862 research outputs found
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
201
Computing the Boolean product of two n\times n Boolean matrices using O(n^2) mechanical operation
We study the problem of determining the Boolean product of two n\times n
Boolean matrices in an unconventional computational model allowing for
mechanical operations. We show that O(n^2) operations are sufficient to compute
the product in this model.Comment: 11 pages, 7 figure
Switching functions whose monotone complexity is nearly quadratic
AbstractA sequence of monotone switching functions hn:{0,1}n→ {0,1}n is constructed, such that the monotone complexity of hn grows faster than Ω(n2 log−2n). Previously the best lower bounds of this nature were several Ω(n32 bounds due to Pratt, Paterson, Mehlhorn and Galil and Savage
Lower Bounds for DeMorgan Circuits of Bounded Negation Width
We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits.
We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width
Formulas vs. Circuits for Small Distance Connectivity
We give the first super-polynomial separation in the power of bounded-depth
boolean formulas vs. circuits. Specifically, we consider the problem Distance
Connectivity, which asks whether two specified nodes in a graph of size
are connected by a path of length at most . This problem is solvable
(by the recursive doubling technique) on {\bf circuits} of depth
and size . In contrast, we show that solving this problem on {\bf
formulas} of depth requires size for all . As corollaries:
(i) It follows that polynomial-size circuits for Distance Connectivity
require depth for all . This matches the
upper bound from recursive doubling and improves a previous lower bound of Beame, Pitassi and Impagliazzo [BIP98].
(ii) We get a tight lower bound of on the size required to
simulate size- depth- circuits by depth- formulas for all and . No lower bound better than
was previously known for any .
Our proof technique is centered on a new notion of pathset complexity, which
roughly speaking measures the minimum cost of constructing a set of (partial)
paths in a universe of size via the operations of union and relational
join, subject to certain density constraints. Half of our proof shows that
bounded-depth formulas solving Distance Connectivity imply upper bounds
on pathset complexity. The other half is a combinatorial lower bound on pathset
complexity
Karchmer-Wigderson Games for Hazard-Free Computation
We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games.
Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity
Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth
We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n-dimensional Boolean vector convolution has Omega(n^{2-4 epsilon}) and-gates. Analogously, any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n x n Boolean matrix product has Omega(n^{3-4 epsilon}) and-gates. We complete our lower-bound trade-offs with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms
Karchmer-Wigderson Games for Hazard-free Computation
We present a Karchmer-Wigderson game to study the complexity of hazard-free
formulas. This new game is both a generalization of the monotone
Karchmer-Wigderson game and an analog of the classical Boolean
Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing
monotone and general games.
Using this game, we prove hazard-free formula size and depth lower bounds
that are provably stronger than those possible by the standard technique of
transferring results from monotone complexity in a black-box fashion. For the
multiplexer function we give (1) a hazard-free formula of optimal size and (2)
an improved low-depth hazard-free formula of almost optimal size and (3) a
hazard-free formula with alternation depth that has optimal depth. We then
use our optimal constructions to obtain an improved universal worst-case
hazard-free formula size upper bound. We see our results as a significant step
towards establishing hazard-free computation as an independent missing link
between Boolean complexity and monotone complexity.Comment: 34 pages, To appear in ITCS 202
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
- …