3,500 research outputs found
Theory and Applications of Robust Optimization
In this paper we survey the primary research, both theoretical and applied,
in the area of Robust Optimization (RO). Our focus is on the computational
attractiveness of RO approaches, as well as the modeling power and broad
applicability of the methodology. In addition to surveying prominent
theoretical results of RO, we also present some recent results linking RO to
adaptable models for multi-stage decision-making problems. Finally, we
highlight applications of RO across a wide spectrum of domains, including
finance, statistics, learning, and various areas of engineering.Comment: 50 page
Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning under structural
restrictions. All these problems involve two tasks: (i) identifying the
structure in the input as required by the restriction, and (ii) using the
identified structure to solve the reasoning task efficiently. We show that for
most of the considered problems, task (i) admits a polynomial-time
preprocessing to a problem kernel whose size is polynomial in a structural
problem parameter of the input, in contrast to task (ii) which does not admit
such a reduction to a problem kernel of polynomial size, subject to a
complexity theoretic assumption. As a notable exception we show that the
consistency problem for the AtMost-NValue constraint admits a polynomial kernel
consisting of a quadratic number of variables and domain values. Our results
provide a firm worst-case guarantees and theoretical boundaries for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541,
arXiv:1104.556
Detecting and counting small subgraphs, and evaluating a parameterized Tutte polynomial: lower bounds via toroidal grids and Cayley graph expanders
Given a graph property , we consider the problem , where the input is a pair of a graph and a positive integer , and the task is to decide whether contains a -edge subgraph that satisfies . Specifically, we study the parameterized complexity of and of its counting problem with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties : the decision problem always admits an FPT algorithm and the counting problem always admits an FPTRAS. For exact counting, we present an exhaustive and explicit criterion on the property which, if satisfied, yields fixed-parameter tractability and otherwise -hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for that run in time for any computable function . As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of . This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). Our methods can also be applied to a parameterized variant of the Tutte Polynomial of a graph , to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, corresponds to the number of -forests in the graph . Our techniques allow us to completely understand the parametrized complexity of computing the evaluation of at every pair of rational coordinates
Pure Parsimony Xor Haplotyping
The haplotype resolution from xor-genotype data has been recently formulated
as a new model for genetic studies. The xor-genotype data is a cheaply
obtainable type of data distinguishing heterozygous from homozygous sites
without identifying the homozygous alleles. In this paper we propose a
formulation based on a well-known model used in haplotype inference: pure
parsimony. We exhibit exact solutions of the problem by providing polynomial
time algorithms for some restricted cases and a fixed-parameter algorithm for
the general case. These results are based on some interesting combinatorial
properties of a graph representation of the solutions. Furthermore, we show
that the problem has a polynomial time k-approximation, where k is the maximum
number of xor-genotypes containing a given SNP. Finally, we propose a heuristic
and produce an experimental analysis showing that it scales to real-world large
instances taken from the HapMap project
Power and Channel Allocation for Non-orthogonal Multiple Access in 5G Systems: Tractability and Computation
Network capacity calls for significant increase for 5G cellular systems. A
promising multi-user access scheme, non-orthogonal multiple access (NOMA) with
successive interference cancellation (SIC), is currently under consideration.
In NOMA, spectrum efficiency is improved by allowing more than one user to
simultaneously access the same frequency-time resource and separating
multi-user signals by SIC at the receiver. These render resource allocation and
optimization in NOMA different from orthogonal multiple access in 4G. In this
paper, we provide theoretical insights and algorithmic solutions to jointly
optimize power and channel allocation in NOMA. For utility maximization, we
mathematically formulate NOMA resource allocation problems. We characterize and
analyze the problems' tractability under a range of constraints and utility
functions. For tractable cases, we provide polynomial-time solutions for global
optimality. For intractable cases, we prove the NP-hardness and propose an
algorithmic framework combining Lagrangian duality and dynamic programming
(LDDP) to deliver near-optimal solutions. To gauge the performance of the
obtained solutions, we also provide optimality bounds on the global optimum.
Numerical results demonstrate that the proposed algorithmic solution can
significantly improve the system performance in both throughput and fairness
over orthogonal multiple access as well as over a previous NOMA resource
allocation scheme.Comment: IEEE Transactions on Wireless Communications, revisio
Data-driven Inverse Optimization with Imperfect Information
In data-driven inverse optimization an observer aims to learn the preferences
of an agent who solves a parametric optimization problem depending on an
exogenous signal. Thus, the observer seeks the agent's objective function that
best explains a historical sequence of signals and corresponding optimal
actions. We focus here on situations where the observer has imperfect
information, that is, where the agent's true objective function is not
contained in the search space of candidate objectives, where the agent suffers
from bounded rationality or implementation errors, or where the observed
signal-response pairs are corrupted by measurement noise. We formalize this
inverse optimization problem as a distributionally robust program minimizing
the worst-case risk that the {\em predicted} decision ({\em i.e.}, the decision
implied by a particular candidate objective) differs from the agent's {\em
actual} response to a random signal. We show that our framework offers rigorous
out-of-sample guarantees for different loss functions used to measure
prediction errors and that the emerging inverse optimization problems can be
exactly reformulated as (or safely approximated by) tractable convex programs
when a new suboptimality loss function is used. We show through extensive
numerical tests that the proposed distributionally robust approach to inverse
optimization attains often better out-of-sample performance than the
state-of-the-art approaches
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