2,978 research outputs found
The complexity of Boolean surjective general-valued CSPs
Valued constraint satisfaction problems (VCSPs) are discrete optimisation
problems with a -valued objective function given as
a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on
labels from and an optimal assignment is required to use both
labels from . Examples include the classical global Min-Cut problem in
graphs and the Minimum Distance problem studied in coding theory.
We establish a dichotomy theorem and thus give a complete complexity
classification of Boolean surjective VCSPs with respect to exact solvability.
Our work generalises the dichotomy for -valued constraint
languages (corresponding to surjective decision CSPs) obtained by Creignou and
H\'ebrard. For the maximisation problem of -valued
surjective VCSPs, we also establish a dichotomy theorem with respect to
approximability.
Unlike in the case of Boolean surjective (decision) CSPs, there appears a
novel tractable class of languages that is trivial in the non-surjective
setting. This newly discovered tractable class has an interesting mathematical
structure related to downsets and upsets. Our main contribution is identifying
this class and proving that it lies on the borderline of tractability. A
crucial part of our proof is a polynomial-time algorithm for enumerating all
near-optimal solutions to a generalised Min-Cut problem, which might be of
independent interest.Comment: v5: small corrections and improved presentatio
Spectrally approximating large graphs with smaller graphs
How does coarsening affect the spectrum of a general graph? We provide
conditions such that the principal eigenvalues and eigenspaces of a coarsened
and original graph Laplacian matrices are close. The achieved approximation is
shown to depend on standard graph-theoretic properties, such as the degree and
eigenvalue distributions, as well as on the ratio between the coarsened and
actual graph sizes. Our results carry implications for learning methods that
utilize coarsening. For the particular case of spectral clustering, they imply
that coarse eigenvectors can be used to derive good quality assignments even
without refinement---this phenomenon was previously observed, but lacked formal
justification.Comment: 22 pages, 10 figure
Minimizing optimal transport for functions with fixed-size nodal sets
Consider the class of zero-mean functions with fixed and
norms and exactly nodal points. Which functions minimize
, the Wasserstein distance between the measures whose densities
are the positive and negative parts? We provide a complete solution to this
minimization problem on the line and the circle, which provides sharp constants
for previously proven ``uncertainty principle''-type inequalities, i.e., lower
bounds on . We further show that, while such
inequalities hold in many metric measure spaces, they are no longer sharp when
the non-branching assumption is violated; indeed, for metric star-graphs, the
optimal lower bound on is not inversely proportional to the size
of the nodal set, . Based on similar reductions, we make connections between
the analogous problem of minimizing for defined on
with an equivalent optimal domain partition
problem
The intrinsic dynamics of optimal transport
The question of which costs admit unique optimizers in the Monge-Kantorovich
problem of optimal transportation between arbitrary probability densities is
investigated. For smooth costs and densities on compact manifolds, the only
known examples for which the optimal solution is always unique require at least
one of the two underlying spaces to be homeomorphic to a sphere. We introduce a
(multivalued) dynamics which the transportation cost induces between the target
and source space, for which the presence or absence of a sufficiently large set
of periodic trajectories plays a role in determining whether or not optimal
transport is necessarily unique. This insight allows us to construct smooth
costs on a pair of compact manifolds with arbitrary topology, so that the
optimal transportation between any pair of probility densities is unique.Comment: 33 pages, 4 figure
A semidefinite programming hierarchy for packing problems in discrete geometry
Packing problems in discrete geometry can be modeled as finding independent
sets in infinite graphs where one is interested in independent sets which are
as large as possible. For finite graphs one popular way to compute upper bounds
for the maximal size of an independent set is to use Lasserre's semidefinite
programming hierarchy. We generalize this approach to infinite graphs. For this
we introduce topological packing graphs as an abstraction for infinite graphs
coming from packing problems in discrete geometry. We show that our hierarchy
converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in
Mathematical Programming Series B special issue on polynomial optimizatio
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