Spanning trees and the complexity of flood-filling games
|
|
|
- Andrea Matthews
- 5 years ago
- Views:
Transcription
1 Spanning trees and the complexity of flood-filling games Kitty Meeks Alex Scott Mathematical Institute University of Oxford FUN 2012, Venice
2 The original Flood-It game
3 The original Flood-It game
4 The original Flood-It game
5 The original Flood-It game
6 The original Flood-It game
7 The original Flood-It game
8 The original Flood-It game
9 The original Flood-It game
10 The Honey-Bee game
11 Generalising to arbitrary graphs
12 Generalising to arbitrary graphs Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com
13 Generalising to arbitrary graphs Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com
14 Generalising to arbitrary graphs Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com
15 Generalising to arbitrary graphs Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com
16 Generalising to arbitrary graphs Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com
17 Generalising to arbitrary graphs Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com
18 Generalising to arbitrary graphs Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com
19 Generalising to arbitrary graphs Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com
20 Existing results concerning Free Flood It Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph.
21 Existing results concerning Free Flood It Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n n or 3 n grids,
22 Existing results concerning Free Flood It Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n n or 3 n grids, trees,
23 Existing results concerning Free Flood It Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n n or 3 n grids, trees, series-parallel graphs.
24 Existing results concerning Free Flood It Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n n or 3 n grids, trees, series-parallel graphs. Free Flood It can be solved in polynomial time when restricted to paths cycles co-comparability graphs
25 Existing results concerning Free Flood It Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n n or 3 n grids, trees, series-parallel graphs. Free Flood It can be solved in polynomial time when restricted to or paths cycles co-comparability graphs if only two colours are used
26 Connecting pairs of vertices G v u
27 Connecting pairs of vertices G v u Using this fact, we can compute in time O( V 3 E c 2 ) the number of moves required to connect any given pair of vertices in a graph G = (V, E) coloured with c colours.
28 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
29 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
30 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
31 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
32 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
33 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
34 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
35 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
36 Spanning trees Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.
37 Proof of spanning trees result: key step A B The number of moves required to flood T with colour d is at most the sum of the numbers of moves required to flood A and B respectively with colour d.
38 This is useless! In general, a graph has an exponential number of spanning trees.
39 This is useless! In general, a graph has an exponential number of spanning trees. Besides, Free Flood It is still NP-hard even on trees.
40 ... or is it?
41 ... or is it? P = NP
42 ... or is it? P = NP
43 ... or is it? Source: finditinscotland.com
44 ... or is it? Source: finditinscotland.com
45 ... or is it? A B The number of moves required to flood G with colour d is at most the sum of the numbers of moves required to flood A and B respectively with colour d.
46 ... or is it? H The number of moves required to flood a subgraph doesn t increase when we play in a larger graph.
47 ... or is it? G H The number of moves required to flood a subgraph doesn t increase when we play in a larger graph.
48 Application I: Graphs with polynomially many connected subgraphs Theorem Free Flood It can be solved in polynomial time on graphs that have only a polynomial number of connected subgraphs. A
49 Application I: Graphs with polynomially many connected subgraphs Theorem Free Flood It can be solved in polynomial time on graphs that have only a polynomial number of connected subgraphs. A A 1 A 2
50 Application I: Graphs with polynomially many connected subgraphs Classes of graphs with only a polynomial number of connected subgraphs include: paths cycles
51 Application I: Graphs with polynomially many connected subgraphs Classes of graphs with only a polynomial number of connected subgraphs include: paths cycles subdivisions of any fixed graph H
52 Application II: Connecting k points Given a coloured graph G and a subset U of at most k vertices, k-linking Flood It is the problem of determining the number of moves required to create a single monochromatic component containing U. Theorem k-linking Flood It can be solved in time O( V k+3 E c 2 2 k ) on a graph G = (V, E) coloured with c colours.
53 Application II: Connecting k points G The number of moves required to connect U is equal to the minimum, taken over all subtrees T of G that contain U, of the number of moves required to flood T.
54 Application II: Connecting k points G
55 Application II: Connecting k points G
56 Conclusions We can analyse flood filling problems by considering only trees.
57 Conclusions We can analyse flood filling problems by considering only trees. This allows us to prove nice complexity results:
58 Conclusions We can analyse flood filling problems by considering only trees. This allows us to prove nice complexity results: Free Flood It is solvable in polynomial time on graphs with polynomially many connected subgraphs.
59 Conclusions We can analyse flood filling problems by considering only trees. This allows us to prove nice complexity results: Free Flood It is solvable in polynomial time on graphs with polynomially many connected subgraphs. k-linking Flood It is solvable in polynomial time on arbitrary graphs (for fixed k).
60 Open Problems Is k-linking Flood It fixed parameter tractable, parameterised by k?
61 Open Problems Is k-linking Flood It fixed parameter tractable, parameterised by k? On what other minor-closed classes of trees is Free Flood It solvable in polynomial time?
62 Open Problems Is k-linking Flood It fixed parameter tractable, parameterised by k? On what other minor-closed classes of trees is Free Flood It solvable in polynomial time? Extremal problems...
63 Open Problems Is k-linking Flood It fixed parameter tractable, parameterised by k? On what other minor-closed classes of trees is Free Flood It solvable in polynomial time? Extremal problems... Does the Loch Ness Monster exist?
64 Thank you
Copyright 2013 Springer Science+Business Media New York
Meeks, K., and Scott, A. (2014) Spanning trees and the complexity of floodfilling games. Theory of Computing Systems, 54 (4). pp. 731-753. ISSN 1432-4350 Copyright 2013 Springer Science+Business Media
July 18, Approximation Algorithms (Travelling Salesman Problem)
Approximation Algorithms (Travelling Salesman Problem) July 18, 2014 The travelling-salesman problem Problem: given complete, undirected graph G = (V, E) with non-negative integer cost c(u, v) for each
Dynamic 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
VIII. NP-completeness
VIII. NP-completeness 1 / 15 NP-Completeness Overview 1. Introduction 2. P and NP 3. NP-complete (NPC): formal definition 4. How to prove a problem is NPC 5. How to solve a NPC problem: approximate algorithms
NP-Complete Problems and Approximation Algorithms
NP-Complete Problems and Approximation Algorithms Efficiency of Algorithms Algorithms that have time efficiency of O(n k ), that is polynomial of the input size, are considered to be tractable or easy
COSE215: Theory of Computation. Lecture 20 P, NP, and NP-Complete Problems
COSE215: Theory of Computation Lecture 20 P, NP, and NP-Complete Problems Hakjoo Oh 2018 Spring Hakjoo Oh COSE215 2018 Spring, Lecture 20 June 6, 2018 1 / 14 Contents 1 P and N P Polynomial-time reductions
EXERCISES SHORTEST PATHS: APPLICATIONS, OPTIMIZATION, VARIATIONS, AND SOLVING THE CONSTRAINED SHORTEST PATH PROBLEM. 1 Applications and Modelling
SHORTEST PATHS: APPLICATIONS, OPTIMIZATION, VARIATIONS, AND SOLVING THE CONSTRAINED SHORTEST PATH PROBLEM EXERCISES Prepared by Natashia Boland 1 and Irina Dumitrescu 2 1 Applications and Modelling 1.1
Tree 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
Parameterised 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
CSC 1700 Analysis of Algorithms: P and NP Problems
CSC 1700 Analysis of Algorithms: P and NP Problems Professor Henry Carter Fall 2016 Recap Algorithmic power is broad but limited Lower bounds determine whether an algorithm can be improved by more than
1 Notation. 2 Sergey Norin OPEN PROBLEMS
OPEN PROBLEMS 1 Notation Throughout, v(g) and e(g) mean the number of vertices and edges of a graph G, and ω(g) and χ(g) denote the maximum cardinality of a clique of G and the chromatic number of G. 2
CSC 8301 Design & Analysis of Algorithms: Lower Bounds
CSC 8301 Design & Analysis of Algorithms: Lower Bounds Professor Henry Carter Fall 2016 Recap Iterative improvement algorithms take a feasible solution and iteratively improve it until optimized Simplex
The Complexity of Computing the Sign of the Tutte Polynomial
The Complexity of Computing the Sign of the Tutte Polynomial Leslie Ann Goldberg (based on joint work with Mark Jerrum) Oxford Algorithms Workshop, October 2012 The Tutte polynomial of a graph G = (V,
COSE215: Theory of Computation. Lecture 21 P, NP, and NP-Complete Problems
COSE215: Theory of Computation Lecture 21 P, NP, and NP-Complete Problems Hakjoo Oh 2017 Spring Hakjoo Oh COSE215 2017 Spring, Lecture 21 June 11, 2017 1 / 11 Contents 1 The classes P and N P Reductions
Tree-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
Matroids 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
The Lefthanded Local Lemma characterizes chordal dependency graphs
The Lefthanded Local Lemma characterizes chordal dependency graphs Wesley Pegden March 30, 2012 Abstract Shearer gave a general theorem characterizing the family L of dependency graphs labeled with probabilities
ECS122A Handout on NP-Completeness March 12, 2018
ECS122A Handout on NP-Completeness March 12, 2018 Contents: I. Introduction II. P and NP III. NP-complete IV. How to prove a problem is NP-complete V. How to solve a NP-complete problem: approximate algorithms
Tractability. Some problems are intractable: as they grow large, we are unable to solve them in reasonable time What constitutes reasonable time?
Tractability Some problems are intractable: as they grow large, we are unable to solve them in reasonable time What constitutes reasonable time?» Standard working definition: polynomial time» On an input
15.1 Matching, Components, and Edge cover (Collaborate with Xin Yu)
15.1 Matching, Components, and Edge cover (Collaborate with Xin Yu) First show l = c by proving l c and c l. For a maximum matching M in G, let V be the set of vertices covered by M. Since any vertex in
NP-Completeness. f(n) \ n n sec sec sec. n sec 24.3 sec 5.2 mins. 2 n sec 17.9 mins 35.
NP-Completeness Reference: Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson, W.H. Freeman and Company, 1979. NP-Completeness 1 General Problems, Input Size and
Contents. Typical techniques. Proving hardness. Constructing efficient algorithms
Contents Typical techniques Proving hardness Constructing efficient algorithms Generating Maximal Independent Sets Consider the generation of all maximal sets of an independence system. Generating Maximal
A Dynamic Programming algorithm for minimizing total cost of duplication in scheduling an outtree with communication delays and duplication
A Dynamic Programming algorithm for minimizing total cost of duplication in scheduling an outtree with communication delays and duplication Claire Hanen Laboratory LIP6 4, place Jussieu F-75 252 Paris
Computer Science 385 Analysis of Algorithms Siena College Spring Topic Notes: Limitations of Algorithms
Computer Science 385 Analysis of Algorithms Siena College Spring 2011 Topic Notes: Limitations of Algorithms We conclude with a discussion of the limitations of the power of algorithms. That is, what kinds
1. How many labeled trees are there on n vertices such that all odd numbered vertices are leaves?
1. How many labeled trees are there on n vertices such that all odd numbered vertices are leaves? This is most easily done by Prüfer codes. The number of times a vertex appears is the degree of the vertex.
Pareto Optimality in Coalition Formation
Pareto Optimality in Coalition Formation Haris Aziz Felix Brandt Paul Harrenstein Technische Universität München IJCAI Workshop on Social Choice and Artificial Intelligence, July 16, 2011 1 / 21 Coalition
Spectral Graph Theory and You: Matrix Tree Theorem and Centrality Metrics
Spectral Graph Theory and You: and Centrality Metrics Jonathan Gootenberg March 11, 2013 1 / 19 Outline of Topics 1 Motivation Basics of Spectral Graph Theory Understanding the characteristic polynomial
Computational Complexity
Computational Complexity Problems, instances and algorithms Running time vs. computational complexity General description of the theory of NP-completeness Problem samples 1 Computational Complexity What
More on NP and Reductions
Indian Institute of Information Technology Design and Manufacturing, Kancheepuram Chennai 600 127, India An Autonomous Institute under MHRD, Govt of India http://www.iiitdm.ac.in COM 501 Advanced Data
Combinatorial and physical content of Kirchhoff polynomials
Combinatorial and physical content of Kirchhoff polynomials Karen Yeats May 19, 2009 Spanning trees Let G be a connected graph, potentially with multiple edges and loops in the sense of a graph theorist.
Disjoint paths in unions of tournaments
Disjoint paths in unions of tournaments Maria Chudnovsky 1 Princeton University, Princeton, NJ 08544, USA Alex Scott Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK Paul Seymour 2 Princeton
ACO 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
Computing Minmax; Dominance
Computing Minmax; Dominance CPSC 532A Lecture 5 Computing Minmax; Dominance CPSC 532A Lecture 5, Slide 1 Lecture Overview 1 Recap 2 Linear Programming 3 Computational Problems Involving Maxmin 4 Domination
P, NP, NP-Complete, and NPhard
P, NP, NP-Complete, and NPhard Problems Zhenjiang Li 21/09/2011 Outline Algorithm time complicity P and NP problems NP-Complete and NP-Hard problems Algorithm time complicity Outline What is this course
An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees
An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees Francesc Rosselló 1, Gabriel Valiente 2 1 Department of Mathematics and Computer Science, Research Institute
CS 320, Fall Dr. Geri Georg, Instructor 320 NP 1
NP CS 320, Fall 2017 Dr. Geri Georg, Instructor georg@colostate.edu 320 NP 1 NP Complete A class of problems where: No polynomial time algorithm has been discovered No proof that one doesn t exist 320
Reconstructing sequences
Reconstructing sequences A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, England. Abstract. We prove that every sequence
Tree-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
Some hard families of parameterised counting problems
Some hard families of parameterised counting problems Mark Jerrum and Kitty Meeks School of Mathematical Sciences, Queen Mary University of London {m.jerrum,k.meeks}@qmul.ac.uk September 2014 Abstract
arxiv: v3 [cs.dm] 18 Jan 2018
![arxiv: v3 [cs.dm] 18 Jan 2018](/thumbs/81/83422292.jpg "arxiv: 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,
COMP-330 Theory of Computation. Fall Prof. Claude Crépeau. Lecture 2 : Regular Expressions & DFAs
COMP-330 Theory of Computation Fall 2017 -- Prof. Claude Crépeau Lecture 2 : Regular Expressions & DFAs COMP 330 Fall 2017: Lectures Schedule 1-2. Introduction 1.5. Some basic mathematics 2-3. Deterministic
R(p, k) = the near regular complete k-partite graph of order p. Let p = sk+r, where s and r are positive integers such that 0 r < k.
MATH3301 EXTREMAL GRAPH THEORY Definition: A near regular complete multipartite graph is a complete multipartite graph with orders of its partite sets differing by at most 1. R(p, k) = the near regular
P vs. NP. Data Structures and Algorithms CSE AU 1
P vs. NP Data Structures and Algorithms CSE 373-18AU 1 Goals for today Define P, NP, and NP-complete Explain the P vs. NP problem -why it s the biggest open problem in CS. -And what to do when a problem
CS 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,
Tractable & Intractable Problems
Tractable & Intractable Problems We will be looking at : What is a P and NP problem NP-Completeness The question of whether P=NP The Traveling Salesman problem again Programming and Data Structures 1 Polynomial
Computational Complexity and Intractability: An Introduction to the Theory of NP. Chapter 9
1 Computational Complexity and Intractability: An Introduction to the Theory of NP Chapter 9 2 Objectives Classify problems as tractable or intractable Define decision problems Define the class P Define
Minimum spanning tree
Minimum spanning tree Jean Cousty MorphoGraph and Imagery 2011 J. Cousty : MorphoGraph and Imagery 1/17 Outline of the lecture 1 Minimum spanning tree 2 Cut theorem for MST 3 Kruskal algorithm J. Cousty
Finite Model Theory and Graph Isomorphism. II.
Finite Model Theory and Graph Isomorphism. II. Anuj Dawar University of Cambridge Computer Laboratory visiting RWTH Aachen Beroun, 13 December 2013 Recapitulation Finite Model Theory aims to study the
Cleaning 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
Easy Problems vs. Hard Problems. CSE 421 Introduction to Algorithms Winter Is P a good definition of efficient? The class P
Easy Problems vs. Hard Problems CSE 421 Introduction to Algorithms Winter 2000 NP-Completeness (Chapter 11) Easy - problems whose worst case running time is bounded by some polynomial in the size of the
Computing Minmax; Dominance
Computing Minmax; Dominance CPSC 532A Lecture 5 Computing Minmax; Dominance CPSC 532A Lecture 5, Slide 1 Lecture Overview 1 Recap 2 Linear Programming 3 Computational Problems Involving Maxmin 4 Domination
CS1800: Strong Induction. Professor Kevin Gold
CS1800: Strong Induction Professor Kevin Gold Mini-Primer/Refresher on Unrelated Topic: Limits This is meant to be a problem about reasoning about quantifiers, with a little practice of other skills, too
Workshop on Algorithmic Graph Structure Theory
Workshop on Algorithmic Graph Structure Theory University of Warsaw, July 14 2017 3 Lectures/Tutorials 3 Contributed Talks Costs: 50 Euro University of Warsaw, July 10-13 2017 On Neighbourhood Complexity
Graph Theory(I): استاذ الماده: أ.م.د. أكرم برزان عطار
استاذ الماده: أ.م.د. أكرم برزان عطار Graph Theory(I): 1. An introduction to Graphs: The Definition of Graphs - Graphs as Models - Matrix Degree - Subgraphs- Paths and Cycles - the Matrix Representation
Data Structures and Algorithms
Data Structures and Algorithms Session 21. April 13, 2009 Instructor: Bert Huang http://www.cs.columbia.edu/~bert/courses/3137 Announcements Homework 5 due next Monday I m out of town Wed to Sun for conference
The Complexity of Maximum. Matroid-Greedoid Intersection and. Weighted Greedoid Maximization
Department of Computer Science Series of Publications C Report C-2004-2 The Complexity of Maximum Matroid-Greedoid Intersection and Weighted Greedoid Maximization Taneli Mielikäinen Esko Ukkonen University
Computational Complexity
Computational Complexity Algorithm performance and difficulty of problems So far we have seen problems admitting fast algorithms flow problems, shortest path, spanning tree... and other problems for which
ICS 252 Introduction to Computer Design
ICS 252 fall 2006 Eli Bozorgzadeh Computer Science Department-UCI References and Copyright Textbooks referred [Mic94] G. De Micheli Synthesis and Optimization of Digital Circuits McGraw-Hill, 1994. [CLR90]
arxiv:math/ v1 [math.co] 17 Apr 2002
![arxiv:math/ v1 [math.co] 17 Apr 2002](/thumbs/88/115735478.jpg "arxiv:math/ v1 [math.co] 17 Apr 2002")
arxiv:math/0204222v1 [math.co] 17 Apr 2002 On Arithmetic Progressions of Cycle Lengths in Graphs Jacques Verstraëte Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences
CMPSCI611: The Matroid Theorem Lecture 5
CMPSCI611: The Matroid Theorem Lecture 5 We first review our definitions: A subset system is a set E together with a set of subsets of E, called I, such that I is closed under inclusion. This means that
Review of unsolvability
Review of unsolvability L L H To prove unsolvability: show a reduction. To prove solvability: show an algorithm. Unsolvable problems (main insight) Turing machine (algorithm) properties Pattern matching
arxiv: v1 [cs.ds] 2 Oct 2018
![arxiv: v1 [cs.ds] 2 Oct 2018](/thumbs/87/96752737.jpg "arxiv: v1 [cs.ds] 2 Oct 2018")
Contracting to a Longest Path in H-Free Graphs Walter Kern 1 and Daniël Paulusma 2 1 Department of Applied Mathematics, University of Twente, The Netherlands w.kern@twente.nl 2 Department of Computer Science,
P, NP, and Beyond. Hengfeng Wei. May 01 May 04, Hengfeng Wei P, NP, and Beyond May 01 May 04, / 29
P, NP, and Beyond Hengfeng Wei hfwei@njueducn May 01 May 04, 2017 Hengfeng Wei (hfwei@njueducn) P, NP, and Beyond May 01 May 04, 2017 1 / 29 P, NP, and Beyond 1 2 Reductions: Tetris is NP-complete P P
arxiv: v1 [math.co] 17 Jul 2017
![arxiv: v1 [math.co] 17 Jul 2017](/thumbs/93/114388492.jpg "arxiv: v1 [math.co] 17 Jul 2017")
Bad News for Chordal Partitions Alex Scott Paul Seymour David R. Wood Abstract. Reed and Seymour [1998] asked whether every graph has a partition into induced connected non-empty bipartite subgraphs such
NP-Completeness. Until now we have been designing algorithms for specific problems
NP-Completeness 1 Introduction Until now we have been designing algorithms for specific problems We have seen running times O(log n), O(n), O(n log n), O(n 2 ), O(n 3 )... We have also discussed lower
Lecture 21: Counting and Sampling Problems
princeton univ. F 14 cos 521: Advanced Algorithm Design Lecture 21: Counting and Sampling Problems Lecturer: Sanjeev Arora Scribe: Today s topic of counting and sampling problems is motivated by computational
Complexity Theory Part II
Complexity Theory Part II Time Complexity The time complexity of a TM M is a function denoting the worst-case number of steps M takes on any input of length n. By convention, n denotes the length of the
Partition is reducible to P2 C max. c. P2 Pj = 1, prec Cmax is solvable in polynomial time. P Pj = 1, prec Cmax is NP-hard
I. Minimizing Cmax (Nonpreemptive) a. P2 C max is NP-hard. Partition is reducible to P2 C max b. P Pj = 1, intree Cmax P Pj = 1, outtree Cmax are both solvable in polynomial time. c. P2 Pj = 1, prec Cmax
The path of order 5 is called the W-graph. Let (X, Y ) be a bipartition of the W-graph with X = 3. Vertices in X are called big vertices, and vertices
The Modular Chromatic Number of Trees Jia-Lin Wang, Jie-Ying Yuan Abstract:The concept of modular coloring and modular chromatic number was proposed by F. Okamoto, E. Salehi and P. Zhang in 2010. They
CS599: Convex and Combinatorial Optimization Fall 2013 Lecture 17: Combinatorial Problems as Linear Programs III. Instructor: Shaddin Dughmi
CS599: Convex and Combinatorial Optimization Fall 2013 Lecture 17: Combinatorial Problems as Linear Programs III Instructor: Shaddin Dughmi Announcements Today: Spanning Trees and Flows Flexibility awarded
CS 350 Algorithms and Complexity
CS 350 Algorithms and Complexity Winter 2019 Lecture 15: Limitations of Algorithmic Power Introduction to complexity theory Andrew P. Black Department of Computer Science Portland State University Lower
arxiv: v1 [cs.ds] 26 Feb 2016
![arxiv: v1 [cs.ds] 26 Feb 2016](/thumbs/71/65629444.jpg "arxiv: v1 [cs.ds] 26 Feb 2016")
On the computational complexity of minimum-concave-cost flow in a two-dimensional grid Shabbir Ahmed, Qie He, Shi Li, George L. Nemhauser arxiv:1602.08515v1 [cs.ds] 26 Feb 2016 Abstract We study the minimum-concave-cost
Polynomial Time Computation. Topics in Logic and Complexity Handout 2. Nondeterministic Polynomial Time. Succinct Certificates.
1 2 Topics in Logic and Complexity Handout 2 Anuj Dawar MPhil Advanced Computer Science, Lent 2010 Polynomial Time Computation P = TIME(n k ) k=1 The class of languages decidable in polynomial time. The
CS 350 Algorithms and Complexity
1 CS 350 Algorithms and Complexity Fall 2015 Lecture 15: Limitations of Algorithmic Power Introduction to complexity theory Andrew P. Black Department of Computer Science Portland State University Lower
Non-Deterministic Time
Non-Deterministic Time Master Informatique 2016 1 Non-Deterministic Time Complexity Classes Reminder on DTM vs NDTM [Turing 1936] (q 0, x 0 ) (q 1, x 1 ) Deterministic (q n, x n ) Non-Deterministic (q
4. How to prove a problem is NPC
The reducibility relation T is transitive, i.e, A T B and B T C imply A T C Therefore, to prove that a problem A is NPC: (1) show that A NP (2) choose some known NPC problem B define a polynomial transformation
A 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
The inefficiency of equilibria
The inefficiency of equilibria Chapters 17,18,19 of AGT 3 May 2010 University of Bergen Outline 1 Reminder 2 Potential games 3 Complexity Formalization Like usual, we consider a game with k players {1,...,
Lecture 10 February 4, 2013
UBC CPSC 536N: Sparse Approximations Winter 2013 Prof Nick Harvey Lecture 10 February 4, 2013 Scribe: Alexandre Fréchette This lecture is about spanning trees and their polyhedral representation Throughout
CS481: Bioinformatics Algorithms
CS481: Bioinformatics Algorithms Can Alkan EA224 calkan@cs.bilkent.edu.tr http://www.cs.bilkent.edu.tr/~calkan/teaching/cs481/ Reminder The TA will hold a few recitation sessions for the students from
Graph 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.......................................
Algorithmic 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
Degrees and Trees. Garth Isaak Lehigh University. 47th SEICCGTC at FAU, March Acknowledgements to: Kathleen Ryan,...
Degrees and Trees Garth Isaak Lehigh University 47th SEICCGTC at FAU, March 2016 Acknowledgements to: Kathleen Ryan,... REU Students (Hannah Alpert, Amy Becker, Jenny Iglesius, James Hilbert) T.S. Michael
Constraint satisfaction problems and dualities
Andrei Krokhin - CSPs and dualities 1 Constraint satisfaction problems and dualities Andrei Krokhin University of Durham Andrei Krokhin - CSPs and dualities 2 The CSP The Constraint Satisfaction (or Homomorphism)
Randomness and Computation
Randomness and Computation or, Randomized Algorithms Mary Cryan School of Informatics University of Edinburgh RC (2018/19) Lecture 11 slide 1 The Probabilistic Method The Probabilistic Method is a nonconstructive
Exponential algorithmic speedup by quantum walk
Exponential algorithmic speedup by quantum walk Andrew Childs MIT Center for Theoretical Physics joint work with Richard Cleve Enrico Deotto Eddie Farhi Sam Gutmann Dan Spielman quant-ph/0209131 Motivation
COMP Analysis of Algorithms & Data Structures
COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Computational Complexity CLRS 34.1-34.4 University of Manitoba COMP 3170 - Analysis of Algorithms & Data Structures 1 / 50 Polynomial
Lecture 14 - P v.s. NP 1
CME 305: Discrete Mathematics and Algorithms Instructor: Professor Aaron Sidford (sidford@stanford.edu) February 27, 2018 Lecture 14 - P v.s. NP 1 In this lecture we start Unit 3 on NP-hardness and approximation
Model-Checking for Successor-Invariant First-Order Formulas on Graph Classes of Bounded Expansion
Model-Checking for Successor-Invariant First-Order Formulas on Graph Classes of Bounded Expansion Jan van den Heuvel Stephan Kreutzer Micha l Pilipczuk Daniel A. Quiroz Roman Rabinovich Sebastian Siebertz
the configuration space of Branched polymers R. Kenyon, P. Winkler
the configuration space of Branched polymers R. Kenyon, P. Winkler artificial blood catalyst recovery Branched polymers (dendrimers) in modern science artificial photosynthesis A branched polymer is a
The embedding problem for partial Steiner triple systems
Late last year several news-items appeared announcing the solution of the 30 year old Lindner s conjecture by UoQ Graduate student Daniel Horsley and his supervisor Darryn Bryant, see e.g., http://www.uq.edu.au/news/?article=8359,
First order logic on Galton-Watson trees
First order logic on Galton-Watson trees Moumanti Podder Georgia Institute of Technology Joint work with Joel Spencer January 9, 2018 Mathematics Seminar, Indian Institute of Science, Bangalore 1 / 20
Computational Intractability 2010/4/15. Lecture 2
Computational Intractability 2010/4/15 Professor: David Avis Lecture 2 Scribe:Naoki Hatta 1 P and NP 1.1 Definition of P and NP Decision problem it requires yes/no answer. Example: X is a set of strings.
Bounded 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
Class Note #20. In today s class, the following four concepts were introduced: decision
Class Note #20 Date: 03/29/2006 [Overall Information] In today s class, the following four concepts were introduced: decision version of a problem, formal language, P and NP. We also discussed the relationship
NP-Completeness. Subhash Suri. May 15, 2018
NP-Completeness Subhash Suri May 15, 2018 1 Computational Intractability The classical reference for this topic is the book Computers and Intractability: A guide to the theory of NP-Completeness by Michael
Lecture 18: P & NP. Revised, May 1, CLRS, pp
Lecture 18: P & NP Revised, May 1, 2003 CLRS, pp.966-982 The course so far: techniques for designing efficient algorithms, e.g., divide-and-conquer, dynamic-programming, greedy-algorithms. What happens
Tree-width and planar minors
Tree-width and planar minors Alexander Leaf and Paul Seymour 1 Princeton University, Princeton, NJ 08544 May 22, 2012; revised March 18, 2014 1 Supported by ONR grant N00014-10-1-0680 and NSF grant DMS-0901075.
Combinatorial Optimization
Combinatorial Optimization Problem set 8: solutions 1. Fix constants a R and b > 1. For n N, let f(n) = n a and g(n) = b n. Prove that f(n) = o ( g(n) ). Solution. First we observe that g(n) 0 for all
Spanning trees on the Sierpinski gasket
Spanning trees on the Sierpinski gasket Shu-Chiuan Chang (1997-2002) Department of Physics National Cheng Kung University Tainan 70101, Taiwan and Physics Division National Center for Theoretical Science