2,166 research outputs found
The power of linear programming for general-valued CSPs
Let , called the domain, be a fixed finite set and let , called
the valued constraint language, be a fixed set of functions of the form
, where different functions might have
different arity . We study the valued constraint satisfaction problem
parametrised by , denoted by VCSP. These are minimisation
problems given by variables and the objective function given by a sum of
functions from , each depending on a subset of the variables.
Finite-valued constraint languages contain functions that take on only rational
values and not infinite values.
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation (BLP). For a valued constraint language , BLP is a decision
procedure for if and only if admits a symmetric fractional
polymorphism of every arity. For a finite-valued constraint language ,
BLP is a decision procedure if and only if admits a symmetric
fractional polymorphism of some arity, or equivalently, if admits a
symmetric fractional polymorphism of arity 2.
Using these results, we obtain tractability of several novel classes of
problems, including problems over valued constraint languages that are: (1)
submodular on arbitrary lattices; (2) -submodular on arbitrary finite
domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors
(arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213)
to appear in SIAM Journal on Computing (SICOMP
A categorical semantics for causal structure
We present a categorical construction for modelling causal structures within
a general class of process theories that include the theory of classical
probabilistic processes as well as quantum theory. Unlike prior constructions
within categorical quantum mechanics, the objects of this theory encode
fine-grained causal relationships between subsystems and give a new method for
expressing and deriving consequences for a broad class of causal structures. We
show that this framework enables one to define families of processes which are
consistent with arbitrary acyclic causal orderings. In particular, one can
define one-way signalling (a.k.a. semi-causal) processes, non-signalling
processes, and quantum -combs. Furthermore, our framework is general enough
to accommodate recently-proposed generalisations of classical and quantum
theory where processes only need to have a fixed causal ordering locally, but
globally allow indefinite causal ordering.
To illustrate this point, we show that certain processes of this kind, such
as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner,
and a classical three-party example due to Baumeler, Feix, and Wolf are all
instances of a certain family of processes we refer to as in
the appropriate category of higher-order causal processes. After defining these
families of causal structures within our framework, we give derivations of
their operational behaviour using simple, diagrammatic axioms.Comment: Extended version of a LICS 2017 paper with the same titl
Ultimate periodicity of b-recognisable sets : a quasilinear procedure
It is decidable if a set of numbers, whose representation in a base b is a
regular language, is ultimately periodic. This was established by Honkala in
1986.
We give here a structural description of minimal automata that accept an
ultimately periodic set of numbers. We then show that it can verified in linear
time if a given minimal automaton meets this description.
This thus yields a O(n log(n)) procedure for deciding whether a general
deterministic automaton accepts an ultimately periodic set of numbers.Comment: presented at DLT 201
Credal Networks under Epistemic Irrelevance
A credal network under epistemic irrelevance is a generalised type of
Bayesian network that relaxes its two main building blocks. On the one hand,
the local probabilities are allowed to be partially specified. On the other
hand, the assessments of independence do not have to hold exactly.
Conceptually, these two features turn credal networks under epistemic
irrelevance into a powerful alternative to Bayesian networks, offering a more
flexible approach to graph-based multivariate uncertainty modelling. However,
in practice, they have long been perceived as very hard to work with, both
theoretically and computationally.
The aim of this paper is to demonstrate that this perception is no longer
justified. We provide a general introduction to credal networks under epistemic
irrelevance, give an overview of the state of the art, and present several new
theoretical results. Most importantly, we explain how these results can be
combined to allow for the design of recursive inference methods. We provide
numerous concrete examples of how this can be achieved, and use these to
demonstrate that computing with credal networks under epistemic irrelevance is
most definitely feasible, and in some cases even highly efficient. We also
discuss several philosophical aspects, including the lack of symmetry, how to
deal with probability zero, the interpretation of lower expectations, the
axiomatic status of graphoid properties, and the difference between updating
and conditioning
Multi-path Summation for Decoding 2D Topological Codes
Fault tolerance is a prerequisite for scalable quantum computing.
Architectures based on 2D topological codes are effective for near-term
implementations of fault tolerance. To obtain high performance with these
architectures, we require a decoder which can adapt to the wide variety of
error models present in experiments. The typical approach to the problem of
decoding the surface code is to reduce it to minimum-weight perfect matching in
a way that provides a suboptimal threshold error rate, and is specialized to
correct a specific error model. Recently, optimal threshold error rates for a
variety of error models have been obtained by methods which do not use
minimum-weight perfect matching, showing that such thresholds can be achieved
in polynomial time. It is an open question whether these results can also be
achieved by minimum-weight perfect matching. In this work, we use belief
propagation and a novel algorithm for producing edge weights to increase the
utility of minimum-weight perfect matching for decoding surface codes. This
allows us to correct depolarizing errors using the rotated surface code,
obtaining a threshold of . This is larger than the threshold
achieved by previous matching-based decoders (), though
still below the known upper bound of .Comment: 19 pages, 13 figures, published in Quantum, available at
https://quantum-journal.org/papers/q-2018-10-19-102
Reintroducing credal networks under epistemic irrelevance
A credal network under epistemic irrelevance is a generalised version of a Bayesian network that loosens its two main building blocks. On the one hand, the local probabilities do not have to be specified exactly. On the other hand, the assumptions of independence do not have to hold exactly. Conceptually, these credal networks are elegant and useful. However, in practice, they have long remained very hard to work with, both theoretically and computationally. This paper provides a general introduction to this type of credal networks and presents some promising new theoretical developments that were recently proved using sets of desirable gambles and lower previsions. We explain these developments in terms of probabilities and expectations, thereby making them more easily accessible to the Bayesian network community
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