A New Approximation Algorithm for the Asymmetric TSP with Triangle Inequality By Markus Bläser
|
|
- Malcolm Dean
- 5 years ago
- Views:
Transcription
1 A New Approximation Algorithm for the Asymmetric TSP with Triangle Inequality By Markus Bläser Presented By: Chris Standish 23 November
2 Outline Problem Definition Frieze s Generic Algorithm of 1982 Bottleneck in Frieze s Algorithm LP Formulation for Cycle Covers Bläsers Generic Algorithm of 2003 Dealing with Fractional LP Fractional Solutions Discussion and Homework 2
3 The Asymmetric Minimum TSP The Asymmetric Minimum Traveling Salesman Problem (ATSP) is: Problem 1 Asymmetric Minimum TSP Given: A complete, directed, edge-weighted graph G(V, E), without self-loops, and an edge weight function w which satisfies the triangle inequality, and which is asymmetric, and non-negative. Find: A Hamiltonian tour of G of minimum weight. A weight function w : E Q 0 which satisfies the triangle inequality means that: w(u, v) w(u, x) + w(x, v) for all distinct u, x, v V. An asymmetric weight function w is such that w(u, v) w(v, u). We will abbreviate w(u, v) as w uv If we consider this problem defined with an arbitrary weight function then this problem is NPO-complete. So we don t expect to be able to find a PTAS unless P = NP [1]. 3
4 Definitions Frieze, et. al. [2] developed an algorithm in 1982 that, until now, has had the best approximation ratio for minimum ATSP. But first we need some definitions: k-cycle - A cycle is called a k-cycle if is has length exactly k. cycle cover - A cycle cover of a directed graph G is a set of directed simple cycles such that each node is part of exactly one cycle, i.e., the set of cycles spans the graph. k-cycle cover - A cycle cover C is called a k-cycle cover if each cycle in C has length at least k. cycle cover weight - Let C = {C 1, C 2,...,C t } be a cycle cover. Then the weight of the cycle cover is w(c) = t i=1 w(c i) where w(c i ) = (u,v) C i w uv partial cycle cover - If C is a collection of node disjoint cycles, but not a spanning one, then C is called a partial cycle cover. 4
5 Frieze s Algorithm Freize s generic algorithm [3] to find an approximate minimum-weighted tour can be described as: Algorithm 1 Generic ATSP Step 1) Create a minimum weight cycle cover C for the graph G Step 2) For each cycle in C remove a maximum-weighted edge to form a set of vertex disjoint paths P. Step 3) Patch together the resulting set of paths in P to produce a tour T of G The algorithm has a performance ratio of 1 log n. That is, it always delivers a solution which is at most 1 log n times the weight of an optimal solution [2]. The goal of this paper is to show that this threshold is not tight. The paper gives an algorithm with a log n performance ratio. 5
6 An Example A PARTIAL 2 CYCLE COVER A TOUR Figure 1: Constructing a tour. Most edges are not shown for clarity. 6
7 Bottleneck in the Generic Algorithm Notice that if we could construct a n -cycle cover we can find an optimal 2 tour directly. So why don t we? The problem with the above algorithm is the difficulty of computing a minimum weight cycle cover. In fact, computing a minimum weight 3-cycle cover is APX-hard, even if the weight function satisfies the triangle inequality [4]. This paper adapts Frieze s algorithm to find a good partial cycle cover instead of a cycle cover. 7
8 Computing Cycle Covers The problem of computing a minimum weight cycle cover can be solved by a relaxed linear program (LP). Let x uv be the variable that represents the edge (u, v). If x uv = 1 then the edge is included in a cycle, if x uv = 0 then it is not. The relaxed linear program can be formulated as: Minimize (u,v) E w uv x uv subject to u x uv = 1 for all v V (in-degree constraints) v x uv = 1 for all u V (out-degree constraints) x uv 0 for all (u, v) E (non-negativity constraints) 8
9 Relaxed LP formulation In this particular LP formulation we encounter something quite nice. The solution matrix X = (x uv) n n to this relaxed LP happens to be totally unimodular. What this results in, is an X that has entries that are either 0 or 1. So the optimal solution is exact and we don t have to deal with fractional x uv values. However, we have not constrained the length of the cycles in the cycle cover. Notice that the larger the value of k, the better our approximate tour will be. This LP can be solved in polynomial time, i.e., O(n 3 ) 9
10 Good Partial Cycle Covers In this paper, the generic ATSP algorithm has been modified to use the notion of a good partial cycle cover, see Figure 1. TSP(G) - Let TSP(G) be the weight of an optimal tour in G. b-good partial cycle cover - is a partial cycle cover which satisfies the condition where α log β b 0 < b 1, α 1 is such that the weight of the partial cycle cover is α TSP(G), and β is such that the number of cycles in the partial cycle cover is β V, 0 < β < 1. The generic algorithm of Bläser is also recursive. 10
11 The b-good Partial Cycle Cover Problem The b-good Partial Cycle Cover Problem is: Problem 2 The b-good Partial Cycle Cover Problem (b-gpcc) Given: A directed graph G(V, E), and an asymmetric edge weight function w that satisfies the triangle inequality. Find: A partial cycle cover C of weight w(c) = α TSP(G), which has β V cycles, and such that α log β b 11
12 Recursive Generic Algorithm The generic algorithm this paper uses to find an approximate tour can be described as: Algorithm 2 Generic Recursive ATSP Step 1) Create a b-good cycle cover C for the graph G Step 2) For each cycle in C, choose one arbitrary node. Let V be the set of nodes consisting of these nodes, together with all the nodes in V that are not contained in any cycle of C. Step 3) Recursively compute a TSP tour T of the graph G induced by V. Step 4) For each cycle in C remove a maximum-weighted edge to form a set of vertex disjoint paths P. Step 5) Patch together the resulting set of paths in P and T to produce a tour T. 12
13 An Example C T T Figure 2: Constructing a tour using a partial cycle cover, and a tour. Not all edges are shown. T is the tour constructed by combining C and T. 13
14 Running Time In the worst case, step 2 reduces the size of the graph by one. That is, only one 2-cycle is found. So the algorithm recursively calls itself at most n = V times. If a b-good cycle cover can be computed in polynomial time, then the generic recursive cycle cover algorithm will run in polynomial time. Theorem 1 If the b-gpcc problem can be solved in polynomial time for some 0 < b 1, then there is a polynomial time (b log n)-approximate algorithm for ATSP. The rest of the paper concentrates on showing that a b-gpcc can be found in polynomial time. 14
15 Improved Approximation Ratio This paper improves the approximation ratio of Frieze, i.e., 1 log n, by solving a relaxed LP with an additional constraint. In addition to the previous constraints in the LP formulation, a 2-cycle constraint is added This constraint eliminates 2-cycles. x uv + x vu 1 for all u v However, the solution matrix X is no longer totally unimodular. So this means we have to deal with fractional solution values x uv. 15
16 Decomposing a Fractional Solution The rest of the paper is devoted to showing how to compute a good partial cycle cover given a fractional solution X. The method is involved so I am just going to give an overview of the main ideas. 16
17 Some Definitions and a Lemma doubly stochastic - A matrix S is called doubly stochastic if all its entries are non-negative, and the entries in each row, and in each column, sum to 1. permutation matrix - A matrix P is called a permutation matrix if each entry has value either 0 or 1, and it is doubly stochastic. Lemma 1 (BIRKHOFF - VON NEUMANN) Every doubly stochastic n n matrix S is a convex combination of at most n 2 permutation matrices. Such a decomposition can be found in polynomial time. Since the solution matrix X is double stochastic (because of the in-degree and out-degree constraints), we can decompose it: X = t α i P i i=1 where t n 2, the α i are non-negative reals such that t i=1 α i = 1, (convexity), and the P i are permutation matrices. 17
18 A Permutation Matrix Notice that every permutation matrix induces a cycle cover of G PERMUTATION MATRIX 2 CYCLE COVER Figure 3: A 2-cycle cover and its corresponding permutation matrix. 18
19 Reducing the Number of Cycle Covers Choose a constant B, such that B is the smallest integer such that B α i is an integer, for each i. Define γ i = B α i. Bläser claims that all the α i are rational. So we can find such a B which will be polynomial in the problem input size. Notice that B can be quite large. So now we can treat each permutation matrix P i as a cycle cover C i, which has a multiplicity γ i. That is, there are γ i copies of the cycle cover C i. The point is that we can work with a set of t cycle covers C i instead of working with all the B cycle covers explicitly. 19
20 Normalization So now lets treat each P i as a C i. (Note C i is a set of edges, P i is a matrix) Let C i C j denote the graph (V, E i E j ). This is the graph formed by the set of the edges in the two cycle covers. ( ) t For each of the possible cycle cover pairs C 2 i C j, we transform some particularly shaped strongly connected components, that are formed by 2-cycles in C i C j, into larger length cycles. This process is called normalization and can be done in polynomial time. 20
21 Normalization Figure 4: A strongly connected component formed by 2-cycles in C i C j. Green(dashed) denotes a 2-cycle in C i, blue(solid) for C j. 21
22 Computing Good Partial Cycle Covers When we normalize the union of two cycle covers, we are eliminating 2- cycles in each of C i and C j, and replacing them with larger length cycles. After we normalize all pairs of cycle covers, we compute a b-good partial cycle cover, where b = 0.999, from the normalized set of cycle covers. Bläser gives a case based analysis, and in each case he shows that we can compute a good partial cycle cover. So we have the following: Theorem 2 There is a polynomial time algorithm for the GPCC problem. Corollary 1 There is a (0.999 log n)-approximate algorithm for the minimum ATSP problem with polynomial running time. 22
23 Main Results Presents a recursive generic algorithm which has a log n performance ratio. Shows that the approximation ratio of 1 log n shown by Frieze, et. al. [2] is not tight. This contrasts with the the set cover problem which has a tight threshold of 1 lnn as shown by Feige [5]. The algorithm does not appear to be practical because its running time depends on the value of B. 23
24 Homework Do one of the following problems: 1) Write the minimum ATSP problem in the 4-tuple notation < I Q, S Q, f Q, opt Q > 2) In a previous slide, we said that if we could construct a minimum weight n-cycle cover for an input graph G(V, E) (for the ATSP problem, where 2 n = V ), we could find an optimal tour directly. Q) - Why is this so? Assume n is even. Hint: There are two cases, 1. There are exactly two n-cycles 2 2. There is exactly one cycle with size greater than n 2 24
25 References [1] M. Bläser. An 8 -Approximation Algorithm for the Asymmetric Maximum 13 TSP. Journal of Algorithms, 50(1):23 48, [2] A.M. Frieze, G. Galbiati, and F. Maffioli. On the Worst-Case Performance of Some Algorithms for the Asymmetric Traveling Salesman Problem. Networks, 12:23 39, [3] M. Lewenstein and M. Sviridenko. Approximating Asymmetric Maximum TSP. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pages , [4] M. Bläser. A New Approximation Algorithm for the Asymmetric TSP With Triangle Inequality. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pages , [5] U. Feige. A Threshold of ln n for Approximating Set Cover. Journal of the ACM, 45: ,
Improved Approximation Algorithms for Metric Maximum ATSP and Maximum 3-Cycle Cover Problems
Improved Approximation Algorithms for Metric Maximum ATSP and Maximum 3-Cycle Cover Problems Markus Bläser 1 Computer Science, Saarland University L. Shankar Ram Institut für Theoretische Informatik, ETH
More informationApproximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs
Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs Haim Kaplan Tel-Aviv University, Israel haimk@post.tau.ac.il Nira Shafrir Tel-Aviv University, Israel shafrirn@post.tau.ac.il
More informationAn Improved Approximation Algorithm for the Asymmetric TSP with Strengthened Triangle Inequality
An Improved Approximation Algorithm for the Asymmetric TSP with Strengthened Triangle Inequality Markus Bläser Bodo Manthey Jiří Sgall Abstract We consider the asymmetric traveling salesperson problem
More informationTravelling Salesman Problem
Travelling Salesman Problem Fabio Furini November 10th, 2014 Travelling Salesman Problem 1 Outline 1 Traveling Salesman Problem Separation Travelling Salesman Problem 2 (Asymmetric) Traveling Salesman
More informationWorst Case Performance of Approximation Algorithm for Asymmetric TSP
Worst Case Performance of Approximation Algorithm for Asymmetric TSP Anna Palbom Department of Numerical Analysis and Computer Science Royal Institute of Technology S-100 44 Stockholm, Sweden E-mail: annap@nada.kth.se
More information2 Notation and Preliminaries
On Asymmetric TSP: Transformation to Symmetric TSP and Performance Bound Ratnesh Kumar Haomin Li epartment of Electrical Engineering University of Kentucky Lexington, KY 40506-0046 Abstract We show that
More informationABHELSINKI UNIVERSITY OF TECHNOLOGY
Approximation Algorithms Seminar 1 Set Cover, Steiner Tree and TSP Siert Wieringa siert.wieringa@tkk.fi Approximation Algorithms Seminar 1 1/27 Contents Approximation algorithms for: Set Cover Steiner
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 informationAlgorithms. Outline! Approximation Algorithms. The class APX. The intelligence behind the hardware. ! Based on
6117CIT - Adv Topics in Computing Sci at Nathan 1 Algorithms The intelligence behind the hardware Outline! Approximation Algorithms The class APX! Some complexity classes, like PTAS and FPTAS! Illustration
More informationMulti-Criteria TSP: Min and Max Combined
Multi-Criteria TSP: Min and Max Combined Bodo Manthey University of Twente, Department of Applied Mathematics P. O. Box 217, 7500 AE Enschede, The Netherlands b.manthey@utwente.nl Abstract. We present
More informationApproximate Pareto Curves for the Asymmetric Traveling Salesman Problem
Approximate Pareto Curves for the Asymmetric Traveling Salesman Problem Bodo Manthey Universität des Saarlandes, Informatik Postfach 550, 6604 Saarbrücken, Germany manthey@cs.uni-sb.de arxiv:07.257v [cs.ds]
More informationCS675: Convex and Combinatorial Optimization Fall 2014 Combinatorial Problems as Linear Programs. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Fall 2014 Combinatorial Problems as Linear Programs Instructor: Shaddin Dughmi Outline 1 Introduction 2 Shortest Path 3 Algorithms for Single-Source Shortest
More informationOn the Integrality Ratio for the Asymmetric Traveling Salesman Problem
On the Integrality Ratio for the Asymmetric Traveling Salesman Problem Moses Charikar Dept. of Computer Science, Princeton University, 35 Olden St., Princeton, NJ 08540 email: moses@cs.princeton.edu Michel
More informationCS675: Convex and Combinatorial Optimization Fall 2016 Combinatorial Problems as Linear and Convex Programs. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Fall 2016 Combinatorial Problems as Linear and Convex Programs Instructor: Shaddin Dughmi Outline 1 Introduction 2 Shortest Path 3 Algorithms for Single-Source
More informationApproximability of Connected Factors
Approximability of Connected Factors Kamiel Cornelissen 1, Ruben Hoeksma 1, Bodo Manthey 1, N. S. Narayanaswamy 2, and C. S. Rahul 2 1 University of Twente, Enschede, The Netherlands {k.cornelissen, r.p.hoeksma,
More informationJuly 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
More informationBounds on the Traveling Salesman Problem
Bounds on the Traveling Salesman Problem Sean Zachary Roberson Texas A&M University MATH 613, Graph Theory A common routing problem is as follows: given a collection of stops (for example, towns, stations,
More informationWelcome to... Problem Analysis and Complexity Theory , 3 VU
Welcome to... Problem Analysis and Complexity Theory 716.054, 3 VU Birgit Vogtenhuber Institute for Software Technology email: bvogt@ist.tugraz.at office: Inffeldgasse 16B/II, room IC02044 slides: http://www.ist.tugraz.at/pact17.html
More informationApproximating the minimum quadratic assignment problems
Approximating the minimum quadratic assignment problems Refael Hassin Asaf Levin Maxim Sviridenko Abstract We consider the well-known minimum quadratic assignment problem. In this problem we are given
More informationApproximation Algorithms for Re-optimization
Approximation Algorithms for Re-optimization DRAFT PLEASE DO NOT CITE Dean Alderucci Table of Contents 1.Introduction... 2 2.Overview of the Current State of Re-Optimization Research... 3 2.1.General Results
More informationSAT, Coloring, Hamiltonian Cycle, TSP
1 SAT, Coloring, Hamiltonian Cycle, TSP Slides by Carl Kingsford Apr. 28, 2014 Sects. 8.2, 8.7, 8.5 2 Boolean Formulas Boolean Formulas: Variables: x 1, x 2, x 3 (can be either true or false) Terms: t
More informationLecture 6 January 21, 2013
UBC CPSC 536N: Sparse Approximations Winter 03 Prof. Nick Harvey Lecture 6 January, 03 Scribe: Zachary Drudi In the previous lecture, we discussed max flow problems. Today, we consider the Travelling Salesman
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 informationThe quest for finding Hamiltonian cycles
The quest for finding Hamiltonian cycles Giang Nguyen School of Mathematical Sciences University of Adelaide Travelling Salesman Problem Given a list of cities and distances between cities, what is the
More informationLectures 6, 7 and part of 8
Lectures 6, 7 and part of 8 Uriel Feige April 26, May 3, May 10, 2015 1 Linear programming duality 1.1 The diet problem revisited Recall the diet problem from Lecture 1. There are n foods, m nutrients,
More informationTopics in Theoretical Computer Science April 08, Lecture 8
Topics in Theoretical Computer Science April 08, 204 Lecture 8 Lecturer: Ola Svensson Scribes: David Leydier and Samuel Grütter Introduction In this lecture we will introduce Linear Programming. It was
More informationON COST MATRICES WITH TWO AND THREE DISTINCT VALUES OF HAMILTONIAN PATHS AND CYCLES
ON COST MATRICES WITH TWO AND THREE DISTINCT VALUES OF HAMILTONIAN PATHS AND CYCLES SANTOSH N. KABADI AND ABRAHAM P. PUNNEN Abstract. Polynomially testable characterization of cost matrices associated
More informationApproximation Hardness of TSP with Bounded Metrics. Abstract. The general asymmetric (and metric) TSP is known
Approximation Hardness of TSP with Bounded Metrics Lars Engebretsen? Marek Karpinski?? Abstract. The general asymmetric (and metric) TSP is known to be approximable only to within an O(log n) factor, and
More informationThe traveling salesman problem
Chapter 58 The traveling salesman problem The traveling salesman problem (TSP) asks for a shortest Hamiltonian circuit in a graph. It belongs to the most seductive problems in combinatorial optimization,
More informationLecture 4: NP and computational intractability
Chapter 4 Lecture 4: NP and computational intractability Listen to: Find the longest path, Daniel Barret What do we do today: polynomial time reduction NP, co-np and NP complete problems some examples
More informationCS Algorithms and Complexity
CS 50 - Algorithms and Complexity Linear Programming, the Simplex Method, and Hard Problems Sean Anderson 2/15/18 Portland State University Table of contents 1. The Simplex Method 2. The Graph Problem
More informationAn O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem
An Olog n/log log n)-approximation algorithm for the asymmetric traveling salesman problem The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story
More informationAlgorithms Design & Analysis. Approximation Algorithm
Algorithms Design & Analysis Approximation Algorithm Recap External memory model Merge sort Distribution sort 2 Today s Topics Hard problem Approximation algorithms Metric traveling salesman problem A
More informationSpanning trees with minimum weighted degrees
Spanning trees with minimum weighted degrees Mohammad Ghodsi Hamid Mahini Kian Mirjalali Shayan Oveis Gharan Amin S. Sayedi R. Morteza Zadimoghaddam Abstract Given a metric graph G, we are concerned with
More informationON TESTING HAMILTONICITY OF GRAPHS. Alexander Barvinok. July 15, 2014
ON TESTING HAMILTONICITY OF GRAPHS Alexander Barvinok July 5, 204 Abstract. Let us fix a function f(n) = o(nlnn) and reals 0 α < β. We present a polynomial time algorithm which, given a directed graph
More informationProblem set 1. (c) Is the Ford-Fulkerson algorithm guaranteed to produce an acyclic maximum flow?
CS261, Winter 2017. Instructor: Ashish Goel. Problem set 1 Electronic submission to Gradescope due 11:59pm Thursday 2/2. Form a group of 2-3 students that is, submit one homework with all of your names.
More informationLecturer: Shuchi Chawla Topic: Inapproximability Date: 4/27/2007
CS880: Approximations Algorithms Scribe: Tom Watson Lecturer: Shuchi Chawla Topic: Inapproximability Date: 4/27/2007 So far in this course, we have been proving upper bounds on the approximation factors
More informationLecture 20: LP Relaxation and Approximation Algorithms. 1 Introduction. 2 Vertex Cover problem. CSCI-B609: A Theorist s Toolkit, Fall 2016 Nov 8
CSCI-B609: A Theorist s Toolkit, Fall 2016 Nov 8 Lecture 20: LP Relaxation and Approximation Algorithms Lecturer: Yuan Zhou Scribe: Syed Mahbub Hafiz 1 Introduction When variables of constraints of an
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 informationDecision Problems TSP. Instance: A complete graph G with non-negative edge costs, and an integer
Decision Problems The theory of NP-completeness deals only with decision problems. Why? Because if a decision problem is hard, then the corresponding optimization problem must be hard too. For example,
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 informationChapter 3: Proving NP-completeness Results
Chapter 3: Proving NP-completeness Results Six Basic NP-Complete Problems Some Techniques for Proving NP-Completeness Some Suggested Exercises 1.1 Six Basic NP-Complete Problems 3-SATISFIABILITY (3SAT)
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 informationToday s Outline. CS 362, Lecture 24. The Reduction. Hamiltonian Cycle. Reduction Wrapup Approximation algorithms for NP-Hard Problems
Today s Outline CS 362, Lecture 24 Jared Saia University of New Mexico Reduction Wrapup Approximation algorithms for NP-Hard Problems 1 Hamiltonian Cycle The Reduction A Hamiltonian Cycle in a graph is
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 informationApproximation of Euclidean k-size cycle cover problem
Croatian Operational Research Review 177 CRORR 5(2014), 177 188 Approximation of Euclidean k-size cycle cover problem Michael Khachay 1, and Katherine Neznakhina 1 1 Krasovsky Institute of Mathematics
More informationWeek Cuts, Branch & Bound, and Lagrangean Relaxation
Week 11 1 Integer Linear Programming This week we will discuss solution methods for solving integer linear programming problems. I will skip the part on complexity theory, Section 11.8, although this is
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 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 informationPolynomial-time Reductions
Polynomial-time Reductions Disclaimer: Many denitions in these slides should be taken as the intuitive meaning, as the precise meaning of some of the terms are hard to pin down without introducing the
More informationLecture 5: Lowerbound on the Permanent and Application to TSP
Math 270: Geometry of Polynomials Fall 2015 Lecture 5: Lowerbound on the Permanent and Application to TSP Lecturer: Zsolt Bartha, Satyai Muheree Scribe: Yumeng Zhang Disclaimer: These notes have not been
More informationON APPROXIMATING MULTI-CRITERIA TSP BODO MANTHEY 1
Symposium on Theoretical Aspects of Computer Science 2009 (Freiburg), pp. 637 648 www.stacs-conf.org ON APPROXIMATING MULTI-CRITERIA TSP BODO MANTHEY Saarland University, Computer Science, Postfach 550,
More informationNP and Computational Intractability
NP and Computational Intractability 1 Polynomial-Time Reduction Desiderata'. Suppose we could solve X in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Reduction.
More informationFundamentals of optimization problems
Fundamentals of optimization problems Dmitriy Serdyuk Ferienakademie in Sarntal 2012 FAU Erlangen-Nürnberg, TU München, Uni Stuttgart September 2012 Overview 1 Introduction Optimization problems PO and
More informationCombinatorial 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
More informationIncreasing the Span of Stars
Increasing the Span of Stars Ning Chen Roee Engelberg C. Thach Nguyen Prasad Raghavendra Atri Rudra Gynanit Singh Department of Computer Science and Engineering, University of Washington, Seattle, WA.
More informationInteger Programming Methods LNMB
Integer Programming Methods LNMB 2017 2018 Dion Gijswijt homepage.tudelft.nl/64a8q/intpm/ Dion Gijswijt Intro IntPM 2017-2018 1 / 24 Organisation Webpage: homepage.tudelft.nl/64a8q/intpm/ Book: Integer
More informationA Randomized Rounding Approach to the Traveling Salesman Problem
A Randomized Rounding Approach to the Traveling Salesman Problem Shayan Oveis Gharan Amin Saberi. Mohit Singh. Abstract For some positive constant ɛ 0, we give a ( 3 2 ɛ 0)-approximation algorithm for
More informationRECAP: Extremal problems Examples
RECAP: Extremal problems Examples Proposition 1. If G is an n-vertex graph with at most n edges then G is disconnected. A Question you always have to ask: Can we improve on this proposition? Answer. NO!
More informationIntroduction to Mathematical Programming IE406. Lecture 21. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 21 Dr. Ted Ralphs IE406 Lecture 21 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2 IE406 Lecture 21 2 Branch and Bound Branch
More information1 Review of Vertex Cover
CS266: Parameterized Algorithms and Complexity Stanford University Lecture 3 Tuesday, April 9 Scribe: Huacheng Yu Spring 2013 1 Review of Vertex Cover In the last lecture, we discussed FPT algorithms for
More information1 Integer Decomposition Property
CS 598CSC: Combinatorial Optimization Lecture date: Feb 2, 2010 Instructor: Chandra Chekuri Scribe: Siva Theja Maguluri Material taken mostly from [1] (Chapter 19). 1 Integer Decomposition Property A polyhedron
More informationThe minimum G c cut problem
The minimum G c cut problem Abstract In this paper we define and study the G c -cut problem. Given a complete undirected graph G = (V ; E) with V = n, edge weighted by w(v i, v j ) 0 and an undirected
More informationInteger Linear Programming (ILP)
Integer Linear Programming (ILP) Zdeněk Hanzálek, Přemysl Šůcha hanzalek@fel.cvut.cz CTU in Prague March 8, 2017 Z. Hanzálek (CTU) Integer Linear Programming (ILP) March 8, 2017 1 / 43 Table of contents
More informationPolynomial-Time Reductions
Reductions 1 Polynomial-Time Reductions Classify Problems According to Computational Requirements Q. Which problems will we be able to solve in practice? A working definition. [von Neumann 1953, Godel
More informationInstitute of Operating Systems and Computer Networks Algorithms Group. Network Algorithms. Tutorial 3: Shortest paths and other stuff
Institute of Operating Systems and Computer Networks Algorithms Group Network Algorithms Tutorial 3: Shortest paths and other stuff Christian Rieck Shortest paths: Dijkstra s algorithm 2 Dijkstra s algorithm
More informationWeek 8. 1 LP is easy: the Ellipsoid Method
Week 8 1 LP is easy: the Ellipsoid Method In 1979 Khachyan proved that LP is solvable in polynomial time by a method of shrinking ellipsoids. The running time is polynomial in the number of variables n,
More information15.081J/6.251J Introduction to Mathematical Programming. Lecture 24: Discrete Optimization
15.081J/6.251J Introduction to Mathematical Programming Lecture 24: Discrete Optimization 1 Outline Modeling with integer variables Slide 1 What is a good formulation? Theme: The Power of Formulations
More informationNP-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
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should
More informationLimits to Approximability: When Algorithms Won't Help You. Note: Contents of today s lecture won t be on the exam
Limits to Approximability: When Algorithms Won't Help You Note: Contents of today s lecture won t be on the exam Outline Limits to Approximability: basic results Detour: Provers, verifiers, and NP Graph
More informationVIII. 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
More informationPart III: Traveling salesman problems
Transportation Logistics Part III: Traveling salesman problems c R.F. Hartl, S.N. Parragh 1/282 Motivation Motivation Why do we study the TSP? c R.F. Hartl, S.N. Parragh 2/282 Motivation Motivation Why
More informationCS 6820 Fall 2014 Lectures, October 3-20, 2014
Analysis of Algorithms Linear Programming Notes CS 6820 Fall 2014 Lectures, October 3-20, 2014 1 Linear programming The linear programming (LP) problem is the following optimization problem. We are given
More informationHardness of Approximation
Hardness of Approximation We have seen several methods to find approximation algorithms for NP-hard problems We have also seen a couple of examples where we could show lower bounds on the achievable approxmation
More informationAsymmetric Traveling Salesman Problem (ATSP): Models
Asymmetric Traveling Salesman Problem (ATSP): Models Given a DIRECTED GRAPH G = (V,A) with V = {,, n} verte set A = {(i, j) : i V, j V} arc set (complete digraph) c ij = cost associated with arc (i, j)
More informationAn Improved Approximation Algorithm for Maximum Edge 2-Coloring in Simple Graphs
An Improved Approximation Algorithm for Maximum Edge 2-Coloring in Simple Graphs Zhi-Zhong Chen Ruka Tanahashi Lusheng Wang Abstract We present a polynomial-time approximation algorithm for legally coloring
More informationNP-Complete Problems. More reductions
NP-Complete Problems More reductions Definitions P: problems that can be solved in polynomial time (typically in n, size of input) on a deterministic Turing machine Any normal computer simulates a DTM
More informationApproximability of the Multiple Stack TSP
Approximability of the Multiple Stack TSP Abstract Sophie Toulouse 1 LIPN (UMR CNRS 7030) - Institut Galilée - Université Paris 13, 99 av. Jean-Baptiste Clément, 93430 Villetaneuse, France STSP seeks a
More informationLecture 5 January 16, 2013
UBC CPSC 536N: Sparse Approximations Winter 2013 Prof. Nick Harvey Lecture 5 January 16, 2013 Scribe: Samira Samadi 1 Combinatorial IPs 1.1 Mathematical programs { min c Linear Program (LP): T x s.t. a
More information16.410/413 Principles of Autonomy and Decision Making
6.4/43 Principles of Autonomy and Decision Making Lecture 8: (Mixed-Integer) Linear Programming for Vehicle Routing and Motion Planning Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute
More informationData Structures in Java
Data Structures in Java Lecture 21: Introduction to NP-Completeness 12/9/2015 Daniel Bauer Algorithms and Problem Solving Purpose of algorithms: find solutions to problems. Data Structures provide ways
More informationImproving on the 1.5-Approximation of. a Smallest 2-Edge Connected Spanning Subgraph. September 25, Abstract
Improving on the 1.5-Approximation of a Smallest 2-Edge Connected Spanning Subgraph J. Cheriyan y A. Seb}o z Z. Szigeti x September 25, 1999 Abstract We give a 17 -approximation algorithm for the following
More informationChapter 7 Matchings and r-factors
Chapter 7 Matchings and r-factors Section 7.0 Introduction Suppose you have your own company and you have several job openings to fill. Further, suppose you have several candidates to fill these jobs and
More informationAn Improved Approximation Algorithm for Requirement Cut
An Improved Approximation Algorithm for Requirement Cut Anupam Gupta Viswanath Nagarajan R. Ravi Abstract This note presents improved approximation guarantees for the requirement cut problem: given an
More informationCS/COE
CS/COE 1501 www.cs.pitt.edu/~nlf4/cs1501/ P vs NP But first, something completely different... Some computational problems are unsolvable No algorithm can be written that will always produce the correct
More informationTractable & 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
More informationA polynomial-time approximation scheme for Euclidean Steiner forest
A polynomial-time approximation scheme for Euclidean Steiner forest Glencora Borradaile Combinatorics and Optimization University of Waterloo glencora@uwaterloo.ca Philip Klein Computer Science Brown University
More informationNew Integer Programming Formulations of the Generalized Travelling Salesman Problem
American Journal of Applied Sciences 4 (11): 932-937, 2007 ISSN 1546-9239 2007 Science Publications New Integer Programming Formulations of the Generalized Travelling Salesman Problem Petrica C. Pop Department
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 informationOkunoye Babatunde O. Department of Pure and Applied Biology, Ladoke Akintola University of Technology, Ogbomoso, Nigeria.
Resolving the decision version of the directed Hamiltonian path (cycle) problem under two special conditions by method of matrix determinant: An overview. Okunoye Babatunde O. Department of Pure and Applied
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 P and NP P: The family of problems that can be solved quickly in polynomial time.
More informationData Structures and Algorithms (CSCI 340)
University of Wisconsin Parkside Fall Semester 2008 Department of Computer Science Prof. Dr. F. Seutter Data Structures and Algorithms (CSCI 340) Homework Assignments The numbering of the problems refers
More informationMinmax Tree Cover in the Euclidean Space
Minmax Tree Cover in the Euclidean Space Seigo Karakawa, Ehab Morsy, Hiroshi Nagamochi Department of Applied Mathematics and Physics Graduate School of Informatics Kyoto University Yoshida Honmachi, Sakyo,
More information5 Flows and cuts in digraphs
5 Flows and cuts in digraphs Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset of ordered pairs of elements of V, which we refer to as arcs. Note that two vertices
More informationON THE INTEGRALITY OF THE UNCAPACITATED FACILITY LOCATION POLYTOPE. 1. Introduction
ON THE INTEGRALITY OF THE UNCAPACITATED FACILITY LOCATION POLYTOPE MOURAD BAÏOU AND FRANCISCO BARAHONA Abstract We study a system of linear inequalities associated with the uncapacitated facility location
More informationChapter 3: Discrete Optimization Integer Programming
Chapter 3: Discrete Optimization Integer Programming Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Sito web: http://home.deib.polimi.it/amaldi/ott-13-14.shtml A.A. 2013-14 Edoardo
More informationImproved Approximations for Cubic Bipartite and Cubic TSP
Improved Approximations for Cubic Bipartite and Cubic TSP Anke van Zuylen Department of Mathematics The College of William and Mary, Williamsburg, VA, 23185, USA anke@wm.edu Abstract We show improved approximation
More informationNP-Complete Reductions 1
x x x 2 x 2 x 3 x 3 x 4 x 4 CS 4407 2 22 32 Algorithms 3 2 23 3 33 NP-Complete Reductions Prof. Gregory Provan Department of Computer Science University College Cork Lecture Outline x x x 2 x 2 x 3 x 3
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 information