Approximation Algorithms: Concepts Approximation algorithm: An algorithm that returns near-optimal solutions (i.e. is "provably good“) is called an approximation.

Slides:



Advertisements
Similar presentations
Weighted Matching-Algorithms, Hamiltonian Cycles and TSP
Advertisements

Instructor Neelima Gupta Table of Contents Approximation Algorithms.
Design and Analysis of Algorithms Approximation algorithms for NP-complete problems Haidong Xue Summer 2012, at GSU.
Great Theoretical Ideas in Computer Science for Some.
Greedy Algorithms Spanning Trees Chapter 16, 23. What makes a greedy algorithm? Feasible –Has to satisfy the problem’s constraints Locally Optimal –The.
Lecture 24 Coping with NPC and Unsolvable problems. When a problem is unsolvable, that's generally very bad news: it means there is no general algorithm.
Approximation Algorithms for TSP
1 The TSP : Approximation and Hardness of Approximation All exact science is dominated by the idea of approximation. -- Bertrand Russell ( )
NP-Complete Problems Polynomial time vs exponential time
9.2 The Traveling Salesman Problem. Let us return to the question of finding a cheapest possible cycle through all the given towns: We have n towns (points)
Combinatorial Algorithms
Great Theoretical Ideas in Computer Science.
Computability and Complexity 23-1 Computability and Complexity Andrei Bulatov Search and Optimization.
1 Discrete Structures & Algorithms Graphs and Trees: II EECE 320.
Approximation Algorithms
Approximation Algorithms: Combinatorial Approaches Lecture 13: March 2.
1 Optimization problems such as MAXSAT, MIN NODE COVER, MAX INDEPENDENT SET, MAX CLIQUE, MIN SET COVER, TSP, KNAPSACK, BINPACKING do not have a polynomial.
CS3381 Des & Anal of Alg ( SemA) City Univ of HK / Dept of CS / Helena Wong 8. Approximation Alg Approximation.
Approximation Algorithms for the Traveling Salesperson Problem.
1 NP-Complete Problems Polynomial time vs exponential time –Polynomial O(n k ), where n is the input size (e.g., number of nodes in a graph, the length.
NP-Complete Problems (Fun part)
1 Combinatorial Dominance Analysis Keywords: Combinatorial Optimization (CO) Approximation Algorithms (AA) Approximation Ratio (a.r) Combinatorial Dominance.
9-1 Chapter 9 Approximation Algorithms. 9-2 Approximation algorithm Up to now, the best algorithm for solving an NP-complete problem requires exponential.
9-1 Chapter 9 Approximation Algorithms. 9-2 Approximation algorithm Up to now, the best algorithm for solving an NP-complete problem requires exponential.
Network Optimization Problems: Models and Algorithms
Approximation Algorithms Motivation and Definitions TSP Vertex Cover Scheduling.
Backtracking.
1 NP-Complete Problems (Fun part) Polynomial time vs exponential time –Polynomial O(n k ), where n is the input size (e.g., number of nodes in a graph,
Approximation Algorithms
Theory of Computing Lecture 10 MAS 714 Hartmut Klauck.
Algorithms for Network Optimization Problems This handout: Minimum Spanning Tree Problem Approximation Algorithms Traveling Salesman Problem.
1 Introduction to Approximation Algorithms. 2 NP-completeness Do your best then.
The Traveling Salesperson Problem Algorithms and Networks.
Chapter 12 Coping with the Limitations of Algorithm Power Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
The Traveling Salesman Problem Approximation
University of Texas at Arlington Srikanth Vadada Kishan Kumar B P Fall CSE 5311 Solving Travelling Salesman Problem for Metric Graphs using MST.
1 The TSP : NP-Completeness Approximation and Hardness of Approximation All exact science is dominated by the idea of approximation. -- Bertrand Russell.
Complexity Classes (Ch. 34) The class P: class of problems that can be solved in time that is polynomial in the size of the input, n. if input size is.
1 Introduction to Approximation Algorithms. 2 NP-completeness Do your best then.
Advanced Algorithm Design and Analysis (Lecture 13) SW5 fall 2004 Simonas Šaltenis E1-215b
Princeton University COS 423 Theory of Algorithms Spring 2001 Kevin Wayne Approximation Algorithms These lecture slides are adapted from CLRS.
The Traveling Salesman Problem Over Seventy Years of Research, and a Million in Cash Presented by Vladimir Coxall.
CSE 589 Part VI. Reading Skiena, Sections 5.5 and 6.8 CLR, chapter 37.
Projects Network Theory VLSI PSM 1. Network 1. Steiner trees
1 Approximation Algorithm Updated on 2012/12/25. 2 Approximation Algorithm Up to now, the best algorithm for solving an NP-complete problem requires exponential.
Approximation Algorithms for TSP Tsvi Kopelowitz 1.
Graph theory and networks. Basic definitions  A graph consists of points called vertices (or nodes) and lines called edges (or arcs). Each edge joins.
LIMITATIONS OF ALGORITHM POWER
Lecture 25 NP Class. P = ? NP = ? PSPACE They are central problems in computational complexity.
Algorithms for hard problems Introduction Juris Viksna, 2015.
Approximation Algorithms Guo QI, Chen Zhenghai, Wang Guanhua, Shen Shiqi, Himeshi De Silva.
Steiner Tree Problem Given: A set S of points in the plane = terminals
Approximation Algorithms
Graphs Definition: a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected.
Approximation Algorithms by bounding the OPT Instructor Neelima Gupta
Instructor Neelima Gupta Table of Contents Introduction to Approximation Algorithms Factor 2 approximation algorithm for TSP Factor.
Traveling Salesman Problem DongChul Kim HwangRyol Ryu.
1 Minimum Spanning Tree: Solving TSP for Metric Graphs using MST Heuristic Soheil Shafiee Shabnam Aboughadareh.
Approximation Algorithms
Optimization problems such as
School of Computing Clemson University Fall, 2012
Hamiltonian Cycle and TSP
Hamiltonian Cycle and TSP
Great Theoretical Ideas in Computer Science
Spanning Trees Discrete Mathematics.
Approximation Algorithms for TSP
Richard Anderson Lecture 28 Coping with NP-Completeness
Minimum Spanning Trees
Approximation Algorithms
Presentation transcript:

Approximation Algorithms: Concepts Approximation algorithm: An algorithm that returns near-optimal solutions (i.e. is "provably good“) is called an approximation algorithm. Performance Ratio (Ratio Bound): We say that an approximation algorithm for the problem has a ratio bound of  (n) if for any instance of size n, the cost C of the solution produced by the approximation algorithm is within a factor of  (n) of the cost C * of an optimal solution: max |I|= n (C/C *,C * /C)   (n)

Approximation Algorithm : Euclidean TSP Euclidean Traveling Salesman Problem: Let C 1,C 2,…,C n be a set of points in the plane corresponding to the location of n cities. Find a minimum-distance Hamiltonian cycle (traveling salesman tour) among them. An MST-based approximation (ratio bound = 2) –Fact: cost(MST) < cost(Tour opt ) Because: 1. SMT is the minimum-cost graph that connects all vertices, and has only n-1 edges. 2. any TSP tour must also connect all vertices, and will have n edges. Notice that a tour can be viewed as a spanning tree (that happens to be a chain) plus another edge.

Approximation Algorithm : Euclidean TSP –Idea: Consider the circuit that consists of a DFS traversal of MST (starting from any city), and includes an edge in the opposite direction whenever the search backtracks. And then we can take shortcut on the tour we get. Next slide is an example: DFS traversal starting from city a produces a circuit a-b-c-b-a-d… We can then use a shortcut c-d to replace original path c-b-a- d. –Note: Being in a metric space (Euclidean is just one possibility) means that the triangle inequality holds, which means that the shortcuts reduce tour cost.

Approximation Algorithm: Euclidean TSP DSF traversal of MSTTaking shortcut from DSF tour. (e.g. replacing a-b-c-b-a-d, by a-b-c-d) Tour Heur  2 * MST  2 * Tour opt abc d

Approximation Algorithm: Euclidean TSP Complexity of given approximation algorithm: The running time is dominated by MST algorithm, which, in the case of Euclidean graphs, is O(nlogn) Performance Ratio of the given approximation algorithm: 2

Approximation Algorithm: Euclidean TSP Improving the conversion from the tree traversal into a TSP tour: (Christofides 1976) –New way to look at previous conversion: we build an Eulerian circuit on top of the tree, by doubling each edge. Then we obtain the TSP tour by taking shortcuts from the Eulerian circuit. –Intuition: Tour_Heur has less cost than the cost of the Eulerian graph. So, if we can start with a lower-cost Eulerian graph, we will get a better bound  Try to get a minimum augmentation on the MST, such that the resulting graph is an Eulerian graph.

Approximation Algorithm: Euclidean TSP –Definition: –What the Eulerian graph requires: every node’s degree is even. –Property of the tree: There must be even number of node with odd-degree. (Because the sum of nodes’ degree in a tree = 2 * # of edges in the tree) –Approach: Add exact one edge for each odd-degree node in the MST. In particular, find a minimum-cost matching among the odd-degree vertices of the MST, and then add an edge between every matched pair. The result is an Eulerian graph, which we then traverse and shortcut exactly as we did with the doubled MST.

Approximation Algorithm: Euclidean TSP The SMT plus the matching: the red line is the Eulerian circuit The tour obtained by taking shortcut from Eulerian circuit

Approximation Algorithm: Euclidean TSP Consider optimal TSP tour. Nodes marked by triangles are odd-degree nodes in MST. The solid line represents the opt TSP tour. The red dashed lines and blue dashed lines represents two possible matching among those odd-degree nodes.. Either total length of blue lines or total length of red lines  0.5 * Tour opt. the minimum matching is no more costly than either the red or blue matching.If we use Min Matching, the distance of Eulerian graph will be no more than 1.5 Tour opt.

Approximation Algorithm: Euclidean TSP Ratio Bound of new approach: 3/2 Complexity: We have O(n^3) min-matching algorithm for general graphs (Gabow 1976 or Lawler 1976) and O(n^2.5(logn)^4) algorithm for Euclidean graph (Vaidya 1988).