51 research outputs found
Membership in moment polytopes is in NP and coNP
We show that the problem of deciding membership in the moment polytope
associated with a finite-dimensional unitary representation of a compact,
connected Lie group is in NP and coNP. This is the first non-trivial result on
the computational complexity of this problem, which naively amounts to a
quadratically-constrained program. Our result applies in particular to the
Kronecker polytopes, and therefore to the problem of deciding positivity of the
stretched Kronecker coefficients. In contrast, it has recently been shown that
deciding positivity of a single Kronecker coefficient is NP-hard in general
[Ikenmeyer, Mulmuley and Walter, arXiv:1507.02955]. We discuss the consequences
of our work in the context of complexity theory and the quantum marginal
problem.Comment: 20 page
Moment cone membership for quivers in strongly polynomial time
In this note we observe that membership in moment cones of spaces of quiver
representations can be decided in strongly polynomial time, for any acyclic
quiver. This generalizes a recent result by Chindris-Collins-Kline for
bipartite quivers. Their approach was to construct "multiplicity polytopes"
with a geometric realization similar to the Knutson-Tao polytopes for tensor
product multiplicities. Here we show that a less geometric but straightforward
variant of their construction leads to such a multiplicity polytope for any
acyclic quiver. Tardos' strongly polynomial time algorithm for combinatorial
linear programming along with the saturation property then implies that moment
cone membership can be decided in strongly polynomial time. The analogous
question for semi-invariants remains open.Comment: 7 page
On vanishing of Kronecker coefficients
We show that the problem of deciding positivity of Kronecker coefficients is
NP-hard. Previously, this problem was conjectured to be in P, just as for the
Littlewood-Richardson coefficients. Our result establishes in a formal way that
Kronecker coefficients are more difficult than Littlewood-Richardson
coefficients, unless P=NP.
We also show that there exists a #P-formula for a particular subclass of
Kronecker coefficients whose positivity is NP-hard to decide. This is an
evidence that, despite the hardness of the positivity problem, there may well
exist a positive combinatorial formula for the Kronecker coefficients. Finding
such a formula is a major open problem in representation theory and algebraic
combinatorics.
Finally, we consider the existence of the partition triples such that the Kronecker coefficient but the
Kronecker coefficient for some integer
. Such "holes" are of great interest as they witness the failure of the
saturation property for the Kronecker coefficients, which is still poorly
understood. Using insight from computational complexity theory, we turn our
hardness proof into a positive result: We show that not only do there exist
many such triples, but they can also be found efficiently. Specifically, we
show that, for any , there exists such that, for all
, there exist partition triples in the
Kronecker cone such that: (a) the Kronecker coefficient
is zero, (b) the height of is , (c) the height of is , and (d) . The proof of the last result
illustrates the effectiveness of the explicit proof strategy of GCT.Comment: 43 pages, 1 figur
Operator scaling with specified marginals
The completely positive maps, a generalization of the nonnegative matrices,
are a well-studied class of maps from matrices to
matrices. The existence of the operator analogues of doubly stochastic scalings
of matrices is equivalent to a multitude of problems in computer science and
mathematics, such rational identity testing in non-commuting variables,
noncommutative rank of symbolic matrices, and a basic problem in invariant
theory (Garg, Gurvits, Oliveira and Wigderson, FOCS, 2016).
We study operator scaling with specified marginals, which is the operator
analogue of scaling matrices to specified row and column sums. We characterize
the operators which can be scaled to given marginals, much in the spirit of the
Gurvits' algorithmic characterization of the operators that can be scaled to
doubly stochastic (Gurvits, Journal of Computer and System Sciences, 2004). Our
algorithm produces approximate scalings in time poly(n,m) whenever scalings
exist. A central ingredient in our analysis is a reduction from the specified
marginals setting to the doubly stochastic setting.
Operator scaling with specified marginals arises in diverse areas of study
such as the Brascamp-Lieb inequalities, communication complexity, eigenvalues
of sums of Hermitian matrices, and quantum information theory. Some of the
known theorems in these areas, several of which had no effective proof, are
straightforward consequences of our characterization theorem. For instance, we
obtain a simple algorithm to find, when they exist, a tuple of Hermitian
matrices with given spectra whose sum has a given spectrum. We also prove new
theorems such as a generalization of Forster's theorem (Forster, Journal of
Computer and System Sciences, 2002) concerning radial isotropic position.Comment: 34 pages, 3 page appendi
On Vanishing of {K}ronecker Coefficients
It is shown that: (1) The problem of deciding positivity of Kronecker coefficients is NP-hard. (2) There exists a positive ()-formula for a subclass of Kronecker coefficients whose positivity is NP-hard to decide. (3) For any , there exists such that, for all , there exist partition triples in the Kronecker cone such that: (a) the Kronecker coefficient is zero, (b) the height of is , (c) the height of is , and (d) . The last result takes a step towards proving the existence of occurrence-based representation-theoretic obstructions in the context of the GCT approach to the permanent vs. determinant problem. Its proof also illustrates the effectiveness of the explicit proof strategy of GCT
Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
We present a polynomial time algorithm to approximately scale tensors of any
format to arbitrary prescribed marginals (whenever possible). This unifies and
generalizes a sequence of past works on matrix, operator and tensor scaling.
Our algorithm provides an efficient weak membership oracle for the associated
moment polytopes, an important family of implicitly-defined convex polytopes
with exponentially many facets and a wide range of applications. These include
the entanglement polytopes from quantum information theory (in particular, we
obtain an efficient solution to the notorious one-body quantum marginal
problem) and the Kronecker polytopes from representation theory (which capture
the asymptotic support of Kronecker coefficients). Our algorithm can be applied
to succinct descriptions of the input tensor whenever the marginals can be
efficiently computed, as in the important case of matrix product states or
tensor-train decompositions, widely used in computational physics and numerical
mathematics.
We strengthen and generalize the alternating minimization approach of
previous papers by introducing the theory of highest weight vectors from
representation theory into the numerical optimization framework. We show that
highest weight vectors are natural potential functions for scaling algorithms
and prove new bounds on their evaluations to obtain polynomial-time
convergence. Our techniques are general and we believe that they will be
instrumental to obtain efficient algorithms for moment polytopes beyond the
ones consider here, and more broadly, for other optimization problems
possessing natural symmetries
Membership in moment cones and quiver semi-invariants for bipartite quivers
Let be a bipartite quiver with vertex set such that the number of
arrows between any two source and sink vertices is constant. Let
be a dimension vector of with positive
integer coordinates, and let be the moment cone associated
to . We show that the membership problem for can
be solved in strongly polynomial time.
As a key step in our approach, we first solve the polytopal problem for
semi-invariants of and its flag-extensions. Specifically, let
be the flag-extension of obtained by attaching a flag of
length at every vertex of , and let be
the extension of to that takes values
along the vertices of the flag for every vertex of .
For an integral weight of , let
be the dimension of the space of semi-invariants of
weight on the representation space of
-dimensional complex representations of .
We show that can be expressed as the number of
lattice points of a certain hive-type polytope. This polytopal description
together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants
allows us to use Tardos's algorithm to solve the membership problem for
in strongly polynomial time.Comment: v2: Fixed the claim about the generic quiver semi-stability problem
(see Remarks 2.8 and 5.5
Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices
We address the computation of the degrees of minors of a noncommutative
symbolic matrix of form where are matrices over a
field , are noncommutative variables, are integer
weights, and is a commuting variable specifying the degree. This problem
extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can
formulate various combinatorial optimization problems. Extending the study by
Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and
polyhedral characterization for the maximum degrees of minors of of all
sizes, and develop a strongly polynomial-time algorithm for computing them.
This algorithm is viewed as a unified algebraization of the classical Hungarian
method for bipartite matching and the weight-splitting algorithm for linear
matroid intersection. As applications, we provide polynomial-time algorithms
for weighted fractional linear matroid matching and linear optimization over
rank-2 Brascamp-Lieb polytopes
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