54 research outputs found
DNF Sparsification and a Faster Deterministic Counting Algorithm
Given a DNF formula on n variables, the two natural size measures are the
number of terms or size s(f), and the maximum width of a term w(f). It is
folklore that short DNF formulas can be made narrow. We prove a converse,
showing that narrow formulas can be sparsified. More precisely, any width w DNF
irrespective of its size can be -approximated by a width DNF with
at most terms.
We combine our sparsification result with the work of Luby and Velikovic to
give a faster deterministic algorithm for approximately counting the number of
satisfying solutions to a DNF. Given a formula on n variables with poly(n)
terms, we give a deterministic time algorithm
that computes an additive approximation to the fraction of
satisfying assignments of f for \epsilon = 1/\poly(\log n). The previous best
result due to Luby and Velickovic from nearly two decades ago had a run-time of
.Comment: To appear in the IEEE Conference on Computational Complexity, 201
Counting Solutions to Polynomial Systems via Reductions
This paper provides both positive and negative results for counting solutions to systems of polynomial equations over a finite field. The general idea is to try to reduce the problem to counting solutions to a single polynomial, where the task is easier. In both cases, simple methods are utilized that we expect will have wider applicability (far beyond algebra).
First, we give an efficient deterministic reduction from approximate counting for a system of (arbitrary) polynomial equations to approximate counting for one equation, over any finite field. We apply this reduction to give a deterministic poly(n,s,log p)/eps^2 time algorithm for approximately counting the fraction of solutions to a system of s quadratic n-variate polynomials over F_p (the finite field of prime order p) to within an additive eps factor, for any prime p. Note that uniform random sampling would already require Omega(s/eps^2) time, so our algorithm behaves as a full derandomization of uniform sampling. The approximate-counting algorithm yields efficient approximate counting for other well-known problems, such as 2-SAT, NAE-3SAT, and 3-Coloring. As a corollary, there is a deterministic algorithm (with analogous running time) for producing solutions to such systems which have at least eps p^n solutions.
Second, we consider the difficulty of exactly counting solutions to a single polynomial of constant degree, over a finite field. (Note that finding a solution in this case is easy.) It has been known for over 20 years that this counting problem is already NP-hard for degree-three polynomials over F_2; however, all known reductions increased the number of variables by a considerable amount. We give a subexponential-time reduction from counting solutions to k-CNF formulas to counting solutions to a degree-k^{O(k)} polynomial (over any finite field of O(1) order) which exactly preserves the number of variables. As a corollary, the Strong Exponential Time Hypothesis (even its weak counting variant #SETH) implies that counting solutions to constant-degree polynomials (even over F_2) requires essentially 2^n time. Similar results hold for counting orthogonal pairs of vectors over F_p
A Simple FPTAS for Counting Edge Covers
An edge cover of a graph is a set of edges such that every vertex has at
least an adjacent edge in it. Previously, approximation algorithm for counting
edge covers is only known for 3 regular graphs and it is randomized. We design
a very simple deterministic fully polynomial-time approximation scheme (FPTAS)
for counting the number of edge covers for any graph. Our main technique is
correlation decay, which is a powerful tool to design FPTAS for counting
problems. In order to get FPTAS for general graphs without degree bound, we
make use of a stronger notion called computationally efficient correlation
decay, which is introduced in [Li, Lu, Yin SODA 2012].Comment: To appear in SODA 201
FPTAS for Counting Monotone CNF
A monotone CNF formula is a Boolean formula in conjunctive normal form where
each variable appears positively. We design a deterministic fully
polynomial-time approximation scheme (FPTAS) for counting the number of
satisfying assignments for a given monotone CNF formula when each variable
appears in at most clauses. Equivalently, this is also an FPTAS for
counting set covers where each set contains at most elements. If we allow
variables to appear in a maximum of clauses (or sets to contain
elements), it is NP-hard to approximate it. Thus, this gives a complete
understanding of the approximability of counting for monotone CNF formulas. It
is also an important step towards a complete characterization of the
approximability for all bounded degree Boolean #CSP problems. In addition, we
study the hypergraph matching problem, which arises naturally towards a
complete classification of bounded degree Boolean #CSP problems, and show an
FPTAS for counting 3D matchings of hypergraphs with maximum degree .
Our main technique is correlation decay, a powerful tool to design
deterministic FPTAS for counting problems defined by local constraints among a
number of variables. All previous uses of this design technique fall into two
categories: each constraint involves at most two variables, such as independent
set, coloring, and spin systems in general; or each variable appears in at most
two constraints, such as matching, edge cover, and holant problem in general.
The CNF problems studied here have more complicated structures than these
problems and require new design and proof techniques. As it turns out, the
technique we developed for the CNF problem also works for the hypergraph
matching problem. We believe that it may also find applications in other CSP or
more general counting problems.Comment: 24 pages, 2 figures. version 1=>2: minor edits, highlighted the
picture of set cover/packing, and an implication of our previous result in 3D
matchin
From DNF Compression to Sunflower Theorems via Regularity
The sunflower conjecture is one of the most well-known open problems in combinatorics. It has several applications in theoretical computer science, one of which is DNF compression, due to Gopalan, Meka and Reingold (Computational Complexity, 2013). In this paper, we show that improved bounds for DNF compression imply improved bounds for the sunflower conjecture, which is the reverse direction of the DNF compression result. The main approach is based on regularity of set systems and a structure-vs-pseudorandomness approach to the sunflower conjecture
Deterministic search for CNF satisfying assignments in almost polynomial time
We consider the fundamental derandomization problem of deterministically
finding a satisfying assignment to a CNF formula that has many satisfying
assignments. We give a deterministic algorithm which, given an -variable
-clause CNF formula that has at least
satisfying assignments, runs in time for
and outputs a satisfying assignment of
. Prior to our work the fastest known algorithm for this problem was simply
to enumerate over all seeds of a pseudorandom generator for CNFs; using the
best known PRGs for CNFs [DETT10], this takes time
even for constant . Our approach is based on a new general
framework relating deterministic search and deterministic approximate counting,
which we believe may find further applications
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Discrete Fourier Analysis and Its Applications
The topic of discrete Fourier analysis has been extensively studied in recent decades. It plays an important role in theoretical computer science and discrete mathematics. One hand it is interesting to study the structure of boolean functions via discrete Fourier analysis. On the other hand, these structural results also provide a huge number of applications in theoretical computer science, including computational complexity, pseudorandomness, cryptography, learning theory. In this dissertation, we extend some more connections between discrete Fourier analysis and theoretical computer science. In particular, we study the following questions.\begin{itemize}\item Robust sensitivity of boolean function. In this part, we study the connection between the Fourier tail bound and the sensitivity tail bound of boolean functions, which is an analogue of the sensitivity conjecture, which was proposed by Nisan \cite{nisan1991crew}.\item DNF sparsification. The disjunctive normal form (or DNF) is a widely used representation of boolean functions. It is very interesting to study the structure of DNFs. There are two natural ways to measure the complexity of DNFs, the width and the size. In this thesis, we study a connection between these two measures. We propose a new approach by combing the swithing lemma (a combinatoric tool) and the hypercontrativity inequality (an analytic inequality). This framework does also suggest a new approach to the famous sunflower conjecture.\item Applications in learning theory. In 1989, the first Fourier-based learning algorithms was introduced by a seminar paper of Linial, Mansour and Nisan \cite{linial1989constant}. Followed by a series of subsequent works, people found that discrete Fourier analysis is powerful to design learning algorithms. One hand sparse Fourier functions are strong enough to approximate a lot of functions, on the other hand sparse Fourier functions are relatively easy to learn. Build on this framework, we give a more efficient algorithm to solve the \emph{population recovery} problem. That is how to recover a unknown distribution from noisy samples.\end{itemize
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