12,805 research outputs found
Linear-Time FPT Algorithms via Network Flow
In the area of parameterized complexity, to cope with NP-Hard problems, we
introduce a parameter k besides the input size n, and we aim to design
algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some
function f(k) and constant d. Though FPT algorithms have been successfully
designed for many problems, typically they are not sufficiently fast because of
huge f(k) and d. In this paper, we give FPT algorithms with small f(k) and d
for many important problems including Odd Cycle Transversal and Almost 2-SAT.
More specifically, we can choose f(k) as a single exponential (4^k) and d as
one, that is, linear in the input size. To the best of our knowledge, our
algorithms achieve linear time complexity for the first time for these
problems. To obtain our algorithms for these problems, we consider a large
class of integer programs, called BIP2. Then we show that, in linear time, we
can reduce BIP2 to Vertex Cover Above LP preserving the parameter k, and we can
compute an optimal LP solution for Vertex Cover Above LP using network flow.
Then, we perform an exhaustive search by fixing half-integral values in the
optimal LP solution for Vertex Cover Above LP. A bottleneck here is that we
need to recompute an LP optimal solution after branching. To address this
issue, we exploit network flow to update the optimal LP solution in linear
time.Comment: 20 page
Fair Assortment Planning
Many online platforms, ranging from online retail stores to social media
platforms, employ algorithms to optimize their offered assortment of items
(e.g., products and contents). These algorithms tend to prioritize the
platforms' short-term goals by solely featuring items with the highest
popularity or revenue. However, this practice can then lead to undesirable
outcomes for the rest of the items, making them leave the platform, and in turn
hurting the platform's long-term goals. Motivated by that, we introduce and
study a fair assortment planning problem, which requires any two items with
similar quality/merits to be offered similar outcomes. We show that the problem
can be formulated as a linear program (LP), called (FAIR), that optimizes over
the distribution of all feasible assortments. To find a near-optimal solution
to (FAIR), we propose a framework based on the Ellipsoid method, which requires
a polynomial-time separation oracle to the dual of the LP. We show that finding
an optimal separation oracle to the dual problem is an NP-complete problem, and
hence we propose a series of approximate separation oracles, which then result
in a -approx. algorithm and a PTAS for the original Problem (FAIR). The
approximate separation oracles are designed by (i) showing the separation
oracle to the dual of the LP is equivalent to solving an infinite series of
parameterized knapsack problems, and (ii) taking advantage of the structure of
the parameterized knapsack problems. Finally, we conduct a case study using the
MovieLens dataset, which demonstrates the efficacy of our algorithms and
further sheds light on the price of fairness.Comment: 86 pages, 7 figure
CURIOUS: Intrinsically Motivated Modular Multi-Goal Reinforcement Learning
In open-ended environments, autonomous learning agents must set their own
goals and build their own curriculum through an intrinsically motivated
exploration. They may consider a large diversity of goals, aiming to discover
what is controllable in their environments, and what is not. Because some goals
might prove easy and some impossible, agents must actively select which goal to
practice at any moment, to maximize their overall mastery on the set of
learnable goals. This paper proposes CURIOUS, an algorithm that leverages 1) a
modular Universal Value Function Approximator with hindsight learning to
achieve a diversity of goals of different kinds within a unique policy and 2)
an automated curriculum learning mechanism that biases the attention of the
agent towards goals maximizing the absolute learning progress. Agents focus
sequentially on goals of increasing complexity, and focus back on goals that
are being forgotten. Experiments conducted in a new modular-goal robotic
environment show the resulting developmental self-organization of a learning
curriculum, and demonstrate properties of robustness to distracting goals,
forgetting and changes in body properties.Comment: Accepted at ICML 201
LP-decodable multipermutation codes
In this paper, we introduce a new way of constructing and decoding
multipermutation codes. Multipermutations are permutations of a multiset that
may consist of duplicate entries. We first introduce a new class of matrices
called multipermutation matrices. We characterize the convex hull of
multipermutation matrices. Based on this characterization, we propose a new
class of codes that we term LP-decodable multipermutation codes. Then, we
derive two LP decoding algorithms. We first formulate an LP decoding problem
for memoryless channels. We then derive an LP algorithm that minimizes the
Chebyshev distance. Finally, we show a numerical example of our algorithm.Comment: This work was supported by NSF and NSERC. To appear at the 2014
Allerton Conferenc
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis
The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods, and model order reduction (MOR) methods have proven to be very effective in combating such high complexity. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the circuit under study as a function of design parameters such as geometrical and substrate features. Traditional MOR techniques perform order reduction only with respect to frequency, and therefore the computation of a new electromagnetic model and the corresponding reduced model are needed each time a design parameter is modified, reducing the CPU efficiency. Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as geometrical layout or substrate characteristics. We propose a novel PMOR technique applicable to PEEC analysis which is based on a parameterization process of matrices generated by the PEEC method and the projection subspace generated by a passivity-preserving MOR method. The proposed PMOR technique guarantees overall stability and passivity of parameterized reduced order models over a user-defined range of design parameter values. Pertinent numerical examples validate the proposed PMOR approach
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