9 Search Results for "Kumar, Nikhil"


Document
APPROX
A Primal-Dual Algorithm for Multicommodity Flows and Multicuts in Treewidth-2 Graphs

Authors: Tobias Friedrich, Davis Issac, Nikhil Kumar, Nadym Mallek, and Ziena Zeif

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)


Abstract
We study the problem of multicommodity flow and multicut in treewidth-2 graphs and prove bounds on the multiflow-multicut gap. In particular, we give a primal-dual algorithm for computing multicommodity flow and multicut in treewidth-2 graphs and prove the following approximate max-flow min-cut theorem: given a treewidth-2 graph, there exists a multicommodity flow of value f with congestion 4, and a multicut of capacity c such that c ≤ 20 f. This implies a multiflow-multicut gap of 80 and improves upon the previous best known bounds for such graphs. Our algorithm runs in polynomial time when all the edges have capacity one. Our algorithm is completely combinatorial and builds upon the primal-dual algorithm of Garg, Vazirani and Yannakakis for multicut in trees and the augmenting paths framework of Ford and Fulkerson.

Cite as

Tobias Friedrich, Davis Issac, Nikhil Kumar, Nadym Mallek, and Ziena Zeif. A Primal-Dual Algorithm for Multicommodity Flows and Multicuts in Treewidth-2 Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 55:1-55:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{friedrich_et_al:LIPIcs.APPROX/RANDOM.2022.55,
  author =	{Friedrich, Tobias and Issac, Davis and Kumar, Nikhil and Mallek, Nadym and Zeif, Ziena},
  title =	{{A Primal-Dual Algorithm for Multicommodity Flows and Multicuts in Treewidth-2 Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{55:1--55:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.55},
  URN =		{urn:nbn:de:0030-drops-171774},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.55},
  annote =	{Keywords: Approximation Algorithms, Multicommodity Flow, Multicut}
}
Document
Online Metric Allocation and Time-Varying Regularization

Authors: Nikhil Bansal and Christian Coester

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)


Abstract
We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let M be an n-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of M. At each time t, a convex monotone cost function c_t: [0,1] → ℝ_+ appears at some point r_t ∈ M. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost c_t(x_{r_t}), where x_{r_t} is the fraction of the resource at r_t at the end of time t. For example, when the cost functions are c_t(x) = α x, this is equivalent to randomized MTS, and when the cost functions are c_t(x) = ∞⋅1_{x < 1/k}, this is equivalent to fractional k-server. Because of an inherent scale-freeness property of the problem, existing techniques for MTS and k-server fail to achieve similar guarantees for metric allocation. To handle this, we consider a generalization of the online multiplicative update method where we decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. We use this to give an O(log n)-competitive algorithm for weighted star metrics. We then show how this corresponds to an extension of the online mirror descent framework to a setting where the regularizer is time-varying. Using this perspective, we further refine the guarantees of our algorithm. We also consider the case of non-convex cost functions. Using a simple 𝓁₂²-regularizer, we give tight bounds of Θ(n) on tree metrics, which imply deterministic and randomized competitive ratios of O(n²) and O(nlog n) respectively on arbitrary metrics.

Cite as

Nikhil Bansal and Christian Coester. Online Metric Allocation and Time-Varying Regularization. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bansal_et_al:LIPIcs.ESA.2022.13,
  author =	{Bansal, Nikhil and Coester, Christian},
  title =	{{Online Metric Allocation and Time-Varying Regularization}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{13:1--13:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.13},
  URN =		{urn:nbn:de:0030-drops-169515},
  doi =		{10.4230/LIPIcs.ESA.2022.13},
  annote =	{Keywords: Online algorithms, competitive analysis, k-server, metrical task systems, mirror descent, regularization}
}
Document
Skeletons and Minimum Energy Scheduling

Authors: Antonios Antoniadis, Gunjan Kumar, and Nikhil Kumar

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)


Abstract
Consider the problem where n jobs, each with a release time, a deadline and a required processing time are to be feasibly scheduled in a single- or multi-processor setting so as to minimize the total energy consumption of the schedule. A processor has two available states: a sleep state where no energy is consumed but also no processing can take place, and an active state which consumes energy at a rate of one, and in which jobs can be processed. Transitioning from the active to the sleep does not incur any further energy cost, but transitioning from the sleep to the active state requires q energy units. Jobs may be preempted and (in the multi-processor case) migrated. The single-processor case of the problem is known to be solvable in polynomial time via an involved dynamic program, whereas the only known approximation algorithm for the multi-processor case attains an approximation factor of 3 and is based on rounding the solution to a linear programming relaxation of the problem. In this work, we present efficient and combinatorial approximation algorithms for both the single- and the multi-processor setting. Before, only an algorithm based on linear programming was known for the multi-processor case. Our algorithms build upon the concept of a skeleton, a basic (and not necessarily feasible) schedule that captures the fact that some processor(s) must be active at some time point during an interval. Finally, we further demonstrate the power of skeletons by providing a 2-approximation algorithm for the multiprocessor case, thus improving upon the recent breakthrough 3-approximation result. Our algorithm is based on a novel rounding scheme of a linear-programming relaxation of the problem which incorporates skeletons.

Cite as

Antonios Antoniadis, Gunjan Kumar, and Nikhil Kumar. Skeletons and Minimum Energy Scheduling. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{antoniadis_et_al:LIPIcs.ISAAC.2021.51,
  author =	{Antoniadis, Antonios and Kumar, Gunjan and Kumar, Nikhil},
  title =	{{Skeletons and Minimum Energy Scheduling}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{51:1--51:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.51},
  URN =		{urn:nbn:de:0030-drops-154849},
  doi =		{10.4230/LIPIcs.ISAAC.2021.51},
  annote =	{Keywords: scheduling, energy-efficiency, approximation algorithms, dynamic programming, combinatorial algorithms}
}
Document
Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four

Authors: Suryajith Chillara

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)


Abstract
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit d^{O(1)}-variate and degree d polynomial P_{d} ∈ VNP such that if any depth four circuit C of bounded formal degree d which computes a polynomial of bounded individual degree O(1), that is functionally equivalent to P_d, then C must have size 2^Ω(√dlog{d}). The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for ACC⁰ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in ACC⁰ can also be computed by algebraic Σ∧ΣΠ circuits (i.e., circuits of the form - sums of powers of polynomials) of 2^(log^O(1) n) size. Thus they argued that a 2^{ω(polylog n)} "functional" lower bound for an explicit polynomial Q against Σ∧ΣΠ circuits would imply a lower bound for the "corresponding Boolean function" of Q against non-uniform ACC⁰. In their work, they ask if their lower bound be extended to Σ∧ΣΠ circuits. In this paper, for large integers n and d such that ω(log²n) ≤ d ≤ n^{0.01}, we show that any Σ∧ΣΠ circuit of bounded individual degree at most O(d/k²) that functionally computes Iterated Matrix Multiplication polynomial IMM_{n,d} (∈ VP) over {0,1}^{n²d} must have size n^Ω(k). Since Iterated Matrix Multiplication IMM_{n,d} over {0,1}^{n²d} is functionally in GapL, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of ACC⁰ from GapL. For the sake of completeness, we also show a syntactic size lower bound against any Σ∧ΣΠ circuit computing IMM_{n,d} (for the same regime of d) which is tight over large fields. Like Forbes, Kumar and Saptharishi [CCC, 2016], we too prove lower bounds against circuits of bounded formal degree which functionally compute IMM_{n,d}, for a slightly larger range of individual degree.

Cite as

Suryajith Chillara. Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{chillara:LIPIcs.FSTTCS.2021.14,
  author =	{Chillara, Suryajith},
  title =	{{Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{14:1--14:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.14},
  URN =		{urn:nbn:de:0030-drops-155251},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.14},
  annote =	{Keywords: Functional Lower Bounds, Boolean Circuit Lower Bounds, Depth Four, Connections to Boolean Complexity, Iterated Matrix Multiplication}
}
Document
Track A: Algorithms, Complexity and Games
On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications

Authors: Sayan Bandyapadhyay, Fedor V. Fomin, and Kirill Simonov

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)


Abstract
Fair clustering is a variant of constrained clustering where the goal is to partition a set of colored points. The fraction of points of each color in every cluster should be more or less equal to the fraction of points of this color in the dataset. This variant was recently introduced by Chierichetti et al. [NeurIPS 2017] and became widely popular. This paper proposes a new construction of coresets for fair k-means and k-median clustering for Euclidean and general metrics based on random sampling. For the Euclidean space ℝ^d, we provide the first coresets whose size does not depend exponentially on the dimension d. The question of whether such constructions exist was asked by Schmidt, Schwiegelshohn, and Sohler [WAOA 2019] and Huang, Jiang, and Vishnoi [NeurIPS 2019]. For general metric, our construction provides the first coreset for fair k-means and k-median. New coresets appear to be a handy tool for designing better approximation and streaming algorithms for fair and other constrained clustering variants. In particular, we obtain - the first fixed-parameter tractable (FPT) PTAS for fair k-means and k-median clustering in ℝ^d. The near-linear time of our PTAS improves over the previous scheme of Böhm, Fazzone, Leonardi, and Schwiegelshohn [ArXiv 2020] with running time n^{poly(k/ε)}; - FPT "true" constant-approximation for metric fair clustering. All previous algorithms for fair k-means and k-median in general metric are bicriteria and violate the fairness constraints; - FPT 3-approximation for lower-bounded k-median improving the best-known 3.736 factor of Bera, Chakrabarty, and Negahbani [ArXiv 2019]; - the first FPT constant-approximations for metric chromatic clustering and 𝓁-Diversity clustering; - near linear-time (in n) PTAS for capacitated and lower-bounded clustering improving over PTAS of Bhattacharya, Jaiswal, and Kumar [TOCS 2018] with super-quadratic running time; - a streaming (1+ε)-approximation for fair k-means and k-median of space complexity polynomial in k, d, ε and log{n} (the previous algorithms have exponential space complexity on either d or k).

Cite as

Sayan Bandyapadhyay, Fedor V. Fomin, and Kirill Simonov. On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{bandyapadhyay_et_al:LIPIcs.ICALP.2021.23,
  author =	{Bandyapadhyay, Sayan and Fomin, Fedor V. and Simonov, Kirill},
  title =	{{On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.23},
  URN =		{urn:nbn:de:0030-drops-140923},
  doi =		{10.4230/LIPIcs.ICALP.2021.23},
  annote =	{Keywords: fair clustering, coresets, approximation algorithms}
}
Document
Multicommodity Flows in Planar Graphs with Demands on Faces

Authors: Nikhil Kumar

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)


Abstract
We consider the problem of multicommodity flows in planar graphs. Seymour [Seymour, 1981] showed that if the union of supply and demand graphs is planar, then the cut condition is also sufficient for routing demands. Okamura-Seymour [Okamura and Seymour, 1981] showed that if the supply graph is planar and all demands are incident on one face, then again the cut condition is sufficient for routing demands. We consider a common generalization of these settings where the end points of each demand are on the same face of the planar graph. We show that if the source sink pairs on each face of the graph are such that sources and sinks appear contiguously on the cycle bounding the face, then the flow cut gap is at most 3. We come up with a notion of approximating demands on a face by convex combination of laminar demands to prove this result.

Cite as

Nikhil Kumar. Multicommodity Flows in Planar Graphs with Demands on Faces. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 41:1-41:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{kumar:LIPIcs.ISAAC.2020.41,
  author =	{Kumar, Nikhil},
  title =	{{Multicommodity Flows in Planar Graphs with Demands on Faces}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{41:1--41:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.41},
  URN =		{urn:nbn:de:0030-drops-133857},
  doi =		{10.4230/LIPIcs.ISAAC.2020.41},
  annote =	{Keywords: Combinatorial Optimization, Multicommodity Flow, Network Design}
}
Document
Dual Half-Integrality for Uncrossable Cut Cover and Its Application to Maximum Half-Integral Flow

Authors: Naveen Garg and Nikhil Kumar

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)


Abstract
Given an edge weighted graph and a forest F, the 2-edge connectivity augmentation problem is to pick a minimum weighted set of edges, E', such that every connected component of E' ∪ F is 2-edge connected. Williamson et al. gave a 2-approximation algorithm (WGMV) for this problem using the primal-dual schema. We show that when edge weights are integral, the WGMV procedure can be modified to obtain a half-integral dual. The 2-edge connectivity augmentation problem has an interesting connection to routing flow in graphs where the union of supply and demand is planar. The half-integrality of the dual leads to a tight 2-approximate max-half-integral-flow min-multicut theorem.

Cite as

Naveen Garg and Nikhil Kumar. Dual Half-Integrality for Uncrossable Cut Cover and Its Application to Maximum Half-Integral Flow. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{garg_et_al:LIPIcs.ESA.2020.55,
  author =	{Garg, Naveen and Kumar, Nikhil},
  title =	{{Dual Half-Integrality for Uncrossable Cut Cover and Its Application to Maximum Half-Integral Flow}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{55:1--55:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.55},
  URN =		{urn:nbn:de:0030-drops-129214},
  doi =		{10.4230/LIPIcs.ESA.2020.55},
  annote =	{Keywords: Combinatorial Optimization, Multicommodity Flow, Network Design}
}
Document
APPROX
A Constant Factor Approximation for Capacitated Min-Max Tree Cover

Authors: Syamantak Das, Lavina Jain, and Nikhil Kumar

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)


Abstract
Given a graph G = (V,E) with non-negative real edge lengths and an integer parameter k, the (uncapacitated) Min-Max Tree Cover problem seeks to find a set of at most k trees which together span V and each tree is a subgraph of G. The objective is to minimize the maximum length among all the trees. In this paper, we consider a capacitated generalization of the above and give the first constant factor approximation algorithm. In the capacitated version, there is a hard uniform capacity (λ) on the number of vertices a tree can cover. Our result extends to the rooted version of the problem, where we are given a set of k root vertices, R and each of the covering trees is required to include a distinct vertex in R as the root. Prior to our work, the only result known was a (2k-1)-approximation algorithm for the special case when the total number of vertices in the graph is kλ [Guttmann-Beck and Hassin, J. of Algorithms, 1997]. Our technique circumvents the difficulty of using the minimum spanning tree of the graph as a lower bound, which is standard for the uncapacitated version of the problem [Even et al.,OR Letters 2004] [Khani et al.,Algorithmica 2010]. Instead, we use Steiner trees that cover λ vertices along with an iterative refinement procedure that ensures that the output trees have low cost and the vertices are well distributed among the trees.

Cite as

Syamantak Das, Lavina Jain, and Nikhil Kumar. A Constant Factor Approximation for Capacitated Min-Max Tree Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{das_et_al:LIPIcs.APPROX/RANDOM.2020.55,
  author =	{Das, Syamantak and Jain, Lavina and Kumar, Nikhil},
  title =	{{A Constant Factor Approximation for Capacitated Min-Max Tree Cover}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{55:1--55:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.55},
  URN =		{urn:nbn:de:0030-drops-126581},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.55},
  annote =	{Keywords: Approximation Algorithms, Graph Algorithms, Min-Max Tree Cover, Vehicle Routing, Steiner Tree}
}
Document
On Reconstructing a Hidden Permutation

Authors: Flavio Chierichetti, Anirban Dasgupta, Ravi Kumar, and Silvio Lattanzi

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
The Mallows model is a classical model for generating noisy perturbations of a hidden permutation, where the magnitude of the perturbations is determined by a single parameter. In this work we consider the following reconstruction problem: given several perturbations of a hidden permutation that are generated according to the Mallows model, each with its own parameter, how to recover the hidden permutation? When the parameters are approximately known and satisfy certain conditions, we obtain a simple algorithm for reconstructing the hidden permutation; we also show that these conditions are nearly inevitable for reconstruction. We then provide an algorithm to estimate the parameters themselves. En route we obtain a precise characterization of the swapping probability in the Mallows model.

Cite as

Flavio Chierichetti, Anirban Dasgupta, Ravi Kumar, and Silvio Lattanzi. On Reconstructing a Hidden Permutation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 604-617, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{chierichetti_et_al:LIPIcs.APPROX-RANDOM.2014.604,
  author =	{Chierichetti, Flavio and Dasgupta, Anirban and Kumar, Ravi and Lattanzi, Silvio},
  title =	{{On Reconstructing a Hidden Permutation}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{604--617},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.604},
  URN =		{urn:nbn:de:0030-drops-47251},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.604},
  annote =	{Keywords: Mallows model; Rank aggregation; Reconstruction}
}
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