737 research outputs found
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
Structured Decompositions: Structural and Algorithmic Compositionality
We introduce structured decompositions: category-theoretic generalizations of
many combinatorial invariants -- including tree-width, layered tree-width,
co-tree-width and graph decomposition width -- which have played a central role
in the study of structural and algorithmic compositionality in both graph
theory and parameterized complexity. Structured decompositions allow us to
generalize combinatorial invariants to new settings (for example decompositions
of matroids) in which they describe algorithmically useful structural
compositionality. As an application of our theory we prove an algorithmic meta
theorem for the Sub_P-composition problem which, when instantiated in the
category of graphs, yields compositional algorithms for NP-hard problems such
as: Maximum Bipartite Subgraph, Maximum Planar Subgraph and Longest Path
Learning Differentiable Programs with Admissible Neural Heuristics
We study the problem of learning differentiable functions expressed as programs in a domain-specific language. Such programmatic models can offer benefits such as composability and interpretability; however, learning them requires optimizing over a combinatorial space of program "architectures". We frame this optimization problem as a search in a weighted graph whose paths encode top-down derivations of program syntax. Our key innovation is to view various classes of neural networks as continuous relaxations over the space of programs, which can then be used to complete any partial program. This relaxed program is differentiable and can be trained end-to-end, and the resulting training loss is an approximately admissible heuristic that can guide the combinatorial search. We instantiate our approach on top of the A-star algorithm and an iteratively deepened branch-and-bound search, and use these algorithms to learn programmatic classifiers in three sequence classification tasks. Our experiments show that the algorithms outperform state-of-the-art methods for program learning, and that they discover programmatic classifiers that yield natural interpretations and achieve competitive accuracy
Learning Differentiable Programs with Admissible Neural Heuristics
We study the problem of learning differentiable functions expressed as
programs in a domain-specific language. Such programmatic models can offer
benefits such as composability and interpretability; however, learning them
requires optimizing over a combinatorial space of program "architectures". We
frame this optimization problem as a search in a weighted graph whose paths
encode top-down derivations of program syntax. Our key innovation is to view
various classes of neural networks as continuous relaxations over the space of
programs, which can then be used to complete any partial program. This relaxed
program is differentiable and can be trained end-to-end, and the resulting
training loss is an approximately admissible heuristic that can guide the
combinatorial search. We instantiate our approach on top of the A-star
algorithm and an iteratively deepened branch-and-bound search, and use these
algorithms to learn programmatic classifiers in three sequence classification
tasks. Our experiments show that the algorithms outperform state-of-the-art
methods for program learning, and that they discover programmatic classifiers
that yield natural interpretations and achieve competitive accuracy.Comment: 9 pages, published in NeurIPS 202
Uniform and Bernoulli measures on the boundary of trace monoids
Trace monoids and heaps of pieces appear in various contexts in
combinatorics. They also constitute a model used in computer science to
describe the executions of asynchronous systems. The design of a natural
probabilistic layer on top of the model has been a long standing challenge. The
difficulty comes from the presence of commuting pieces and from the absence of
a global clock. In this paper, we introduce and study the class of Bernoulli
probability measures that we claim to be the simplest adequate probability
measures on infinite traces. For this, we strongly rely on the theory of trace
combinatorics with the M\"obius polynomial in the key role. These new measures
provide a theoretical foundation for the probabilistic study of concurrent
systems.Comment: 34 pages, 5 figures, 27 reference
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
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