2 Search Results for "Goyal, Navin"

Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.23
Online Carpooling Using Expander Decompositions

Authors: Anupam Gupta, Ravishankar Krishnaswamy, Amit Kumar, and Sahil Singla

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We consider the online carpooling problem: given n vertices, a sequence of edges arrive over time. When an edge e_t = (u_t, v_t) arrives at time step t, the algorithm must orient the edge either as v_t → u_t or u_t → v_t, with the objective of minimizing the maximum discrepancy of any vertex, i.e., the absolute difference between its in-degree and out-degree. Edges correspond to pairs of persons wanting to ride together, and orienting denotes designating the driver. The discrepancy objective then corresponds to every person driving close to their fair share of rides they participate in. In this paper, we design efficient algorithms which can maintain polylog(n,T) maximum discrepancy (w.h.p) over any sequence of T arrivals, when the arriving edges are sampled independently and uniformly from any given graph G. This provides the first polylogarithmic bounds for the online (stochastic) carpooling problem. Prior to this work, the best known bounds were O(√{n log n})-discrepancy for any adversarial sequence of arrivals, or O(log log n)-discrepancy bounds for the stochastic arrivals when G is the complete graph. The technical crux of our paper is in showing that the simple greedy algorithm, which has provably good discrepancy bounds when the arriving edges are drawn uniformly at random from the complete graph, also has polylog discrepancy when G is an expander graph. We then combine this with known expander-decomposition results to design our overall algorithm.
Cite as

Anupam Gupta, Ravishankar Krishnaswamy, Amit Kumar, and Sahil Singla. Online Carpooling Using Expander Decompositions. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2020.23,
  author =	{Gupta, Anupam and Krishnaswamy, Ravishankar and Kumar, Amit and Singla, Sahil},
  title =	{{Online Carpooling Using Expander Decompositions}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{23:1--23:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.23},
  URN =		{urn:nbn:de:0030-drops-132647},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.23},
  annote =	{Keywords: Online Algorithms, Discrepancy Minimization, Carpooling}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2012.58
Lower Bounds for the Average and Smoothed Number of Pareto Optima

Authors: Navin Goyal and Luis Rademacher

Published in: LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Abstract
Smoothed analysis of multiobjective 0-1 linear optimization has drawn considerable attention recently. In this literature, the number of Pareto-optimal solutions (i.e., solutions with the property that no other solution is at least as good in all the coordinates and better in at least one) for multiobjective optimization problems is the central object of study. In this paper, we prove several lower bounds for the expected number of Pareto optima. Our basic result is a lower bound of Omega_d(n^{d-1}) for optimization problems with d objectives and $n$ variables under fairly general conditions on the distributions of the linear objectives. Our proof relates the problem of lower bounding the number of Pareto optima to results in discrete geometry and geometric probability connected to arrangements of hyperplanes. We use our basic result to derive (1) To our knowledge, the first lower bound for natural multiobjective optimization problems. We illustrate this for the maximum spanning tree problem with randomly chosen edge weights. Our technique is sufficiently flexible to yield such lower bounds for other standard objective functions studied in this setting (such as multiobjective shortest path, TSP tour, matching). (2) Smoothed lower bound of min(Omega_d( n^{d-1.5} phi^{(d-\log d) (1-Theta(1/phi))}), 2^{Theta(n)}) for the 0-1 knapsack problem with d profits for phi-semirandom distributions for a version of the knapsack problem. This improves the recent lower bound of Brunsch and Röglin.
Cite as

Navin Goyal and Luis Rademacher. Lower Bounds for the Average and Smoothed Number of Pareto Optima. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 58-69, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{goyal_et_al:LIPIcs.FSTTCS.2012.58,
  author =	{Goyal, Navin and Rademacher, Luis},
  title =	{{Lower Bounds for the Average and Smoothed Number of Pareto Optima}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)},
  pages =	{58--69},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-47-7},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{18},
  editor =	{D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.58},
  URN =		{urn:nbn:de:0030-drops-38488},
  doi =		{10.4230/LIPIcs.FSTTCS.2012.58},
  annote =	{Keywords: geometric probability, smoothed analysis, multiobjective optimization, Pareto optimal, knapsack}
}
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