44,691 research outputs found
Quantitative Approximation of the Probability Distribution of a Markov Process by Formal Abstractions
The goal of this work is to formally abstract a Markov process evolving in
discrete time over a general state space as a finite-state Markov chain, with
the objective of precisely approximating its state probability distribution in
time, which allows for its approximate, faster computation by that of the
Markov chain. The approach is based on formal abstractions and employs an
arbitrary finite partition of the state space of the Markov process, and the
computation of average transition probabilities between partition sets. The
abstraction technique is formal, in that it comes with guarantees on the
introduced approximation that depend on the diameters of the partitions: as
such, they can be tuned at will. Further in the case of Markov processes with
unbounded state spaces, a procedure for precisely truncating the state space
within a compact set is provided, together with an error bound that depends on
the asymptotic properties of the transition kernel of the original process. The
overall abstraction algorithm, which practically hinges on piecewise constant
approximations of the density functions of the Markov process, is extended to
higher-order function approximations: these can lead to improved error bounds
and associated lower computational requirements. The approach is practically
tested to compute probabilistic invariance of the Markov process under study,
and is compared to a known alternative approach from the literature.Comment: 29 pages, Journal of Logical Methods in Computer Scienc
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
In this paper we show a new way of constructing deterministic polynomial-time
approximation algorithms for computing complex-valued evaluations of a large
class of graph polynomials on bounded degree graphs. In particular, our
approach works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
Computing an Approximately Optimal Agreeable Set of Items
We study the problem of finding a small subset of items that is
\emph{agreeable} to all agents, meaning that all agents value the subset at
least as much as its complement. Previous work has shown worst-case bounds,
over all instances with a given number of agents and items, on the number of
items that may need to be included in such a subset. Our goal in this paper is
to efficiently compute an agreeable subset whose size approximates the size of
the smallest agreeable subset for a given instance. We consider three
well-known models for representing the preferences of the agents: ordinal
preferences on single items, the value oracle model, and additive utilities. In
each of these models, we establish virtually tight bounds on the approximation
ratio that can be obtained by algorithms running in polynomial time.Comment: A preliminary version appeared in Proceedings of the 26th
International Joint Conference on Artificial Intelligence (IJCAI), 201
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
NP-hardness of circuit minimization for multi-output functions
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive.
In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators.
Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions
Counting Independent Sets and Colorings on Random Regular Bipartite Graphs
We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. In the weighted case, for all sufficiently large integers Delta and weight parameters lambda = Omega~ (1/(Delta)), we also obtain an FPTAS on almost every Delta-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all q >= 3 and sufficiently large integers Delta=Delta(q), there is an FPTAS to count the number of q-colorings on almost every Delta-regular bipartite graph
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