305,321 research outputs found
The family constrained network problem
AbstractAn extension of the classical fixed charge transportation problem is developed that allows a wide variety of practical production, distribution, and inventory planning models to be addressed. Computational results are presented for problems with up to one thousand network constraints and five thousand network variables
A Laplacian Approach to -Norm Minimization
We propose a novel differentiable reformulation of the linearly-constrained
minimization problem, also known as the basis pursuit problem. The
reformulation is inspired by the Laplacian paradigm of network theory and leads
to a new family of gradient-based methods for the solution of
minimization problems. We analyze the iteration complexity of a natural
solution approach to the reformulation, based on a multiplicative weights
update scheme, as well as the iteration complexity of an accelerated gradient
scheme. The results can be seen as bounds on the complexity of iteratively
reweighted least squares (IRLS) type methods of basis pursuit
Matroidal Degree-Bounded Minimum Spanning Trees
We consider the minimum spanning tree (MST) problem under the restriction
that for every vertex v, the edges of the tree that are adjacent to v satisfy a
given family of constraints. A famous example thereof is the classical
degree-constrained MST problem, where for every vertex v, a simple upper bound
on the degree is imposed. Iterative rounding/relaxation algorithms became the
tool of choice for degree-bounded network design problems. A cornerstone for
this development was the work of Singh and Lau, who showed for the
degree-bounded MST problem how to find a spanning tree violating each degree
bound by at most one unit and with cost at most the cost of an optimal solution
that respects the degree bounds.
However, current iterative rounding approaches face several limits when
dealing with more general degree constraints. In particular, when several
constraints are imposed on the edges adjacent to a vertex v, as for example
when a partition of the edges adjacent to v is given and only a fixed number of
elements can be chosen out of each set of the partition, current approaches
might violate each of the constraints by a constant, instead of violating all
constraints together by at most a constant number of edges. Furthermore, it is
also not clear how previous iterative rounding approaches can be used for
degree constraints where some edges are in a super-constant number of
constraints.
We extend iterative rounding/relaxation approaches both on a conceptual level
as well as aspects involving their analysis to address these limitations. This
leads to an efficient algorithm for the degree-constrained MST problem where
for every vertex v, the edges adjacent to v have to be independent in a given
matroid. The algorithm returns a spanning tree T of cost at most OPT, such that
for every vertex v, it suffices to remove at most 8 edges from T to satisfy the
matroidal degree constraint at v
Distributionally Robust Statistical Verification with Imprecise Neural Networks
A particularly challenging problem in AI safety is providing guarantees on
the behavior of high-dimensional autonomous systems. Verification approaches
centered around reachability analysis fail to scale, and purely statistical
approaches are constrained by the distributional assumptions about the sampling
process. Instead, we pose a distributionally robust version of the statistical
verification problem for black-box systems, where our performance guarantees
hold over a large family of distributions. This paper proposes a novel approach
based on a combination of active learning, uncertainty quantification, and
neural network verification. A central piece of our approach is an ensemble
technique called Imprecise Neural Networks, which provides the uncertainty to
guide active learning. The active learning uses an exhaustive neural-network
verification tool Sherlock to collect samples. An evaluation on multiple
physical simulators in the openAI gym Mujoco environments with
reinforcement-learned controllers demonstrates that our approach can provide
useful and scalable guarantees for high-dimensional systems
Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable
Graph editing problems have a long history and have been widely studied, with applications in biochemistry and complex network analysis. They generally ask whether an input graph can be modified by inserting and deleting vertices and edges to a graph with the desired property. We consider the problem \textsc{Graph-Edit-to-NDL} (GEN) where the goal is to modify to a graph with a given neighbourhood degree list (NDL). The NDL lists the degrees of the neighbours of vertices in a graph, and is a stronger invariant than the degree sequence, which lists the degrees of vertices.
We show \textsc{Graph-Edit-to-NDL} is NP-complete and study its parameterized complexity. In parameterized complexity, a problem is said to be fixed-parameter tractable with respect to a parameter if it has a solution whose running time is a function that is polynomial in the input size but possibly superpolynomial in the parameter.
Golovach and Mertzios [ICSSR, 2016] studied editing to a graph with a given degree sequence and showed the problem is fixed-parameter tractable when parameterized by , where is the maximum degree of the input graph and is the number of edits. We prove \textsc{Graph-Edit-to-NDL} is fixed-parameter tractable when parameterized by .
Furthermore, we consider a harder problem \textsc{Constrained-Graph-Edit-to-NDL} (CGEN) that imposes constraints on the NDLs of intermediate graphs produced in the sequence. We adapt our FPT algorithm for \textsc{Graph-Edit-to-NDL} to solve \textsc{Constrained-Graph-Edit-to-NDL}, which proves \textsc{Constrained-Graph-Edit-to-NDL} is also fixed-parameter tractable when parameterized by .
Our results imply that, for graph properties that can be expressed as properties of NDLs, editing to a graph with such a property is fixed-parameter tractable when parameterized by . We show that this family of graph properties includes some well-known graph measures used in complex network analysis
Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable
Graph editing problems have a long history and have been widely studied, with applications in biochemistry and complex network analysis. They generally ask whether an input graph can be modified by inserting and deleting vertices and edges to a graph with the desired property. We consider the problem \textsc{Graph-Edit-to-NDL} (GEN) where the goal is to modify to a graph with a given neighbourhood degree list (NDL). The NDL lists the degrees of the neighbours of vertices in a graph, and is a stronger invariant than the degree sequence, which lists the degrees of vertices.
We show \textsc{Graph-Edit-to-NDL} is NP-complete and study its parameterized complexity. In parameterized complexity, a problem is said to be fixed-parameter tractable with respect to a parameter if it has a solution whose running time is a function that is polynomial in the input size but possibly superpolynomial in the parameter.
Golovach and Mertzios [ICSSR, 2016] studied editing to a graph with a given degree sequence and showed the problem is fixed-parameter tractable when parameterized by , where is the maximum degree of the input graph and is the number of edits. We prove \textsc{Graph-Edit-to-NDL} is fixed-parameter tractable when parameterized by .
Furthermore, we consider a harder problem \textsc{Constrained-Graph-Edit-to-NDL} (CGEN) that imposes constraints on the NDLs of intermediate graphs produced in the sequence. We adapt our FPT algorithm for \textsc{Graph-Edit-to-NDL} to solve \textsc{Constrained-Graph-Edit-to-NDL}, which proves \textsc{Constrained-Graph-Edit-to-NDL} is also fixed-parameter tractable when parameterized by .
Our results imply that, for graph properties that can be expressed as properties of NDLs, editing to a graph with such a property is fixed-parameter tractable when parameterized by . We show that this family of graph properties includes some well-known graph measures used in complex network analysis
Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction
We consider a family of open star-shaped domains made of a finite number of non intersecting semi-infinite strips of small thickness and of a central region whose diameter is of the same order of thickness, that may be called the junction. When the thickness tends to 0, the domains tend to a union of half-lines sharing an endpoint. This set is termed "network". We study infinite horizon optimal control problems in which the state is constrained to remain in the star-shaped domains. In the above mentioned strips the running cost may have a fast variation w.r.t. the transverse coordinate. When the thickness tends to 0 we prove that the value function tends to the solution of a Hamilton-Jacobi equation on the network, which may also be related to an optimal control problem. One difficulty is to find the transmission condition at the junction node in the limit problem. For passing to the limit, we use the method of the perturbed test-functions of Evans, which requires constructing suitable correctors. This is another difficulty since the domain is unbounded
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