338 research outputs found
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
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
The Ising Partition Function: Zeros and Deterministic Approximation
We study the problem of approximating the partition function of the
ferromagnetic Ising model in graphs and hypergraphs. Our first result is a
deterministic approximation scheme (an FPTAS) for the partition function in
bounded degree graphs that is valid over the entire range of parameters
(the interaction) and (the external field), except for the case
(the "zero-field" case). A randomized algorithm (FPRAS)
for all graphs, and all , has long been known. Unlike most other
deterministic approximation algorithms for problems in statistical physics and
counting, our algorithm does not rely on the "decay of correlations" property.
Rather, we exploit and extend machinery developed recently by Barvinok, and
Patel and Regts, based on the location of the complex zeros of the partition
function, which can be seen as an algorithmic realization of the classical
Lee-Yang approach to phase transitions. Our approach extends to the more
general setting of the Ising model on hypergraphs of bounded degree and edge
size, where no previous algorithms (even randomized) were known for a wide
range of parameters. In order to achieve this extension, we establish a tight
version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a
classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results
on optimality of univariate Lee-Yang theorem
FPTAS for #BIS with Degree Bounds on One Side
Counting the number of independent sets for a bipartite graph (#BIS) plays a
crucial role in the study of approximate counting. It has been conjectured that
there is no fully polynomial-time (randomized) approximation scheme
(FPTAS/FPRAS) for #BIS, and it was proved that the problem for instances with a
maximum degree of is already as hard as the general problem. In this paper,
we obtain a surprising tractability result for a family of #BIS instances. We
design a very simple deterministic fully polynomial-time approximation scheme
(FPTAS) for #BIS when the maximum degree for one side is no larger than .
There is no restriction for the degrees on the other side, which do not even
have to be bounded by a constant. Previously, FPTAS was only known for
instances with a maximum degree of for both sides.Comment: 15 pages, to appear in STOC 2015; Improved presentations from
previous version
Correlation decay and partition function zeros: Algorithms and phase transitions
We explore connections between the phenomenon of correlation decay and the
location of Lee-Yang and Fisher zeros for various spin systems. In particular
we show that, in many instances, proofs showing that weak spatial mixing on the
Bethe lattice (infinite -regular tree) implies strong spatial mixing on
all graphs of maximum degree can be lifted to the complex plane,
establishing the absence of zeros of the associated partition function in a
complex neighborhood of the region in parameter space corresponding to strong
spatial mixing. This allows us to give unified proofs of several recent results
of this kind, including the resolution by Peters and Regts of the Sokal
conjecture for the partition function of the hard core lattice gas. It also
allows us to prove new results on the location of Lee-Yang zeros of the
anti-ferromagnetic Ising model.
We show further that our methods extend to the case when weak spatial mixing
on the Bethe lattice is not known to be equivalent to strong spatial mixing on
all graphs. In particular, we show that results on strong spatial mixing in the
anti-ferromagnetic Potts model can be lifted to the complex plane to give new
zero-freeness results for the associated partition function. This extension
allows us to give the first deterministic FPTAS for counting the number of
-colorings of a graph of maximum degree provided only that . This matches the natural bound for randomized algorithms obtained by
a straightforward application of Markov chain Monte Carlo. We also give an
improved version of this result for triangle-free graphs
Zeros of ferromagnetic 2-spin systems
We study zeros of the partition functions of ferromagnetic 2-state spin
systems in terms of the external field, and obtain new zero-free regions of
these systems via a refinement of Asano's and Ruelle's contraction method. The
strength of our results is that they do not depend on the maximum degree of the
underlying graph. Via Barvinok's method, we also obtain new efficient and
deterministic approximate counting algorithms. In certain regimes, our
algorithm outperforms all other methods such as Markov chain Monte Carlo and
correlation decay
Fisher Zeros and Correlation Decay in the Ising Model
The Ising model originated in statistical physics as a means of studying phase transitions in magnets, and has been the object of intensive study for almost a century. Combinatorially, it can be viewed as a natural distribution over cuts in a graph, and it has also been widely studied in computer science, especially in the context of approximate counting and sampling. In this paper, we study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the "interaction parameter"; these are known as Fisher zeros in light of their introduction by Fisher in 1965. While the zeros of the partition function as a polynomial in the "field" parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy, or the logarithm of the partition function) and decay of correlations with distance. We also discuss the consequences of our result for efficient deterministic approximation of the partition function. Our proof relies heavily on algorithmic techniques, notably Weitz\u27s self-avoiding walk tree, and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics
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Approximate counting, phase transitions and geometry of polynomials
In classical statistical physics, a phase transition is understood by studying the geometry (the zero-set) of an associated polynomial (the partition function). In this thesis, we will show that one can exploit this notion of phase transitions algorithmically, and conversely exploit the analysis of algorithms to understand phase transitions. As applications, we give efficient deterministic approximation algorithms (FPTAS) for counting -colorings, and for computing the partition function of the Ising model
Virtual variable sampling discrete fourier transform based selective odd-order harmonic repetitive control of DC/AC converters
This paper proposes a frequency adaptive discrete Fourier transform (DFT) based repetitive control (RC) scheme for DC/AC converters. By generating infinite magnitude on the interested harmonics, the DFT-based RC offers a selective harmonic scheme to eliminate waveform distortion. The traditional DFT-based selective harmonic RC, however, is sensitive to frequency fluctuation since even very small frequency fluctuation leads to a severe magnitude decrease. To address the problem, virtual variable sampling method, which creates an adjustable virtual delay unit to closely approximate a variable sampling delay, is proposed to enable the DFT-based selective harmonic RC to be frequency adaptive. Moreover, a selective odd-order harmonic DFT filter is developed to deal with the dominant odd order harmonic. Because it halves the number of sampling delays in the DFT filter, the system transient response gets nearly 50% improvement. A comprehensive series of experiments of the proposed VVS DFT-based selective odd-order harmonic RC controlled programmable AC power source under frequency variations are presented to verify the effectiveness of the proposed method
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