2,046 research outputs found
Submodular Minimization Under Congruency Constraints
Submodular function minimization (SFM) is a fundamental and efficiently
solvable problem class in combinatorial optimization with a multitude of
applications in various fields. Surprisingly, there is only very little known
about constraint types under which SFM remains efficiently solvable. The
arguably most relevant non-trivial constraint class for which polynomial SFM
algorithms are known are parity constraints, i.e., optimizing only over sets of
odd (or even) cardinality. Parity constraints capture classical combinatorial
optimization problems like the odd-cut problem, and they are a key tool in a
recent technique to efficiently solve integer programs with a constraint matrix
whose subdeterminants are bounded by two in absolute value.
We show that efficient SFM is possible even for a significantly larger class
than parity constraints, by introducing a new approach that combines techniques
from Combinatorial Optimization, Combinatorics, and Number Theory. In
particular, we can show that efficient SFM is possible over all sets (of any
given lattice) of cardinality r mod m, as long as m is a constant prime power.
This covers generalizations of the odd-cut problem with open complexity status,
and with relevance in the context of integer programming with higher
subdeterminants. To obtain our results, we establish a connection between the
correctness of a natural algorithm, and the inexistence of set systems with
specific combinatorial properties. We introduce a general technique to disprove
the existence of such set systems, which allows for obtaining extensions of our
results beyond the above-mentioned setting. These extensions settle two open
questions raised by Geelen and Kapadia [Combinatorica, 2017] in the context of
computing the girth and cogirth of certain types of binary matroids
Query-Efficient Locally Decodable Codes of Subexponential Length
We develop the algebraic theory behind the constructions of Yekhanin (2008)
and Efremenko (2009), in an attempt to understand the ``algebraic niceness''
phenomenon in . We show that every integer ,
where , and are prime, possesses the same good algebraic property as
that allows savings in query complexity. We identify 50 numbers of this
form by computer search, which together with 511, are then applied to gain
improvements on query complexity via Itoh and Suzuki's composition method. More
precisely, we construct a -query LDC for every positive
integer and a -query
LDC for every integer , both of length , improving the
queries used by Efremenko (2009) and queries used by Itoh and
Suzuki (2010).
We also obtain new efficient private information retrieval (PIR) schemes from
the new query-efficient LDCs.Comment: to appear in Computational Complexit
A Class of -Invariant Topological Phases of Interacting Electrons
We describe a class of parity- and time-reversal-invariant topological states
of matter which can arise in correlated electron systems in 2+1-dimensions.
These states are characterized by particle-like excitations exhibiting exotic
braiding statistics. and invariance are maintained by a `doubling' of
the low-energy degrees of freedom which occurs naturally without doubling the
underlying microscopic degrees of freedom. The simplest examples have been the
subject of considerable interest as proposed mechanisms for high-
superconductivity. One is the `doubled' version (i.e. two opposite-chirality
copies) of the U(1) chiral spin liquid. The second example corresponds to
gauge theory, which describes a scenario for spin-charge separation. Our main
concern, with an eye towards applications to quantum computation, are richer
models which support non-Abelian statistics. All of these models, richer or
poorer, lie in a tightly-organized discrete family. The physical inference is
that a material manifesting the gauge theory or a doubled chiral spin
liquid might be easily altered to one capable of universal quantum computation.
These phases of matter have a field-theoretic description in terms of gauge
theories which, in their infrared limits, are topological field theories. We
motivate these gauge theories using a parton model or slave-fermion
construction and show how they can be solved exactly. The structure of the
resulting Hilbert spaces can be understood in purely combinatorial terms. The
highly-constrained nature of this combinatorial construction, phrased in the
language of the topology of curves on surfaces, lays the groundwork for a
strategy for constructing microscopic lattice models which give rise to these
phases.Comment: Typos fixed, references adde
Unconventional Fusion and Braiding of Topological Defects in a Lattice Model
We demonstrate the semiclassical nature of symmetry twist defects that differ
from quantum deconfined anyons in a true topological phase by examining
non-abelian crystalline defects in an abelian lattice model. An underlying
non-dynamical ungauged S3-symmetry labels the quasi-extensive defects by group
elements and gives rise to order dependent fusion. A central subgroup of local
Wilson observables distinguishes defect-anyon composites by species, which can
mutate through abelian anyon tunneling by tuning local defect phase parameters.
We compute a complete consistent set of primitive basis transformations, or
F-symbols, and study braiding and exchange between commuting defects. This
suggests a modified spin-statistics theorem for defects and non-modular group
structures unitarily represented by the braiding S and exchange T matrices.
Non-abelian braiding operations in a closed system represent the sphere braid
group projectively by a non-trivial central extension that relates the
underlying symmetry.Comment: 44 pages, 43 figure
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