1,359 research outputs found
On the complexity of computing Kronecker coefficients
We study the complexity of computing Kronecker coefficients
. We give explicit bounds in terms of the number of parts
in the partitions, their largest part size and the smallest second
part of the three partitions. When , i.e. one of the partitions
is hook-like, the bounds are linear in , but depend exponentially on
. Moreover, similar bounds hold even when . By a separate
argument, we show that the positivity of Kronecker coefficients can be decided
in time for a bounded number of parts and without
restriction on . Related problems of computing Kronecker coefficients when
one partition is a hook, and computing characters of are also considered.Comment: v3: incorporated referee's comments; accepted to Computational
Complexit
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
For each , we count diagonally and antidiagonally
symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal
number of 's along the diagonal and the antidiagonal, as well as
DASASMs of fixed odd order with a minimal number of 's along the diagonal
and the antidiagonal. In these enumerations, we encounter product formulas that
have previously appeared in plane partition or alternating sign matrix
counting, namely for the number of all alternating sign matrices, the number of
cyclically symmetric plane partitions in a given box, and the number of
vertically and horizontally symmetric ASMs. We also prove several refinements.
For instance, in the case of DASASMs with a maximal number of 's along the
diagonal and the antidiagonal, these considerations lead naturally to the
definition of alternating sign triangles. These are new objects that are
equinumerous with ASMs, and we are able to prove a two parameter refinement of
this fact, involving the number of 's and the inversion number on the ASM
side. To prove our results, we extend techniques to deal with triangular
six-vertex configurations that have recently successfully been applied to
settle Robbins' conjecture on the number of all DASASMs of odd order.
Importantly, we use a general solution of the reflection equation to prove the
symmetry of the partition function in the spectral parameters. In all of our
cases, we derive determinant or Pfaffian formulas for the partition functions,
which we then specialize in order to obtain the product formulas for the
various classes of extreme odd DASASMs under consideration.Comment: 41 pages, several minor improvements in response to referee's
comments. Final version. Matches published version except for very minor
change
On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect
We investigate the problem of computing tensor product multiplicities for
complex semisimple Lie algebras. Even though computing these numbers is #P-hard
in general, we show that if the rank of the Lie algebra is assumed fixed, then
there is a polynomial time algorithm, based on counting the lattice points in
polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based
on the ellipsoid algorithm, to decide when the coefficients are nonzero in
polynomial time for arbitrary rank. Our experiments show that the lattice point
algorithm is superior in practice to the standard techniques for computing
multiplicities when the weights have large entries but small rank. Using an
implementation of this algorithm, we provide experimental evidence for
conjectured generalizations of the saturation property of
Littlewood--Richardson coefficients. One of these conjectures seems to be valid
for types B_n, C_n, and D_n.Comment: 21 pages, 6 table
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