
partition number respectively.Comment: 24 pages, 3 figure
On Disperser/Lifting Properties of the Index and Inner-Product Functions
Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated "lifted" function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. A number of important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, which is a universal gadget for lifting, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was recently shown by Lovett, Meka, Mertz, Pitassi and Zhang [Shachar Lovett et al., 2022] using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with an Index function of that size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following;
- The conjecture of Lovett et al. is false when the size of the Index gadget is less than logarithmic in N.
- The same limitation applies to the Inner-Product function. More precisely, the Inner-Product function, which is known to satisfy the disperser property at size O(log N), also does not have this property when its size is less than log N.
- Notwithstanding the above, we prove a lifting theorem that applies to Index gadgets of any size at least 4 and yields lower bounds for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs.
- Using a modification of the same idea with improved lifting parameters we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this, in turn, to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(?) refutation size, which yields many new exponential lower bounds on such proofs
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