Hierarchies. 1. Lovasz-Schrijver (LS), LS+ 2. Sherali Adams 3. Lasserre 4. Mixed Hierarchy (recently used) Idea: P = conv(subset S of 0,1 n )
|
|
- Suzan Arnold
- 5 years ago
- Views:
Transcription
1 Hierarchies
2 Today 1. Some more familiarity with Hierarchies 2. Examples of some basic upper and lower bounds 3. Survey of recent results (possible material for future talks)
3 Hierarchies 1. Lovasz-Schrijver (LS), LS+ 2. Sherali Adams 3. Lasserre 4. Mixed Hierarchy (recently used) Idea: P = conv(subset S of 0,1 n ) x = x 1,, x n P = prob. distribution over S Want to approximate P by putting linear constraints on x i. Hierarchies try to capture some kind of local correlations
4 Sherali Adams Variable x S intended to model π i S x i x i 2 = x i, so x S = x S {i} if i S Take LP, multiply each constraint by Π i I x i Π j J 1 x j for all I J = S S t x S should satisfy additional constraints Applying this to constraint 1 >= 0 x 12 0 x 1 x 12 0 x 2 x x 1 x 2 + x 12 0 Same as: M S (x) PSD and M S (g l * x) PSD
5 Alternate Description x S,α α 0,1 S Non-negativity x S,α 0 Consistency: x 12, 00 + x 12, 01 = x 1, 0 = x 13, 00 + x 13, 01 Sum up to 1. Pf: x S,α = T S α 1 (0) 1 T x S α 1 1 T (inclusion-exclusion) Alternate view: Define a probability distribution D(S) on each subset S. D S and D T consistent on S T Sherali Adams solution local distributions D(S) that satisfy LP constraints. (will make more sense later)
6 Knapsack Single constraint: Max i p i x i s. t. i a i x i B Sherali Adams: Multiply by Π i I x i Π j J 1 x j for all I J = S S t a i = 1 B = 1.99 x x n 1.99 Can determine x i,j = 0. How? Yet LP has integrality gap about 2, even for linear rounds.
7 Knapsack Gap x x n 2 ε Consider solution x i = p = x S = 0 for 2 S αn 2 ε n 1+α 1 ε Claim: Solution valid for αn rounds of SA. Proof: 1) Valid distribution on S. 2) Constraints: Suffices to check multiplication with π i S 1 x i only i S x i 2 ε (1 x i ) i S
8 Max independent set x i + x j 1 for i, j E Recovers x ij = 0 for edges. (Does not seem to capture clique constraints like Lasserre will do?)
9 Max-cut Basic LP. x ij variables i, j V Max (i,j) E x ij x ij + x jk x ik x ij + x jk + x ki 2 Perhaps could add more valid inequalities E.g. For every set of 5 vertices, at most 6 edges in cut. [Kenyon-Mathieu, de la Vega 07] Gap ½ + ε after log Ω 1 n rounds. Idea: Construct consistent distribution D S on up to 2t +3 vertices (i.e. valid combination of cuts). But objective far from any cut. [Charikar Makarychev Makarychev 09] Even after n γ rounds
10 Lasserre Hierarchy x S variables as in Sherali Adams Multiply by Π i I x i Π j S\I (1 x j ) for all I S, S t Put the constraint that M t x is PSD (previously M S x PSD) Has the following nice implication
11 Recall: Y PSD iff y ij = < v i, v j > for some vectors v 1,, v n (Y = V V T, Cholesky decomposition) M t x matrix with entries are indexed by S,T: x S T So x S T = < v S, v T > Stated as follows: Linear constraints on x_s obtained from g l and from constraint 1>= 0 < v S, v S > = < v T, v T > if S S = T T V φ 2 = 1
12 Interesting consequence x i = < v i, v φ > = < v i, v i > In other words, v i v φ 2 2 = v φ 2 4 Could have also expressed things using x S,α x S,α = < v φ, v S,α > V S,α = ( 1) T T α 1 0 v α 1 1 T
13 Knapsack Let us look at the previous example Max x x n x x n 1.9 Lasserre objective: v i 2 i Claim: Objective 1 As previously x ij = 0. Now implies < v i, v j > = 0 Suppose objective = t >1. 2 Consider i v i tv 0
14 Max independent set One round of Lasserre implies clique inequalities i C x i 1 C: clique As previously x ij = 0. Now implies < v i, v j > = 0 So, i v φ, v i v i 2 1 (as v_i/ v_i orthonormal) But x i = < v i, v φ > = v i 2
15 Max-cut Goemans Williamson SDP get (first level of Lasserre)
16 Mixed Hierarchy Lasserre powerful, but Sherali adams also very useful. Dense max-cut Problems on bounded treewidth graphs Mixed k-hierarchy: Sherali Adams k levels But SDP only on level 1, i.e. X ij = < v i, v j >, x i = < v i, v φ >, 1 =< v φ, v φ > Raghavendra 08: Assuming UGC, best algorithm for any Max-k-CSP is one k-level mixed hierarchy. Idea: Any integrality gap instance can be converted to NP-Hardness proof.
17 Partial Survey How to use the power of hierarchies. How to show that they do not help. Progress on long-standing questions. 1. Sparsest Cut Find S, s.t. min E[S,V\S]/( S V\S ) LP does not give better than log n min x e e E s.t. x ij = 1 and x ij + x jk x ik ARV: Using SDPs (2 nd level hierarchy) get log n Natural formulation + triangle inequalities 2 x ij = v i v j (embedding l_2 square metrics into l_1) (known integrality gaps still far from this)
18 Coloring 3-colorable graphs Using n 0.5 colors (Wigderson) n 0.4 [Blum 91] (blum coloring tools) n 0.25 [Karger Motwani Sudan] n 3/14 [Blum Karger] Sequence of improvements Use ARV structure on space of solutions Recent improvement on Blum coloring tools (Thorup Kawarabayashi 12) Current best around n Belief: n ε approximation using O 1 ε rounds of Lasserre Recently Arora-Ge show... Ratios for special classes Use Local-Global Correlation ideas recently developed.
19 Dense k-subgraph improved best known bound n 1/4 Use Sherali Adams Cannot beat n 1/4 using SA (SODA 12) Strong Lasserre lower bounds: n ε rounds but n γ hardness Unique Games: Local-global correlations Performance of Lasserre rounding related to spectrum of graph. General Techniques: Decomposition Lemma in context of Knapsack Directed Steiner Tree [Rothvoss]
20 How to show lower bounds. Nice techniques developed SA: Max-cut: any local constraint involves at most 2t+3 vertices. If x_s obtained from actual distribution on cuts on V[S] no constraint can be violated. Goal: Design distributions s.t. locally good, but globally bad. LS: (1,x) N(P) if protection matrix Y s.t. (Y 0 Y i )/(1 x i ) P and Y i x i Y 0 = (1, x), Y ii = x i, P x N t (P) if terms above in t-1 th level. Design a lower bound can be viewed as a game against adversary.
linear programming and approximate constraint satisfaction
linear programming and approximate constraint satisfaction Siu On Chan MSR New England James R. Lee University of Washington Prasad Raghavendra UC Berkeley David Steurer Cornell results MAIN THEOREM: Any
More informationIntroduction to LP and SDP Hierarchies
Introduction to LP and SDP Hierarchies Madhur Tulsiani Princeton University Local Constraints in Approximation Algorithms Linear Programming (LP) or Semidefinite Programming (SDP) based approximation algorithms
More informationIntegrality Gaps for Sherali Adams Relaxations
Integrality Gaps for Sherali Adams Relaxations Moses Charikar Princeton University Konstantin Makarychev IBM T.J. Watson Research Center Yury Makarychev Microsoft Research Abstract We prove strong lower
More informationLift-and-Project Techniques and SDP Hierarchies
MFO seminar on Semidefinite Programming May 30, 2010 Typical combinatorial optimization problem: max c T x s.t. Ax b, x {0, 1} n P := {x R n Ax b} P I := conv(k {0, 1} n ) LP relaxation Integral polytope
More informationSDP Relaxations for MAXCUT
SDP Relaxations for MAXCUT from Random Hyperplanes to Sum-of-Squares Certificates CATS @ UMD March 3, 2017 Ahmed Abdelkader MAXCUT SDP SOS March 3, 2017 1 / 27 Overview 1 MAXCUT, Hardness and UGC 2 LP
More informationapproximation algorithms I
SUM-OF-SQUARES method and approximation algorithms I David Steurer Cornell Cargese Workshop, 201 meta-task encoded as low-degree polynomial in R x example: f(x) = i,j n w ij x i x j 2 given: functions
More informationA Linear Round Lower Bound for Lovasz-Schrijver SDP Relaxations of Vertex Cover
A Linear Round Lower Bound for Lovasz-Schrijver SDP Relaxations of Vertex Cover Grant Schoenebeck Luca Trevisan Madhur Tulsiani Abstract We study semidefinite programming relaxations of Vertex Cover arising
More informationApproximating Sparsest Cut in Graphs of Bounded Treewidth
Approximating Sparsest Cut in Graphs of Bounded Treewidth Eden Chlamtac 1, Robert Krauthgamer 1, and Prasad Raghavendra 2 1 Weizmann Institute of Science, Rehovot, Israel. {eden.chlamtac,robert.krauthgamer}@weizmann.ac.il
More informationTight Integrality Gaps for Lovasz-Schrijver LP Relaxations of Vertex Cover and Max Cut
Tight Integrality Gaps for Lovasz-Schrijver LP Relaxations of Vertex Cover and Max Cut Grant Schoenebeck Luca Trevisan Madhur Tulsiani Abstract We study linear programming relaxations of Vertex Cover and
More informationPartitioning Algorithms that Combine Spectral and Flow Methods
CS369M: Algorithms for Modern Massive Data Set Analysis Lecture 15-11/11/2009 Partitioning Algorithms that Combine Spectral and Flow Methods Lecturer: Michael Mahoney Scribes: Kshipra Bhawalkar and Deyan
More informationSum-of-Squares Method, Tensor Decomposition, Dictionary Learning
Sum-of-Squares Method, Tensor Decomposition, Dictionary Learning David Steurer Cornell Approximation Algorithms and Hardness, Banff, August 2014 for many problems (e.g., all UG-hard ones): better guarantees
More informationIntroduction to Semidefinite Programming I: Basic properties a
Introduction to Semidefinite Programming I: Basic properties and variations on the Goemans-Williamson approximation algorithm for max-cut MFO seminar on Semidefinite Programming May 30, 2010 Semidefinite
More informationNotes on the decomposition result of Karlin et al. [2] for the hierarchy of Lasserre by M. Laurent, December 13, 2012
Notes on the decomposition result of Karlin et al. [2] for the hierarchy of Lasserre by M. Laurent, December 13, 2012 We present the decomposition result of Karlin et al. [2] for the hierarchy of Lasserre
More informationCutting Planes for First Level RLT Relaxations of Mixed 0-1 Programs
Cutting Planes for First Level RLT Relaxations of Mixed 0-1 Programs 1 Cambridge, July 2013 1 Joint work with Franklin Djeumou Fomeni and Adam N. Letchford Outline 1. Introduction 2. Literature Review
More informationUnique Games Conjecture & Polynomial Optimization. David Steurer
Unique Games Conjecture & Polynomial Optimization David Steurer Newton Institute, Cambridge, July 2013 overview Proof Complexity Approximability Polynomial Optimization Quantum Information computational
More informationLower bounds on the size of semidefinite relaxations. David Steurer Cornell
Lower bounds on the size of semidefinite relaxations David Steurer Cornell James R. Lee Washington Prasad Raghavendra Berkeley Institute for Advanced Study, November 2015 overview of results unconditional
More informationOptimal Sherali-Adams Gaps from Pairwise Independence
Optimal Sherali-Adams Gaps from Pairwise Independence Konstantinos Georgiou Avner Magen Madhur Tulsiani {cgeorg,avner}@cs.toronto.edu, madhurt@cs.berkeley.edu Abstract This work considers the problem of
More informationLecture : Lovász Theta Body. Introduction to hierarchies.
Strong Relaations for Discrete Optimization Problems 20-27/05/6 Lecture : Lovász Theta Body. Introduction to hierarchies. Lecturer: Yuri Faenza Scribes: Yuri Faenza Recall: stable sets and perfect graphs
More informationSherali-Adams relaxations of the matching polytope
Sherali-Adams relaxations of the matching polytope Claire Mathieu Alistair Sinclair November 17, 2008 Abstract We study the Sherali-Adams lift-and-project hierarchy of linear programming relaxations of
More informationApproximating Sparsest Cut in Graphs of Bounded Treewidth
Approximating Sparsest Cut in Graphs of Bounded Treewidth Eden Chlamtac Weizmann Institute Robert Krauthgamer Weizmann Institute Prasad Raghavendra Microsoft Research New England Abstract We give the first
More informationCuts for mixed 0-1 conic programs
Cuts for mixed 0-1 conic programs G. Iyengar 1 M. T. Cezik 2 1 IEOR Department Columbia University, New York. 2 GERAD Université de Montréal, Montréal TU-Chemnitz Workshop on Integer Programming and Continuous
More informationImproved Hardness of Approximating Chromatic Number
Improved Hardness of Approximating Chromatic Number Sangxia Huang KTH Royal Institute of Technology Stockholm, Sweden sangxia@csc.kth.se April 13, 2013 Abstract We prove that for sufficiently large K,
More informationA better approximation ratio for the Vertex Cover problem
A better approximation ratio for the Vertex Cover problem George Karakostas Dept. of Computing and Software McMaster University October 5, 004 Abstract We reduce the approximation factor for Vertex Cover
More informationLimitations of Linear and Semidefinite Programs by Grant Robert Schoenebeck
Limitations of Linear and Semidefinite Programs by Grant Robert Schoenebeck A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science
More informationApproximation Algorithms for the k-set Packing Problem
Approximation Algorithms for the k-set Packing Problem Marek Cygan Institute of Informatics University of Warsaw 20th October 2016, Warszawa Marek Cygan Approximation Algorithms for the k-set Packing Problem
More informationImproving Integrality Gaps via Chvátal-Gomory Rounding
Improving Integrality Gaps via Chvátal-Gomory Rounding Mohit Singh Kunal Talwar June 23, 2010 Abstract In this work, we study the strength of the Chvátal-Gomory cut generating procedure for several hard
More informationarxiv: v1 [cs.ds] 6 Oct 2011
Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph Aditya Bhaskara Moses Charikar Venkatesan Guruswami Aravindan Vijayaraghavan Yuan Zhou arxiv:1110.1360v1 [cs.ds] 6 Oct 2011
More informationCSC Linear Programming and Combinatorial Optimization Lecture 12: The Lift and Project Method
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 12: The Lift and Project Method Notes taken by Stefan Mathe April 28, 2007 Summary: Throughout the course, we have seen the importance
More informationPostprint.
http://www.diva-portal.org Postprint This is the accepted version of a paper presented at 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and
More informationApproximation & Complexity
Summer school on semidefinite optimization Approximation & Complexity David Steurer Cornell University Part 1 September 6, 2012 Overview Part 1 Unique Games Conjecture & Basic SDP Part 2 SDP Hierarchies:
More informationIntegrality Gaps of Linear and Semi-definite Programming Relaxations for Knapsack
Integrality Gaps of Linear and Semi-definite Programming Relaxations for Knapsack Anna R. Karlin Claire Mathieu C. Thach Nguyen Abstract Recent years have seen an explosion of interest in lift and project
More informationLecture 5. Max-cut, Expansion and Grothendieck s Inequality
CS369H: Hierarchies of Integer Programming Relaxations Spring 2016-2017 Lecture 5. Max-cut, Expansion and Grothendieck s Inequality Professor Moses Charikar Scribes: Kiran Shiragur Overview Here we derive
More informationPositive Semi-definite programing and applications for approximation
Combinatorial Optimization 1 Positive Semi-definite programing and applications for approximation Guy Kortsarz Combinatorial Optimization 2 Positive Sem-Definite (PSD) matrices, a definition Note that
More informationCSC Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming Notes taken by Mike Jamieson March 28, 2005 Summary: In this lecture, we introduce semidefinite programming
More informationNotice that lemma 4 has nothing to do with 3-colorability. To obtain a better result for 3-colorable graphs, we need the following observation.
COMPSCI 632: Approximation Algorithms November 1, 2017 Lecturer: Debmalya Panigrahi Lecture 18 Scribe: Feng Gui 1 Overview In this lecture, we examine graph coloring algorithms. We first briefly discuss
More informationInteger Linear Programs
Lecture 2: Review, Linear Programming Relaxations Today we will talk about expressing combinatorial problems as mathematical programs, specifically Integer Linear Programs (ILPs). We then see what happens
More informationMINI-TUTORIAL ON SEMI-ALGEBRAIC PROOF SYSTEMS. Albert Atserias Universitat Politècnica de Catalunya Barcelona
MINI-TUTORIAL ON SEMI-ALGEBRAIC PROOF SYSTEMS Albert Atserias Universitat Politècnica de Catalunya Barcelona Part I CONVEX POLYTOPES Convex polytopes as linear inequalities Polytope: P = {x R n : Ax b}
More informationHow hard is it to find a good solution?
How hard is it to find a good solution? Simons Institute Open Lecture November 4, 2013 Research Area: Complexity Theory Given a computational problem, find an efficient algorithm that solves it. Goal of
More informationUnique Games and Small Set Expansion
Proof, beliefs, and algorithms through the lens of sum-of-squares 1 Unique Games and Small Set Expansion The Unique Games Conjecture (UGC) (Khot [2002]) states that for every ɛ > 0 there is some finite
More informationCMPUT 675: Approximation Algorithms Fall 2014
CMPUT 675: Approximation Algorithms Fall 204 Lecture 25 (Nov 3 & 5): Group Steiner Tree Lecturer: Zachary Friggstad Scribe: Zachary Friggstad 25. Group Steiner Tree In this problem, we are given a graph
More informationApproximating maximum satisfiable subsystems of linear equations of bounded width
Approximating maximum satisfiable subsystems of linear equations of bounded width Zeev Nutov The Open University of Israel Daniel Reichman The Open University of Israel Abstract We consider the problem
More informationarxiv: v2 [cs.ds] 11 Jun 2012
Directed Steiner Tree and the Lasserre Hierarchy Thomas Rothvoß arxiv:1111.5473v2 [cs.ds] 11 Jun 2012 M.I.T. rothvoss@math.mit.edu June 13, 2012 Abstract The goal for the DIRECTED STEINER TREE problem
More informationMax Cut (1994; 1995; Goemans, Williamson)
Max Cut (1994; 1995; Goemans, Williamson) Alantha Newman Max-Planck-Institut für Informatik http://www.mpi-inf.mpg.de/alantha 1 Index Terms Graph partitioning, Approximation algorithms. 2 Synonyms Maximum
More informationLecture 20: Course summary and open problems. 1 Tools, theorems and related subjects
CSE 533: The PCP Theorem and Hardness of Approximation (Autumn 2005) Lecture 20: Course summary and open problems Dec. 7, 2005 Lecturer: Ryan O Donnell Scribe: Ioannis Giotis Topics we discuss in this
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 11 Luca Trevisan February 29, 2016
U.C. Berkeley CS294: Spectral Methods and Expanders Handout Luca Trevisan February 29, 206 Lecture : ARV In which we introduce semi-definite programming and a semi-definite programming relaxation of sparsest
More informationSubsampling Semidefinite Programs and Max-Cut on the Sphere
Electronic Colloquium on Computational Complexity, Report No. 129 (2009) Subsampling Semidefinite Programs and Max-Cut on the Sphere Boaz Barak Moritz Hardt Thomas Holenstein David Steurer November 29,
More informationSDP gaps for 2-to-1 and other Label-Cover variants
SDP gaps for -to-1 and other Label-Cover variants Venkatesan Guruswami 1, Subhash Khot, Ryan O Donnell 1, Preyas Popat, Madhur Tulsiani 3, and Yi Wu 1 1 Computer Science Department Carnegie Mellon University
More informationOn the efficient approximability of constraint satisfaction problems
On the efficient approximability of constraint satisfaction problems July 13, 2007 My world Max-CSP Efficient computation. P Polynomial time BPP Probabilistic Polynomial time (still efficient) NP Non-deterministic
More informationDownloaded 05/14/13 to Redistribution subject to SIAM license or copyright; see
SIAM J. COMPUT. Vol. 39, No. 8, pp. 3553 3570 c 2010 Society for Industrial and Applied Mathematics INTEGRALITY GAPS OF 2 o(1) FOR VERTEX COVER SDPs IN THE LOVÁSZ SCHRIJVER HIERARCHY KONSTANTINOS GEORGIOU,
More informationApproximate Hypergraph Coloring. 1 Introduction. Noga Alon 1 Pierre Kelsen 2 Sanjeev Mahajan 3 Hariharan Ramesh 4
Approximate Hypergraph Coloring Noga Alon 1 Pierre Kelsen 2 Sanjeev Mahajan Hariharan Ramesh 4 Abstract A coloring of a hypergraph is a mapping of vertices to colors such that no hyperedge is monochromatic.
More informationarxiv:cs/ v1 [cs.cc] 7 Sep 2006
Approximation Algorithms for the Bipartite Multi-cut problem arxiv:cs/0609031v1 [cs.cc] 7 Sep 2006 Sreyash Kenkre Sundar Vishwanathan Department Of Computer Science & Engineering, IIT Bombay, Powai-00076,
More informationHypergraph Matching by Linear and Semidefinite Programming. Yves Brise, ETH Zürich, Based on 2010 paper by Chan and Lau
Hypergraph Matching by Linear and Semidefinite Programming Yves Brise, ETH Zürich, 20110329 Based on 2010 paper by Chan and Lau Introduction Vertex set V : V = n Set of hyperedges E Hypergraph matching:
More informationIntegrality gaps of semidefinite programs for Vertex Cover and relations to l 1 embeddability of negative type metrics
Integrality gaps of semidefinite programs for Vertex Cover and relations to l 1 embeddability of negative type metrics Hamed Hatami, Avner Magen, and Evangelos Markakis Department of Computer Science,
More informationComplexity Classes V. More PCPs. Eric Rachlin
Complexity Classes V More PCPs Eric Rachlin 1 Recall from last time Nondeterminism is equivalent to having access to a certificate. If a valid certificate exists, the machine accepts. We see that problems
More informationTight Size-Degree Lower Bounds for Sums-of-Squares Proofs
Tight Size-Degree Lower Bounds for Sums-of-Squares Proofs Massimo Lauria KTH Royal Institute of Technology (Stockholm) 1 st Computational Complexity Conference, 015 Portland! Joint work with Jakob Nordström
More informationSome Open Problems in Approximation Algorithms
Some Open Problems in Approximation Algorithms David P. Williamson School of Operations Research and Information Engineering Cornell University November 2, 2010 Egerváry Research Group on Combinatorial
More informationCORC REPORT Approximate fixed-rank closures of covering problems
CORC REPORT 2003-01 Approximate fixed-rank closures of covering problems Daniel Bienstock and Mark Zuckerberg (Columbia University) January, 2003 version 2004-August-25 Abstract Consider a 0/1 integer
More informationHierarchical Clustering via Spreading Metrics
Journal of Machine Learning Research 18 2017) 1-35 Submitted 2/17; Revised 5/17; Published 8/17 Hierarchical Clustering via Spreading Metrics Aurko Roy College of Computing Georgia Institute of Technology
More informationSome Open Problems in Approximation Algorithms
Some Open Problems in Approximation Algorithms David P. Williamson School of Operations Research and Information Engineering Cornell University February 28, 2011 University of Bonn Bonn, Germany David
More informationNear-Optimal Algorithms for Maximum Constraint Satisfaction Problems
Near-Optimal Algorithms for Maximum Constraint Satisfaction Problems Moses Charikar Konstantin Makarychev Yury Makarychev Princeton University Abstract In this paper we present approximation algorithms
More informationInformation Theory + Polyhedral Combinatorics
Information Theory + Polyhedral Combinatorics Sebastian Pokutta Georgia Institute of Technology ISyE, ARC Joint work Gábor Braun Information Theory in Complexity Theory and Combinatorics Simons Institute
More informationarxiv: v15 [math.oc] 19 Oct 2016
LP formulations for mixed-integer polynomial optimization problems Daniel Bienstock and Gonzalo Muñoz, Columbia University, December 2014 arxiv:1501.00288v15 [math.oc] 19 Oct 2016 Abstract We present a
More informationGraph Sparsifiers: A Survey
Graph Sparsifiers: A Survey Nick Harvey UBC Based on work by: Batson, Benczur, de Carli Silva, Fung, Hariharan, Harvey, Karger, Panigrahi, Sato, Spielman, Srivastava and Teng Approximating Dense Objects
More informationPart II Strong lift-and-project cutting planes. Vienna, January 2012
Part II Strong lift-and-project cutting planes Vienna, January 2012 The Lovász and Schrijver M(K, K) Operator Let K be a given linear system in 0 1 variables. for any pair of inequalities αx β 0 and α
More informationA simple LP relaxation for the Asymmetric Traveling Salesman Problem
A simple LP relaxation for the Asymmetric Traveling Salesman Problem Thành Nguyen Cornell University, Center for Applies Mathematics 657 Rhodes Hall, Ithaca, NY, 14853,USA thanh@cs.cornell.edu Abstract.
More information1 The independent set problem
ORF 523 Lecture 11 Spring 2016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Tuesday, March 29, 2016 When in doubt on the accuracy of these notes, please cross chec with the instructor
More informationLabel Cover Algorithms via the Log-Density Threshold
Label Cover Algorithms via the Log-Density Threshold Jimmy Wu jimmyjwu@stanford.edu June 13, 2017 1 Introduction Since the discovery of the PCP Theorem and its implications for approximation algorithms,
More informationHow to Play Unique Games against a Semi-Random Adversary
How to Play Unique Games against a Semi-Random Adversary Study of Semi-Random Models of Unique Games Alexandra Kolla Microsoft Research Konstantin Makarychev IBM Research Yury Makarychev TTIC Abstract
More informationORIE 6334 Spectral Graph Theory December 1, Lecture 27 Remix
ORIE 6334 Spectral Graph Theory December, 06 Lecturer: David P. Williamson Lecture 7 Remix Scribe: Qinru Shi Note: This is an altered version of the lecture I actually gave, which followed the structure
More informationDynamic Programming on Trees. Example: Independent Set on T = (V, E) rooted at r V.
Dynamic Programming on Trees Example: Independent Set on T = (V, E) rooted at r V. For v V let T v denote the subtree rooted at v. Let f + (v) be the size of a maximum independent set for T v that contains
More informationOverview. 1 Introduction. 2 Preliminary Background. 3 Unique Game. 4 Unique Games Conjecture. 5 Inapproximability Results. 6 Unique Game Algorithms
Overview 1 Introduction 2 Preliminary Background 3 Unique Game 4 Unique Games Conjecture 5 Inapproximability Results 6 Unique Game Algorithms 7 Conclusion Antonios Angelakis (NTUA) Theory of Computation
More informationRounding Lasserre SDPs using column selection and spectrum-based approximation schemes for graph partitioning and Quadratic IPs
Rounding Lasserre SDPs using column selection and spectrum-based approximation schemes for graph partitioning and Quadratic IPs Venkatesan Guruswami guruswami@cmu.edu Computer Science Department, Carnegie
More informationApproximability of Constraint Satisfaction Problems
Approximability of Constraint Satisfaction Problems Venkatesan Guruswami Carnegie Mellon University October 2009 Venkatesan Guruswami (CMU) Approximability of CSPs Oct 2009 1 / 1 Constraint Satisfaction
More informationLecture 1: April 24, 2013
TTIC/CMSC 31150 Mathematical Toolkit Spring 2013 Madhur Tulsiani Lecture 1: April 24, 2013 Scribe: Nikita Mishra 1 Applications of dimensionality reduction In this lecture we talk about various useful
More informationSherali-Adams Relaxations of the Matching Polytope
Sherali-Adams Relaxations of the Matching Polytope Claire Mathieu Computer Science Department Brown University Providence, RI 091 claire@cs.brown.edu Alistair Sinclair Computer Science Division University
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 12 Luca Trevisan October 3, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analysis Handout 1 Luca Trevisan October 3, 017 Scribed by Maxim Rabinovich Lecture 1 In which we begin to prove that the SDP relaxation exactly recovers communities
More informationPolynomial integrality gaps for strong SDP relaxations of Densest k-subgraph
Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph Aditya Bhaskara Moses Charikar Venkatesan Guruswami Aravindan Vijayaraghavan Yuan Zhou July 12, 2011 Abstract The Densest k-subgraph
More informationA Union of Euclidean Spaces is Euclidean. Konstantin Makarychev, Northwestern Yury Makarychev, TTIC
Union of Euclidean Spaces is Euclidean Konstantin Makarychev, Northwestern Yury Makarychev, TTIC MS Meeting, New York, May 7, 2017 Problem by ssaf Naor Suppose that metric space (X, d) is a union of two
More informationApplications of semidefinite programming in Algebraic Combinatorics
Applications of semidefinite programming in Algebraic Combinatorics Tohoku University The 23rd RAMP Symposium October 24, 2011 We often want to 1 Bound the value of a numerical parameter of certain combinatorial
More informationApproximation Algorithms and Hardness of Approximation May 14, Lecture 22
Approximation Algorithms and Hardness of Approximation May 4, 03 Lecture Lecturer: Alantha Newman Scribes: Christos Kalaitzis The Unique Games Conjecture The topic of our next lectures will be the Unique
More informationBreaking the Rectangle Bound Barrier against Formula Size Lower Bounds
Breaking the Rectangle Bound Barrier against Formula Size Lower Bounds Kenya Ueno The Young Researcher Development Center and Graduate School of Informatics, Kyoto University kenya@kuis.kyoto-u.ac.jp Abstract.
More informationLecture 2: November 9
Semidefinite programming and computational aspects of entanglement IHP Fall 017 Lecturer: Aram Harrow Lecture : November 9 Scribe: Anand (Notes available at http://webmitedu/aram/www/teaching/sdphtml)
More informationOn the Optimality of Some Semidefinite Programming-Based. Approximation Algorithms under the Unique Games Conjecture. A Thesis presented
On the Optimality of Some Semidefinite Programming-Based Approximation Algorithms under the Unique Games Conjecture A Thesis presented by Seth Robert Flaxman to Computer Science in partial fulfillment
More informationGraph Partitioning and Semi-definite Programming Hierarchies
Graph Partitioning and Semi-definite Programming Hierarchies Ali Kemal Sinop CMU-CS-12-121 May 15, 2012 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Thesis Committee: Venkatesan
More informationACO Comprehensive Exam 19 March Graph Theory
1. Graph Theory Let G be a connected simple graph that is not a cycle and is not complete. Prove that there exist distinct non-adjacent vertices u, v V (G) such that the graph obtained from G by deleting
More informationOptimization Exercise Set n.5 :
Optimization Exercise Set n.5 : Prepared by S. Coniglio translated by O. Jabali 2016/2017 1 5.1 Airport location In air transportation, usually there is not a direct connection between every pair of airports.
More information8 Approximation Algorithms and Max-Cut
8 Approximation Algorithms and Max-Cut 8. The Max-Cut problem Unless the widely believed P N P conjecture is false, there is no polynomial algorithm that can solve all instances of an NP-hard problem.
More informationSolving the MWT. Recall the ILP for the MWT. We can obtain a solution to the MWT problem by solving the following ILP:
Solving the MWT Recall the ILP for the MWT. We can obtain a solution to the MWT problem by solving the following ILP: max subject to e i E ω i x i e i C E x i {0, 1} x i C E 1 for all critical mixed cycles
More informationarxiv: v1 [cs.ds] 29 Aug 2015
APPROXIMATING (UNWEIGHTED) TREE AUGMENTATION VIA LIFT-AND-PROJECT, PART I: STEMLESS TAP JOSEPH CHERIYAN AND ZHIHAN GAO arxiv:1508.07504v1 [cs.ds] 9 Aug 015 Abstract. In Part I, we study a special case
More informationTopic: Balanced Cut, Sparsest Cut, and Metric Embeddings Date: 3/21/2007
CS880: Approximations Algorithms Scribe: Tom Watson Lecturer: Shuchi Chawla Topic: Balanced Cut, Sparsest Cut, and Metric Embeddings Date: 3/21/2007 In the last lecture, we described an O(log k log D)-approximation
More informationA NEARLY-TIGHT SUM-OF-SQUARES LOWER BOUND FOR PLANTED CLIQUE
A NEARLY-TIGHT SUM-OF-SQUARES LOWER BOUND FOR PLANTED CLIQUE Boaz Barak Sam Hopkins Jon Kelner Pravesh Kothari Ankur Moitra Aaron Potechin on the market! PLANTED CLIQUE ("DISTINGUISHING") Input: graph
More informationLocal Constraints in Combinatorial Optimization. Madhur Tulsiani
Local Constraints in Combinatorial Optimization by Madhur Tulsiani A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the
More informationBasic Properties of Metric and Normed Spaces
Basic Properties of Metric and Normed Spaces Computational and Metric Geometry Instructor: Yury Makarychev The second part of this course is about metric geometry. We will study metric spaces, low distortion
More informationOptimization of Submodular Functions Tutorial - lecture I
Optimization of Submodular Functions Tutorial - lecture I Jan Vondrák 1 1 IBM Almaden Research Center San Jose, CA Jan Vondrák (IBM Almaden) Submodular Optimization Tutorial 1 / 1 Lecture I: outline 1
More informationOn the Rank of Cutting-Plane Proof Systems
On the Rank of Cutting-Plane Proof Systems Sebastian Pokutta 1 and Andreas S. Schulz 2 1 H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA. Email:
More informationRandom Lifts of Graphs
27th Brazilian Math Colloquium, July 09 Plan of this talk A brief introduction to the probabilistic method. A quick review of expander graphs and their spectrum. Lifts, random lifts and their properties.
More informationLecture 13: Spectral Graph Theory
CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 13: Spectral Graph Theory Lecturer: Shayan Oveis Gharan 11/14/18 Disclaimer: These notes have not been subjected to the usual scrutiny reserved
More informationLecture 4: January 26
10-725/36-725: Conve Optimization Spring 2015 Lecturer: Javier Pena Lecture 4: January 26 Scribes: Vipul Singh, Shinjini Kundu, Chia-Yin Tsai Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationCS286.2 Lecture 8: A variant of QPCP for multiplayer entangled games
CS286.2 Lecture 8: A variant of QPCP for multiplayer entangled games Scribe: Zeyu Guo In the first lecture, we saw three equivalent variants of the classical PCP theorems in terms of CSP, proof checking,
More informationPartitioning Metric Spaces
Partitioning Metric Spaces Computational and Metric Geometry Instructor: Yury Makarychev 1 Multiway Cut Problem 1.1 Preliminaries Definition 1.1. We are given a graph G = (V, E) and a set of terminals
More information