3,877 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
Studies in Efficient Discrete Algorithms
This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity
From Uncertainty Data to Robust Policies for Temporal Logic Planning
We consider the problem of synthesizing robust disturbance feedback policies
for systems performing complex tasks. We formulate the tasks as linear temporal
logic specifications and encode them into an optimization framework via
mixed-integer constraints. Both the system dynamics and the specifications are
known but affected by uncertainty. The distribution of the uncertainty is
unknown, however realizations can be obtained. We introduce a data-driven
approach where the constraints are fulfilled for a set of realizations and
provide probabilistic generalization guarantees as a function of the number of
considered realizations. We use separate chance constraints for the
satisfaction of the specification and operational constraints. This allows us
to quantify their violation probabilities independently. We compute disturbance
feedback policies as solutions of mixed-integer linear or quadratic
optimization problems. By using feedback we can exploit information of past
realizations and provide feasibility for a wider range of situations compared
to static input sequences. We demonstrate the proposed method on two robust
motion-planning case studies for autonomous driving
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