Chordal structure in computer algebra: Permanents
|
|
- Marcia Cooper
- 5 years ago
- Views:
Transcription
1 Chordal structure in computer algebra: Permanents Diego Cifuentes Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology Joint work with Pablo A. Parrilo (MIT) SIAM Conference on Discrete Mathematics Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 1 / 21
2 Algebraic tools in discrete math Several problems from discrete mathematics can be approached with tools from computer algebra. Sensor network localization: Find positions, given a few known fixed anchors and pairwise distances. x i x j 2 = d 2 ij x i a k 2 = e 2 ij ij A ik B This is a system of quadratic polynomial equations. Can be solved using Gröbner bases. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 2 / 21
3 Algebraic tools in discrete math Several problems from discrete mathematics can be approached with tools from computer algebra. Lattice walks Let A Z n consist of integer vectors. Consider a graph with vertex set N n in which α, β are adjacent if α β A. Describe the connected components. This can be solved using a primary decomposition. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 3 / 21
4 Algebraic tools in discrete math Several problems from discrete mathematics can be approached with tools from computer algebra. Permanents (this talk) Given a n n matrix M, compute Perm(M) := π M i,π(i) i sum over permutations π. Important case: sparse matrices. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 4 / 21
5 Algebraic tools in discrete math Many more problems can be naturally approached with algebraic tools: graph colorings, independent sets, Hamiltonian cycles, crypto problems, etc. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 5 / 21
6 Algebraic tools in discrete math Many more problems can be naturally approached with algebraic tools: graph colorings, independent sets, Hamiltonian cycles, crypto problems, etc. However these problems are NP-hard, and these algebraic methods do not work well for relatively small problems. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 5 / 21
7 Algebraic tools in discrete math Many more problems can be naturally approached with algebraic tools: graph colorings, independent sets, Hamiltonian cycles, crypto problems, etc. However these problems are NP-hard, and these algebraic methods do not work well for relatively small problems. Idea: exploit the graphical structure (chordality) of the problem! Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 5 / 21
8 Algebraic tools in discrete math Many more problems can be naturally approached with algebraic tools: graph colorings, independent sets, Hamiltonian cycles, crypto problems, etc. However these problems are NP-hard, and these algebraic methods do not work well for relatively small problems. Idea: exploit the graphical structure (chordality) of the problem! Complexity aspects? Identify families of tractable instances. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 5 / 21
9 Permanent computation General facts: (Ryser 63) Best general and exact method; complexity O(n 2 n ). (Valiant 79) Computing the permanent of a matrix is #P-hard. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 6 / 21
10 Permanent computation General facts: (Ryser 63) Best general and exact method; complexity O(n 2 n ). (Valiant 79) Computing the permanent of a matrix is #P-hard. Approximation algorithms: (Jerrum,Sinclair,Vigoda 04) FPRAS for nonnegative matrices. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 6 / 21
11 Permanent computation General facts: (Ryser 63) Best general and exact method; complexity O(n 2 n ). (Valiant 79) Computing the permanent of a matrix is #P-hard. Approximation algorithms: (Jerrum,Sinclair,Vigoda 04) FPRAS for nonnegative matrices. Tractable under special structure: (Fisher,Kasteleyn,Temperley 67) Bipartite graph is planar. (Barvinok 96) Rank is bounded. (Courcelle et al 01) Treewidth is bounded. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 6 / 21
12 Graph abstractions of a matrix The sparsity structure of a matrix M can be represented with a graph. Let a 1,..., a n denote the rows and x 1,..., x n denote the columns. Consider these abstractions: G, bipartite adjacency graph: (a i, x j ) E iff M i,j 0 G s, (symmetrized) adjacency graph: (i, j) E iff M i,j + M j,i 0 G X, column graph: (x j, x k ) E iff exists a i such that M i,j M i,k 0. Equivalently, the adjacency graph of M T M. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 7 / 21
13 Graph abstractions of a matrix x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 a a a a a a a a a Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 8 / 21
14 Tree decompositions Definition: A tree decomposition of a graph G = (X, E) is a pair (T, χ), where T is a tree and χ : T {0, 1} X assigns some χ(t) X to each node t of T, that satisfies the following conditions. i. The union of {χ(t)} t T is the whole vertex set X. ii. For every (x i, x j ) E, there is some node t of T with x i, x j χ(t). iii. For every x i X the set {t : x i χ(t)} forms a subtree of T. The width ω of the decomposition is the largest χ(t). Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 9 / 21
15 Chordality and treewidth The treewidth of G is the minimum width among all possible tree decompositions. The treewidth ω(g) of a graph G can be though of as a measure of complexity: the smaller ω(g), the simpler the graph (closer to a tree). Meta-theorem: NP-complete problems are easy on graphs of small treewidth. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
16 Chordality and treewidth The treewidth of G is the minimum width among all possible tree decompositions. The treewidth ω(g) of a graph G can be though of as a measure of complexity: the smaller ω(g), the simpler the graph (closer to a tree). Meta-theorem: NP-complete problems are easy on graphs of small treewidth. Alternatively, we can define treewidth in terms of chordal completions of the graph (chordal completions are in correspondance with tree decompositions). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
17 Treewidth of matrix graphs Simple fact: tw(g) 2 tw(g s ). tw(g) tw(g X ) + 1. For a fixed tw(g) both tw(g s ), tw(g X ) are unbounded. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
18 Treewidth of matrix graphs Simple fact: tw(g) 2 tw(g s ). tw(g) tw(g X ) + 1. For a fixed tw(g) both tw(g s ), tw(g X ) are unbounded. Previous tree decomposition methods are primarily based on the symmetrized graph G s (Courcelle et al., Flarup et al., Meer), and they do not scale well with the treewidth O(n 2 O(ω2) ). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
19 Our results A tree decomposition method to compute permanents, based on the bipartite graph G, with complexity Õ(n 2ω ). The algorithm naturally extends to higher dimensional problems: mixed discriminants, hyperdeterminants. Complexity Õ(n2 + n 3 ω ). Hardness results for the case of mixed volumes. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
20 Our results HDet tensor MVol n polytopes MDisc n matrices MVol n zonotopes (known) Hard Easy Hard (small ω) Easy (small ω) (this paper) Det matrix Perm matrix Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
21 Basic idea For block matrices ( ) ( ) A A 0 A Perm = Perm(A) Perm(B) = Perm 0 B B B Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
22 Permanent expansion: column graph Similarly, for the matrix A 1,1 0 A 1,3 A 1,4 0 A 2,1 0 A 2,3 A 2,4 0 M = 0 C 3,2 C 3,3 C 3,4 0 0 B 4,2 B 4,3 0 B 4,5 0 B 5,2 B 5,3 0 B 5,5 we have the following formula ( ) A1,1 A Perm(M) = Perm 1,3 A 2,1 A 2,3 ( ) A1,1 A + Perm 1,4 A 2,1 A 2,4 ( ) A1,1 A + Perm 1,4 A 2,1 A 2,4 ) Perm (C 3,4 Perm ) Perm (C 3,3 Perm ) Perm (C 3,2 Perm ( ) B4,2 B 4,5 B 5,2 B 5,5 ( B4,2 ) B 4,5 B 5,2 B 5,5 ( ) B4,3 B 4,5. B 5,3 B 5,5 To evaluate we need 14 multiplications: four 2 2 permanetns. Compare with: 4 5! = 480 multiplications using definition. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
23 Permanent expansion: column graph This simple formula exists because its column graph has a simple structure M 1,1 0 M 1,3 M 1,4 0 M 2,1 0 M 2,3 M 2,4 0 0 M 3,2 M 3,3 M 3,4 0 0 M 4,2 M 4,3 0 M 4,5 0 M 5,2 M 5,3 0 M 5,5 Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
24 Results Lemma: Let (T, χ) be a tree decomposition of G X. Let c 1,..., c k be the children of a node t T. Then perm(a Tt, Y ) = k perm(a t, Y t ) perm(a Tcj, Y cj ) Y j=1 where the sum is over all Y = (Y t, Y c1,..., Y ck ) such that: Y = Y t (Y c1 Y ck ) χ(t cj ) \ χ(t) Y cj χ(t cj ) Y t χ(t). Lemma: The above formula can be evaluated in Õ(k 2ω ). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
25 Results Lemma: Let (T, χ) be a tree decomposition of G X. Let c 1,..., c k be the children of a node t T. Then perm(a Tt, Y ) = k perm(a t, Y t ) perm(a Tcj, Y cj ) Y j=1 where the sum is over all Y = (Y t, Y c1,..., Y ck ) such that: Y = Y t (Y c1 Y ck ) χ(t cj ) \ χ(t) Y cj χ(t cj ) Y t χ(t). Lemma: The above formula can be evaluated in Õ(k 2ω ). Proof: Rewrite the equation as P t (Ȳ ) = k Q t (Ȳ t ) Q cj (Ȳ cj ) Ȳ t Ȳ c1 Ȳ ck =Ȳ j=1 and use the fast subset convolution (Björklund et al. 07). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
26 Permanent expansion: bipartite graph Perm(M) = perm({a 1, a 2}, {x 1, x 4}) perm({a 3, a 4, a 5}, {x 2, x 3, x 5}) + perm({a 1, a 2, a 3}, {x 1, x 3, x 4}) perm({a 4, a 5}, {x 2, x 5}) M 3,3 perm({a 1, a 2}, {x 1, x 4}) perm({a 4, a 5}, {x 2, x 5}) perm({a 1, a 2, a 3}, {x 1, x 3, x 4}) = M 3,3 perm({a 1, a 2}, {x 1, x 4}) + M 3,4 perm({a 1, a 2}, {x 1, x 3}) perm({a 3, a 4, a 5}, {x 2, x 3, x 5}) = M 3,3 perm({a 4, a 5}, {x 2, x 5}) + M 3,2 perm({a 4, a 5}, {x 3, x 5}) To evaluate we need 16 multiplications. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
27 Results Theorem: Let M be a matrix with associated bipartite graph G. Then we can compute Perm(M) in Õ(n 2ω ) where ω = tw(g). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
28 Results Theorem: Let M be a matrix with associated bipartite graph G. Then we can compute Perm(M) in Õ(n 2ω ) where ω = tw(g). The mixed discriminant of n matrices of size n n is Disc(A 1,..., A n ) := sgn(π)sgn(ρ) π,ρ i A i π(i),ρ(i) where the sum is over pairs of permutations π, ρ. Theorem: Let M be a list of matrices with associated tripartite graph G. Then we can compute Disc(M) in Õ(n2 + n 3 ω ), where ω = tw(g). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
29 Further generalizations The first Cayley hyperdeterminant is the simplest generalization of the determinant to multidimensional arrays. Theorem: Let M be a square d-dimensional tensor of length n, with d-partite graph G. We can compute its hyperdeterminant in Õ(n2 + n 3 ω ). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
30 Further generalizations The first Cayley hyperdeterminant is the simplest generalization of the determinant to multidimensional arrays. Theorem: Let M be a square d-dimensional tensor of length n, with d-partite graph G. We can compute its hyperdeterminant in Õ(n2 + n 3 ω ). The mixed volume is a geometric generalization of determinants and permanents to arbitrary convex bodies on R n. Theorem: Computing mixed volumes of n zonotopes remains #P-hard when their associated graph has bounded treewidth. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
31 Summary HDet tensor MVol n polytopes MDisc n matrices MVol n zonotopes (known) Hard Easy Hard (small ω) Easy (small ω) (this paper) Det matrix Perm matrix D. Cifuentes, P.A. Parrilo, An efficient tree decomposition method for permanents and mixed discriminants. arxiv: D. Cifuentes, P.A. Parrilo, Exploiting chordal structure in polynomial ideals: a Gröbner basis approach. arxiv: D. Cifuentes, P.A. Parrilo, Chordal networks of polynomial ideals. arxiv: Thanks for your attention! Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21
Chordal structure and polynomial systems
Chordal structure and polynomial systems Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology Joint work with
More informationSta$s$cal models and graphs
Sta$s$cal models and graphs AFOSR FA9550-11-1-0305 Fundamental methodology and applica$ons Convex op$miza$on and graphs as underlying engines Examples: Sparse graph es$ma$on, latent-variable modeling,
More informationGraph structure in polynomial systems: chordal networks
Graph structure in polynomial systems: chordal networks Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
More informationChordal networks of polynomial ideals
Chordal networks of polynomial ideals Diego Cifuentes Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology Joint work with Pablo
More informationGraph structure in polynomial systems: chordal networks
Graph structure in polynomial systems: chordal networks Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
More informationExploiting Chordal Structure in Polynomial Ideals: A Gröbner Bases Approach
Exploiting Chordal Structure in Polynomial Ideals: A Gröbner Bases Approach The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation
More informationThe Complexity of the Permanent and Related Problems
The Complexity of the Permanent and Related Problems Tim Abbott and Alex Schwendner May 9, 2007 Contents 1 The Permanent 2 1.1 Variations on the Determinant...................... 2 1.2 Graph Interpretation...........................
More informationBounded Treewidth Graphs A Survey German Russian Winter School St. Petersburg, Russia
Bounded Treewidth Graphs A Survey German Russian Winter School St. Petersburg, Russia Andreas Krause krausea@cs.tum.edu Technical University of Munich February 12, 2003 This survey gives an introduction
More informationTree-width. September 14, 2015
Tree-width Zdeněk Dvořák September 14, 2015 A tree decomposition of a graph G is a pair (T, β), where β : V (T ) 2 V (G) assigns a bag β(n) to each vertex of T, such that for every v V (G), there exists
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 informationFinding Paths with Minimum Shared Edges in Graphs with Bounded Treewidth
Finding Paths with Minimum Shared Edges in Graphs with Bounded Treewidth Z.-Q. Ye 1, Y.-M. Li 2, H.-Q. Lu 3 and X. Zhou 4 1 Zhejiang University, Hanzhou, Zhejiang, China 2 Wenzhou University, Wenzhou,
More informationThe Graph Realization Problem
The Graph Realization Problem via Semi-Definite Programming A. Y. Alfakih alfakih@uwindsor.ca Mathematics and Statistics University of Windsor The Graph Realization Problem p.1/21 The Graph Realization
More informationMa/CS 6b Class 20: Spectral Graph Theory
Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Recall: Parity of a Permutation S n the set of permutations of 1,2,, n. A permutation σ S n is even if it can be written as a composition of an
More informationOn the chordality of polynomial sets in triangular decomposition in top-down style
On the chordality of polynomial sets in triangular decomposition in top-down style Chenqi Mou joint work with Yang Bai LMIB School of Mathematics and Systems Science Beihang University, China ISSAC 2018,
More informationA tree-decomposed transfer matrix for computing exact partition functions for arbitrary graphs
A tree-decomposed transfer matrix for computing exact partition functions for arbitrary graphs Andrea Bedini 1 Jesper L. Jacobsen 2 1 MASCOS, The University of Melbourne, Melbourne 2 LPTENS, École Normale
More informationMa/CS 6b Class 20: Spectral Graph Theory
Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Eigenvalues and Eigenvectors A an n n matrix of real numbers. The eigenvalues of A are the numbers λ such that Ax = λx for some nonzero vector x
More informationClassical Complexity and Fixed-Parameter Tractability of Simultaneous Consecutive Ones Submatrix & Editing Problems
Classical Complexity and Fixed-Parameter Tractability of Simultaneous Consecutive Ones Submatrix & Editing Problems Rani M. R, Mohith Jagalmohanan, R. Subashini Binary matrices having simultaneous consecutive
More informationSampling Contingency Tables
Sampling Contingency Tables Martin Dyer Ravi Kannan John Mount February 3, 995 Introduction Given positive integers and, let be the set of arrays with nonnegative integer entries and row sums respectively
More informationAlgorithms and Theory of Computation. Lecture 22: NP-Completeness (2)
Algorithms and Theory of Computation Lecture 22: NP-Completeness (2) Xiaohui Bei MAS 714 November 8, 2018 Nanyang Technological University MAS 714 November 8, 2018 1 / 20 Set Cover Set Cover Input: a set
More informationDichotomy Theorems for Counting Problems. Jin-Yi Cai University of Wisconsin, Madison. Xi Chen Columbia University. Pinyan Lu Microsoft Research Asia
Dichotomy Theorems for Counting Problems Jin-Yi Cai University of Wisconsin, Madison Xi Chen Columbia University Pinyan Lu Microsoft Research Asia 1 Counting Problems Valiant defined the class #P, and
More informationNonlinear Discrete Optimization
Nonlinear Discrete Optimization Technion Israel Institute of Technology http://ie.technion.ac.il/~onn Billerafest 2008 - conference in honor of Lou Billera's 65th birthday (Update on Lecture Series given
More informationRobust Principal Component Analysis
ELE 538B: Mathematics of High-Dimensional Data Robust Principal Component Analysis Yuxin Chen Princeton University, Fall 2018 Disentangling sparse and low-rank matrices Suppose we are given a matrix M
More informationNonnegative Matrices I
Nonnegative Matrices I Daisuke Oyama Topics in Economic Theory September 26, 2017 References J. L. Stuart, Digraphs and Matrices, in Handbook of Linear Algebra, Chapter 29, 2006. R. A. Brualdi and H. J.
More informationSemidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5 Instructor: Farid Alizadeh Scribe: Anton Riabov 10/08/2001 1 Overview We continue studying the maximum eigenvalue SDP, and generalize
More informationTree Decompositions and Tree-Width
Tree Decompositions and Tree-Width CS 511 Iowa State University December 6, 2010 CS 511 (Iowa State University) Tree Decompositions and Tree-Width December 6, 2010 1 / 15 Tree Decompositions Definition
More informationThe Strong Largeur d Arborescence
The Strong Largeur d Arborescence Rik Steenkamp (5887321) November 12, 2013 Master Thesis Supervisor: prof.dr. Monique Laurent Local Supervisor: prof.dr. Alexander Schrijver KdV Institute for Mathematics
More informationACO Comprehensive Exam October 18 and 19, Analysis of Algorithms
Consider the following two graph problems: 1. Analysis of Algorithms Graph coloring: Given a graph G = (V,E) and an integer c 0, a c-coloring is a function f : V {1,,...,c} such that f(u) f(v) for all
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 informationPCPs and Inapproximability Gap-producing and Gap-Preserving Reductions. My T. Thai
PCPs and Inapproximability Gap-producing and Gap-Preserving Reductions My T. Thai 1 1 Hardness of Approximation Consider a maximization problem Π such as MAX-E3SAT. To show that it is NP-hard to approximation
More informationSemidefinite programming lifts and sparse sums-of-squares
1/15 Semidefinite programming lifts and sparse sums-of-squares Hamza Fawzi (MIT, LIDS) Joint work with James Saunderson (UW) and Pablo Parrilo (MIT) Cornell ORIE, October 2015 Linear optimization 2/15
More informationOn the local stability of semidefinite relaxations
On the local stability of semidefinite relaxations Diego Cifuentes Department of Mathematics Massachusetts Institute of Technology Joint work with Sameer Agarwal (Google), Pablo Parrilo (MIT), Rekha Thomas
More informationDimension reduction for semidefinite programming
1 / 22 Dimension reduction for semidefinite programming Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
More informationComputation of the Joint Spectral Radius with Optimization Techniques
Computation of the Joint Spectral Radius with Optimization Techniques Amir Ali Ahmadi Goldstine Fellow, IBM Watson Research Center Joint work with: Raphaël Jungers (UC Louvain) Pablo A. Parrilo (MIT) R.
More informationA Fixed-Parameter Algorithm for Max Edge Domination
A Fixed-Parameter Algorithm for Max Edge Domination Tesshu Hanaka and Hirotaka Ono Department of Economic Engineering, Kyushu University, Fukuoka 812-8581, Japan ono@cscekyushu-uacjp Abstract In a graph,
More informationAcyclic and Oriented Chromatic Numbers of Graphs
Acyclic and Oriented Chromatic Numbers of Graphs A. V. Kostochka Novosibirsk State University 630090, Novosibirsk, Russia X. Zhu Dept. of Applied Mathematics National Sun Yat-Sen University Kaohsiung,
More informationACO Comprehensive Exam October 14 and 15, 2013
1. Computability, Complexity and Algorithms (a) Let G be the complete graph on n vertices, and let c : V (G) V (G) [0, ) be a symmetric cost function. Consider the following closest point heuristic for
More informationTopics in Graph Theory
Topics in Graph Theory September 4, 2018 1 Preliminaries A graph is a system G = (V, E) consisting of a set V of vertices and a set E (disjoint from V ) of edges, together with an incidence function End
More informationProblem 1. Let f n (x, y), n Z, be the sequence of rational functions in two variables x and y given by the initial conditions.
18.217 Problem Set (due Monday, December 03, 2018) Solve as many problems as you want. Turn in your favorite solutions. You can also solve and turn any other claims that were given in class without proofs,
More informationCS 301: Complexity of Algorithms (Term I 2008) Alex Tiskin Harald Räcke. Hamiltonian Cycle. 8.5 Sequencing Problems. Directed Hamiltonian Cycle
8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE,
More informationAdvanced Combinatorial Optimization Feb 13 and 25, Lectures 4 and 6
18.438 Advanced Combinatorial Optimization Feb 13 and 25, 2014 Lectures 4 and 6 Lecturer: Michel X. Goemans Scribe: Zhenyu Liao and Michel X. Goemans Today, we will use an algebraic approach to solve the
More informationLattice polygons. P : lattice polygon in R 2 (vertices Z 2, no self-intersections)
Lattice polygons P : lattice polygon in R 2 (vertices Z 2, no self-intersections) A, I, B A = area of P I = # interior points of P (= 4) B = #boundary points of P (= 10) Pick s theorem Georg Alexander
More informationTensors. Lek-Heng Lim. Statistics Department Retreat. October 27, Thanks: NSF DMS and DMS
Tensors Lek-Heng Lim Statistics Department Retreat October 27, 2012 Thanks: NSF DMS 1209136 and DMS 1057064 L.-H. Lim (Stat Retreat) Tensors October 27, 2012 1 / 20 tensors on one foot a tensor is a multilinear
More informationAn Algorithmist s Toolkit September 24, Lecture 5
8.49 An Algorithmist s Toolkit September 24, 29 Lecture 5 Lecturer: Jonathan Kelner Scribe: Shaunak Kishore Administrivia Two additional resources on approximating the permanent Jerrum and Sinclair s original
More informationHanna Furmańczyk EQUITABLE COLORING OF GRAPH PRODUCTS
Opuscula Mathematica Vol. 6 No. 006 Hanna Furmańczyk EQUITABLE COLORING OF GRAPH PRODUCTS Abstract. A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a
More informationSpectra of Adjacency and Laplacian Matrices
Spectra of Adjacency and Laplacian Matrices Definition: University of Alicante (Spain) Matrix Computing (subject 3168 Degree in Maths) 30 hours (theory)) + 15 hours (practical assignment) Contents 1. Spectra
More informationComplexity of Inference in Graphical Models
Complexity of Inference in Graphical Models Venkat Chandrasekaran 1, Nathan Srebro 2, and Prahladh Harsha 3 1 Laboratory for Information and Decision Systems Department of Electrical Engineering and Computer
More informationA Geometric Approach to Graph Isomorphism
A Geometric Approach to Graph Isomorphism Pawan Aurora and Shashank K Mehta Indian Institute of Technology, Kanpur - 208016, India {paurora,skmehta}@cse.iitk.ac.in Abstract. We present an integer linear
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 19
832: Algebraic Combinatorics Lionel Levine Lecture date: April 2, 20 Lecture 9 Notes by: David Witmer Matrix-Tree Theorem Undirected Graphs Let G = (V, E) be a connected, undirected graph with n vertices,
More informationarxiv: v1 [cs.dm] 22 Jan 2008
On the expressive power of permanents and perfect matchings of matrices of bounded pathwidth/cliquewidth Uffe Flarup 1, Laurent Lyaudet 2 arxiv:0801.3408v1 [cs.dm] 22 Jan 2008 1 Department of Mathematics
More informationFine Grained Counting Complexity I
Fine Grained Counting Complexity I Holger Dell Saarland University and Cluster of Excellence (MMCI) & Simons Institute for the Theory of Computing 1 50 Shades of Fine-Grained #W[1] W[1] fixed-parameter
More informationRobust and Optimal Control, Spring 2015
Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5-303B) G. Sum of Squares (SOS) G.1 SOS Program: SOS/PSD and SDP G.2 Duality, valid ineqalities and Cone G.3 Feasibility/Optimization
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 informationTractable Approximations of Graph Isomorphism.
Tractable Approximations of Graph Isomorphism. Anuj Dawar University of Cambridge Computer Laboratory visiting RWTH Aachen joint work with Bjarki Holm Darmstadt, 5 February 2014 Graph Isomorphism Graph
More informationApproximability and Parameterized Complexity of Consecutive Ones Submatrix Problems
Proc. 4th TAMC, 27 Approximability and Parameterized Complexity of Consecutive Ones Submatrix Problems Michael Dom, Jiong Guo, and Rolf Niedermeier Institut für Informatik, Friedrich-Schiller-Universität
More informationCleaning Interval Graphs
Cleaning Interval Graphs Dániel Marx and Ildikó Schlotter Department of Computer Science and Information Theory, Budapest University of Technology and Economics, H-1521 Budapest, Hungary. {dmarx,ildi}@cs.bme.hu
More informationMa/CS 6b Class 23: Eigenvalues in Regular Graphs
Ma/CS 6b Class 3: Eigenvalues in Regular Graphs By Adam Sheffer Recall: The Spectrum of a Graph Consider a graph G = V, E and let A be the adjacency matrix of G. The eigenvalues of G are the eigenvalues
More information1 Matroid intersection
CS 369P: Polyhedral techniques in combinatorial optimization Instructor: Jan Vondrák Lecture date: October 21st, 2010 Scribe: Bernd Bandemer 1 Matroid intersection Given two matroids M 1 = (E, I 1 ) and
More informationarxiv: v3 [cs.dm] 18 Jan 2018
On H-topological intersection graphs Steven Chaplick 1, Martin Töpfer 2, Jan Voborník 3, and Peter Zeman 3 1 Lehrstuhl für Informatik I, Universität Würzburg, Germany, www1.informatik.uni-wuerzburg.de/en/staff,
More informationThe Maximum Likelihood Threshold of a Graph
The Maximum Likelihood Threshold of a Graph Elizabeth Gross and Seth Sullivant San Jose State University, North Carolina State University August 28, 2014 Seth Sullivant (NCSU) Maximum Likelihood Threshold
More informationConvex sets, conic matrix factorizations and conic rank lower bounds
Convex sets, conic matrix factorizations and conic rank lower bounds Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute
More informationarxiv: v2 [math.co] 7 Jan 2019
Clique-Width for Hereditary Graph Classes Konrad K. Dabrowski, Matthew Johnson and Daniël Paulusma arxiv:1901.00335v2 [math.co] 7 Jan 2019 Abstract Clique-width is a well-studied graph parameter owing
More informationCost-Constrained Matchings and Disjoint Paths
Cost-Constrained Matchings and Disjoint Paths Kenneth A. Berman 1 Department of ECE & Computer Science University of Cincinnati, Cincinnati, OH Abstract Let G = (V, E) be a graph, where the edges are weighted
More informationPositive semidefinite rank
1/15 Positive semidefinite rank Hamza Fawzi (MIT, LIDS) Joint work with João Gouveia (Coimbra), Pablo Parrilo (MIT), Richard Robinson (Microsoft), James Saunderson (Monash), Rekha Thomas (UW) DIMACS Workshop
More informationTropical decomposition of symmetric tensors
Tropical decomposition of symmetric tensors Melody Chan University of California, Berkeley mtchan@math.berkeley.edu December 11, 008 1 Introduction In [], Comon et al. give an algorithm for decomposing
More informationThe partial-fractions method for counting solutions to integral linear systems
The partial-fractions method for counting solutions to integral linear systems Matthias Beck, MSRI www.msri.org/people/members/matthias/ arxiv: math.co/0309332 Vector partition functions A an (m d)-integral
More informationALGEBRA: From Linear to Non-Linear. Bernd Sturmfels University of California at Berkeley
ALGEBRA: From Linear to Non-Linear Bernd Sturmfels University of California at Berkeley John von Neumann Lecture, SIAM Annual Meeting, Pittsburgh, July 13, 2010 Undergraduate Linear Algebra All undergraduate
More informationMIT Algebraic techniques and semidefinite optimization February 14, Lecture 3
MI 6.97 Algebraic techniques and semidefinite optimization February 4, 6 Lecture 3 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture, we will discuss one of the most important applications
More informationCombinatorial Optimization
Combinatorial Optimization 2017-2018 1 Maximum matching on bipartite graphs Given a graph G = (V, E), find a maximum cardinal matching. 1.1 Direct algorithms Theorem 1.1 (Petersen, 1891) A matching M is
More informationPreliminaries and Complexity Theory
Preliminaries and Complexity Theory Oleksandr Romanko CAS 746 - Advanced Topics in Combinatorial Optimization McMaster University, January 16, 2006 Introduction Book structure: 2 Part I Linear Algebra
More informationTree-chromatic number
Tree-chromatic number Paul Seymour 1 Princeton University, Princeton, NJ 08544 November 2, 2014; revised June 25, 2015 1 Supported by ONR grant N00014-10-1-0680 and NSF grant DMS-1265563. Abstract Let
More informationRobin Thomas and Peter Whalen. School of Mathematics Georgia Institute of Technology Atlanta, Georgia , USA
Odd K 3,3 subdivisions in bipartite graphs 1 Robin Thomas and Peter Whalen School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332-0160, USA Abstract We prove that every internally
More informationParameterised Subgraph Counting Problems
Parameterised Subgraph Counting Problems Kitty Meeks University of Glasgow University of Strathclyde, 27th May 2015 Joint work with Mark Jerrum (QMUL) What is a counting problem? Decision problems Given
More informationDIRAC S THEOREM ON CHORDAL GRAPHS
Seminar Series in Mathematics: Algebra 200, 1 7 DIRAC S THEOREM ON CHORDAL GRAPHS Let G be a finite graph. Definition 1. The graph G is called chordal, if each cycle of length has a chord. Theorem 2 (Dirac).
More informationGraph coloring, perfect graphs
Lecture 5 (05.04.2013) Graph coloring, perfect graphs Scribe: Tomasz Kociumaka Lecturer: Marcin Pilipczuk 1 Introduction to graph coloring Definition 1. Let G be a simple undirected graph and k a positive
More informationTree-width and algorithms
Tree-width and algorithms Zdeněk Dvořák September 14, 2015 1 Algorithmic applications of tree-width Many problems that are hard in general become easy on trees. For example, consider the problem of finding
More informationSparsity of Matrix Canonical Forms. Xingzhi Zhan East China Normal University
Sparsity of Matrix Canonical Forms Xingzhi Zhan zhan@math.ecnu.edu.cn East China Normal University I. Extremal sparsity of the companion matrix of a polynomial Joint work with Chao Ma The companion matrix
More informationThe maximal stable set problem : Copositive programming and Semidefinite Relaxations
The maximal stable set problem : Copositive programming and Semidefinite Relaxations Kartik Krishnan Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 USA kartis@rpi.edu
More informationLecture 3: Voronoi Diagrams & VCP Preliminaries July 24, 2013
Blockcourse on High-Dimensional Computational Geometry Spring 2013 Lecture 3: Voronoi Diagrams & VCP Preliminaries July 24, 2013 Wolfgang Mulzer, Helmut Alt Scribes: Simon Tippenhauer, Terese Haimberger
More informationA multivariate interlace polynomial
A multivariate interlace polynomial Bruno Courcelle LaBRI, Université Bordeaux 1 and CNRS General objectives : Logical descriptions of graph polynomials Application to their computations Systematic construction
More informationDistance Geometry-Matrix Completion
Christos Konaxis Algs in Struct BioInfo 2010 Outline Outline Tertiary structure Incomplete data Tertiary structure Measure diffrence of matched sets Def. Root Mean Square Deviation RMSD = 1 n x i y i 2,
More informationSome Results on Matchgates and Holographic Algorithms. Jin-Yi Cai Vinay Choudhary University of Wisconsin, Madison. NSF CCR and CCR
Some Results on Matchgates and Holographic Algorithms Jin-Yi Cai Vinay Choudhary University of Wisconsin, Madison NSF CCR-0208013 and CCR-0511679. 1 #P Counting problems: #SAT: How many satisfying assignments
More informationAlgorithmic Graph Theory (SS2016)
Algorithmic Graph Theory (SS2016) Chapter 4 Treewidth and more Walter Unger Lehrstuhl für Informatik 1 19:00, June 15, 2016 4 Inhaltsverzeichnis Walter Unger 15.6.2016 19:00 SS2016 Z Contents I 1 Bandwidth
More informationRECAP How to find a maximum matching?
RECAP How to find a maximum matching? First characterize maximum matchings A maximal matching cannot be enlarged by adding another edge. A maximum matching of G is one of maximum size. Example. Maximum
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 informationMathematical Programs Linear Program (LP)
Mathematical Programs Linear Program (LP) Integer Program (IP) Can be efficiently solved e.g., by Ellipsoid Method Cannot be efficiently solved Cannot be efficiently solved assuming P NP Combinatorial
More informationMATHEMATICAL ENGINEERING TECHNICAL REPORTS. Exact SDP Relaxations with Truncated Moment Matrix for Binary Polynomial Optimization Problems
MATHEMATICAL ENGINEERING TECHNICAL REPORTS Exact SDP Relaxations with Truncated Moment Matrix for Binary Polynomial Optimization Problems Shinsaku SAKAUE, Akiko TAKEDA, Sunyoung KIM, and Naoki ITO METR
More informationBounds on parameters of minimally non-linear patterns
Bounds on parameters of minimally non-linear patterns P.A. CrowdMath Department of Mathematics Massachusetts Institute of Technology Massachusetts, U.S.A. crowdmath@artofproblemsolving.com Submitted: Jan
More informationPrecoloring extension on chordal graphs
Precoloring extension on chordal graphs Dániel Marx 17th August 2004 Abstract In the precoloring extension problem (PrExt) we are given a graph with some of the vertices having a preassigned color and
More informationEfficient Reassembling of Graphs, Part 1: The Linear Case
Efficient Reassembling of Graphs, Part 1: The Linear Case Assaf Kfoury Boston University Saber Mirzaei Boston University Abstract The reassembling of a simple connected graph G = (V, E) is an abstraction
More informationCollective Tree Spanners of Unit Disk Graphs with Applications to Compact and Low Delay Routing Labeling Schemes. F.F. Dragan
Collective Tree Spanners of Unit Disk Graphs with Applications to Compact and Low Delay Routing Labeling Schemes F.F. Dragan (joint work with Yang Xiang and Chenyu Yan) Kent State University, USA Well-known
More informationMath 671: Tensor Train decomposition methods
Math 671: Eduardo Corona 1 1 University of Michigan at Ann Arbor December 8, 2016 Table of Contents 1 Preliminaries and goal 2 Unfolding matrices for tensorized arrays The Tensor Train decomposition 3
More informationMatroids and submodular optimization
Matroids and submodular optimization Attila Bernáth Research fellow, Institute of Informatics, Warsaw University 23 May 2012 Attila Bernáth () Matroids and submodular optimization 23 May 2012 1 / 10 About
More informationLecture 14: Random Walks, Local Graph Clustering, Linear Programming
CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 14: Random Walks, Local Graph Clustering, Linear Programming Lecturer: Shayan Oveis Gharan 3/01/17 Scribe: Laura Vonessen Disclaimer: These
More informationComputing branchwidth via efficient triangulations and blocks
Computing branchwidth via efficient triangulations and blocks Fedor Fomin Frédéric Mazoit Ioan Todinca Abstract Minimal triangulations and potential maximal cliques are the main ingredients for a number
More informationLecture 22: Counting
CS 710: Complexity Theory 4/8/2010 Lecture 22: Counting Instructor: Dieter van Melkebeek Scribe: Phil Rydzewski & Chi Man Liu Last time we introduced extractors and discussed two methods to construct them.
More informationarxiv: v2 [cs.dm] 2 Feb 2016
The VC-Dimension of Graphs with Respect to k-connected Subgraphs Andrea Munaro Laboratoire G-SCOP, Université Grenoble Alpes arxiv:130.6500v [cs.dm] Feb 016 Abstract We study the VC-dimension of the set
More informationApproximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials
Electronic Colloquium on Computational Complexity, Report No. 124 (2010) Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials Zhixiang Chen Bin
More informationApproximating the Permanent via Nonabelian Determinants
Approximating the Permanent via Nonabelian Determinants Cristopher Moore University of New Mexico & Santa Fe Institute joint work with Alexander Russell University of Connecticut Determinant and Permanent
More informationGraph Minor Theory. Sergey Norin. March 13, Abstract Lecture notes for the topics course on Graph Minor theory. Winter 2017.
Graph Minor Theory Sergey Norin March 13, 2017 Abstract Lecture notes for the topics course on Graph Minor theory. Winter 2017. Contents 1 Background 2 1.1 Minors.......................................
More informationSeparation Techniques for Constrained Nonlinear 0 1 Programming
Separation Techniques for Constrained Nonlinear 0 1 Programming Christoph Buchheim Computer Science Department, University of Cologne and DEIS, University of Bologna MIP 2008, Columbia University, New
More information