79 research outputs found
A Linear-time Algorithm for Sparsification of Unweighted Graphs
Given an undirected graph and an error parameter , the {\em
graph sparsification} problem requires sampling edges in and giving the
sampled edges appropriate weights to obtain a sparse graph with
the following property: the weight of every cut in is within a
factor of of the weight of the corresponding cut in . If
is unweighted, an -time algorithm for constructing
with edges in expectation, and an
-time algorithm for constructing with edges in expectation have recently been developed
(Hariharan-Panigrahi, 2010). In this paper, we improve these results by giving
an -time algorithm for constructing with edges in expectation, for unweighted graphs. Our algorithm is
optimal in terms of its time complexity; further, no efficient algorithm is
known for constructing a sparser . Our algorithm is Monte-Carlo,
i.e. it produces the correct output with high probability, as are all efficient
graph sparsification algorithms
Online Service with Delay
In this paper, we introduce the online service with delay problem. In this
problem, there are points in a metric space that issue service requests
over time, and a server that serves these requests. The goal is to minimize the
sum of distance traveled by the server and the total delay in serving the
requests. This problem models the fundamental tradeoff between batching
requests to improve locality and reducing delay to improve response time, that
has many applications in operations management, operating systems, logistics,
supply chain management, and scheduling.
Our main result is to show a poly-logarithmic competitive ratio for the
online service with delay problem. This result is obtained by an algorithm that
we call the preemptive service algorithm. The salient feature of this algorithm
is a process called preemptive service, which uses a novel combination of
(recursive) time forwarding and spatial exploration on a metric space. We hope
this technique will be useful for related problems such as reordering buffer
management, online TSP, vehicle routing, etc. We also generalize our results to
servers.Comment: 30 pages, 11 figures, Appeared in 49th ACM Symposium on Theory of
Computing (STOC), 201
A General Framework for Learning-Augmented Online Allocation
Online allocation is a broad class of problems where items arriving online
have to be allocated to agents who have a fixed utility/cost for each assigned
item so to maximize/minimize some objective. This framework captures a broad
range of fundamental problems such as the Santa Claus problem (maximizing
minimum utility), Nash welfare maximization (maximizing geometric mean of
utilities), makespan minimization (minimizing maximum cost), minimization of
-norms, and so on. We focus on divisible items (i.e., fractional
allocations) in this paper. Even for divisible items, these problems are
characterized by strong super-constant lower bounds in the classical worst-case
online model.
In this paper, we study online allocations in the {\em learning-augmented}
setting, i.e., where the algorithm has access to some additional
(machine-learned) information about the problem instance. We introduce a {\em
general} algorithmic framework for learning-augmented online allocation that
produces nearly optimal solutions for this broad range of maximization and
minimization objectives using only a single learned parameter for every agent.
As corollaries of our general framework, we improve prior results of Lattanzi
et al. (SODA 2020) and Li and Xian (ICML 2021) for learning-augmented makespan
minimization, and obtain the first learning-augmented nearly-optimal algorithms
for the other objectives such as Santa Claus, Nash welfare,
-minimization, etc. We also give tight bounds on the resilience of our
algorithms to errors in the learned parameters, and study the learnability of
these parameters
Optimization problems in network connectivity
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 115-120).Besides being one of the principal driving forces behind research in algorithmic theory for more than five decades, network optimization has assumed increased significance in recent times with the advent and widespread use of a variety of large-scale real-life networks. The primary goal of such networks is to connect vertices (representing a variety of real-life entities) in a robust and inexpensive manner, and to store and retrieve such connectivity information efficiently. In this thesis, we present efficient algorithms aimed at achieving these broad goals. The main results presented in this thesis are as follows. -- Cactus Construction. We give a near-linear time Monte Carlo algorithm for constructing a cactus representation of all the minimum cuts in an undirected graph. -- Cut Sparsification. A cut sparsifier of an undirected graph is a sparse graph on the same set of vertices that preserves its cut values up to small errors. We give new combinatorial and algorithmic results for constructing cut sparsifiers. -- Online Steiner Tree. Given an undirected graph as input, the goal of the Steiner tree problem is to select its minimum cost subgraph that connects a designated subset of vertices. We give the first online algorithm for the Steiner tree problem that has a poly-logarithmic competitive ratio when the input graph has both node and edge costs. -- Network Activation Problems. In the design of real-life wireless networks, a typical objective is to select one among a possible set of parameter values at each node such that the set of activated links satisfy some desired connectivity properties. We formalize this as the network activation model, and give approximation algorithms for various fundamental network design problems in this model.by Debmalya Panigrahi.Ph.D
Online Set Cover with Set Requests
We consider a generic online allocation problem that generalizes the classical online set cover framework by considering requests comprising a set of elements rather than a single element. This problem has multiple applications in cloud computing, crowd sourcing, facility planning, etc. Formally, it is an online covering problem where each online step comprises an offline covering problem. In addition, the covering sets are capacitated, leading to packing constraints. We give a randomized algorithm for this problem that has a nearly tight competitive ratio in both objectives: overall cost and maximum capacity violation. Our main technical tool is an online algorithm for packing/covering LPs with nested constraints, which may be of interest in other applications as well
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