introduction to approximation algorithms
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Approximation Algorithms
Jhoirene B Clemente
Algorithms and Complexity LabDepartment of Computer Science
University of the Philippines Diliman
October 14, 2014
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Overview
1. Combinatorial Optimization Problems2. NP-Hard Problems
2.1 Clique2.2 Independent Set Problem2.3 Vertex Cover2.4 Traveling Salesman Problem
3. Approaches in Solving Hard Problems4. Approximation Algorithms5. Approximable Problems6. Inapproximable Problems
50
ApproximationAlgorithms
Jhoirene B Clemente
3 Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Optimization Problems[Papadimitriou and Steiglitz, 1998]
Definition (Instance of an Optimization Problem)An instance of an optimization problem is a pair (F , c), where Fis any set, the domain of feasible points; c is the cost function, amapping
c : F → R
The problem is to find an f ∈ F for which
c(f ) ≤ c(y) ∀ y ∈ F
Such a point f is called a globally optimal solution to the giveninstance, or, when no confusion can arise, simply an optimalsolution.
Definition (Optimization Problem)An optimization problem is a set of instances of an optimizationproblem.
50
ApproximationAlgorithms
Jhoirene B Clemente
3 Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Optimization Problems[Papadimitriou and Steiglitz, 1998]
Definition (Instance of an Optimization Problem)An instance of an optimization problem is a pair (F , c), where Fis any set, the domain of feasible points; c is the cost function, amapping
c : F → R
The problem is to find an f ∈ F for which
c(f ) ≤ c(y) ∀ y ∈ F
Such a point f is called a globally optimal solution to the giveninstance, or, when no confusion can arise, simply an optimalsolution.
Definition (Optimization Problem)An optimization problem is a set of instances of an optimizationproblem.
50
ApproximationAlgorithms
Jhoirene B Clemente
4 Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Optimization Problems[Papadimitriou and Steiglitz, 1998]
Two categories1. with continuous variables, where we look for a set of real
numbers or a function2. with discrete variables, which we call combinatorial, where
we look for an object from a finite, or possibly countablyinfinite set, typically an integer, set, permutation, or graph.
50
ApproximationAlgorithms
Jhoirene B Clemente
5 Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Combinatorial Optimization Problem
Definition (Combinatorial Optimization Problem[Papadimitriou and Steiglitz, 1998])An optimization problem Π = (DΠ,RΠ, costΠ, goalΠ) consists of1. A set of valid instances DΠ. Let I ∈ DΠ, denote an input
instance.2. Each I ∈ DΠ has a set of feasible solutions, RΠ(I ).3. Objective function, costΠ, that assigns a nonnegative
rational number to each pair (I ,SOL), where I is an instanceand SOL is a feasible solution to I.
4. Either minimization or maximization problem:goalΠ ∈ {min,max}.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
6 Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
NP-Hard Problems
1. Clique Problem2. Independent Set Problem3. Vertex Cover Problem4. Traveling Salesman Problem
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization Problems
7 Clique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Max Clique Problem
Definition (Max Clique Problem)Given a complete weighted graph, find the largest clique
TheoremMax Clique is NP-hard [Garey and Johnson, 1979].
PropositionThe decision variant of MAX-SAT is NP-Complete[Garey and Johnson, 1979].
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization Problems
7 Clique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Max Clique Problem
Definition (Max Clique Problem)Given a complete weighted graph, find the largest clique
TheoremMax Clique is NP-hard [Garey and Johnson, 1979].
PropositionThe decision variant of MAX-SAT is NP-Complete[Garey and Johnson, 1979].
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization Problems
7 Clique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Max Clique Problem
Definition (Max Clique Problem)Given a complete weighted graph, find the largest clique
TheoremMax Clique is NP-hard [Garey and Johnson, 1979].
PropositionThe decision variant of MAX-SAT is NP-Complete[Garey and Johnson, 1979].
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization Problems
8 Clique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Max Clique Reduction from SAT
Example
(a ∨ b ∨ c) ∧ (b ∨ c ∨ d) ∧ (a ∨ c ∨ d) ∧ (a ∨ b ∨ d)
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization Problems
9 Clique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Max Clique Reduction from SAT
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
10 Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Independent Set
Definition (Max Independent Set Problem)Given a complete weighted graph, find the largest independent set
TheoremIndependent Set Problem is NP-hard [Garey and Johnson, 1979].
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
10 Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Independent Set
Definition (Max Independent Set Problem)Given a complete weighted graph, find the largest independent set
TheoremIndependent Set Problem is NP-hard [Garey and Johnson, 1979].
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
11 Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Vertex Cover Problem
Definition (Min Vertex Cover Problem)Given a complete weighted graph, find the minimum vertex cover.
TheoremVertex Cover Problem is NP-hard [Garey and Johnson, 1979].
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
11 Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Vertex Cover Problem
Definition (Min Vertex Cover Problem)Given a complete weighted graph, find the minimum vertex cover.
TheoremVertex Cover Problem is NP-hard [Garey and Johnson, 1979].
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
12 Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Traveling Salesman Problem (TSP)
DefinitionINPUT: Edge weighted Graph G = (V ,E)OUTPUT: Minimum cost Hamiltonian Cycle.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
13 Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Approaches in Solving NP-hard Problems
I Exact algorithm always obtain the optimal solutionI Approximation algorithm settle for good enough solutions.
Goodness of solution is guaranteed and measured usingapproximation ratio.
I Heuristic algorithms produce solutions, which are notguaranteed to be close to the optimum. The performance ofheuristics is often evaluated empirically.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
14 Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Approaches in Solving NP-hard Problems
Design technique have a well specified structure that evenprovides a framework for possible implementations
I Dynamic ProgrammingI Greedy SchemaI Divide and conquerI Branch and BoundI Local searchI . . .
Concepts formulate ideas and rough frameworks about how toattack hard algorithmic problems .
I Approximation AlgorithmsI Parameterized AlgorithmsI Randomized AlgorithmsI . . .
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
14 Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Approaches in Solving NP-hard Problems
Design technique have a well specified structure that evenprovides a framework for possible implementations
I Dynamic ProgrammingI Greedy SchemaI Divide and conquerI Branch and BoundI Local searchI . . .
Concepts formulate ideas and rough frameworks about how toattack hard algorithmic problems .
I Approximation AlgorithmsI Parameterized AlgorithmsI Randomized AlgorithmsI . . .
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
15 ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Approximation Algorithms
Definition (Approximation Algorithm[Williamson and Shmoy, 2010] )An ρ-approximation algorithm for an optimization problem is apolynomial-time algorithm that for all instances of the problemproduces a solution whose value is within a factor of ρ of thevalue of an optimal solution.Given an problem instance I with an optimal solution Opt(I ), i.e.the cost function cost(Opt(I )) is minimum/maximum.
I An algorithm for a minimization problem is calledρ-approximative algorithm for some ρ > 1, if the algorithmobtains a maximum cost of ρ · cost(Opt(I )), for any inputinstance I .
I An algorithm for a maximization problem is calledρ-approximative algorithm, for some ρ < 1, if the algorithmobtains a minimum cost of ρ · cost(Opt(I )), for any inputinstance I .
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
15 ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Approximation Algorithms
Definition (Approximation Algorithm[Williamson and Shmoy, 2010] )An ρ-approximation algorithm for an optimization problem is apolynomial-time algorithm that for all instances of the problemproduces a solution whose value is within a factor of ρ of thevalue of an optimal solution.Given an problem instance I with an optimal solution Opt(I ), i.e.the cost function cost(Opt(I )) is minimum/maximum.
I An algorithm for a minimization problem is calledρ-approximative algorithm for some ρ > 1, if the algorithmobtains a maximum cost of ρ · cost(Opt(I )), for any inputinstance I .
I An algorithm for a maximization problem is calledρ-approximative algorithm, for some ρ < 1, if the algorithmobtains a minimum cost of ρ · cost(Opt(I )), for any inputinstance I .
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
15 ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Approximation Algorithms
Definition (Approximation Algorithm[Williamson and Shmoy, 2010] )An ρ-approximation algorithm for an optimization problem is apolynomial-time algorithm that for all instances of the problemproduces a solution whose value is within a factor of ρ of thevalue of an optimal solution.Given an problem instance I with an optimal solution Opt(I ), i.e.the cost function cost(Opt(I )) is minimum/maximum.
I An algorithm for a minimization problem is calledρ-approximative algorithm for some ρ > 1, if the algorithmobtains a maximum cost of ρ · cost(Opt(I )), for any inputinstance I .
I An algorithm for a maximization problem is calledρ-approximative algorithm, for some ρ < 1, if the algorithmobtains a minimum cost of ρ · cost(Opt(I )), for any inputinstance I .
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
16 ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Approximation Ratio
I Minimization, ρ > 1
cost(Opt(I )) ≤ cost(SOL) ≤ ρ cost(Opt(I ))
I Maximization, ρ < 1
ρ cost(Opt(I )) ≤ cost(SOL) ≤ cost(Opt(I ))
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
17 ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Approximation Ratio
I Additive Approximation Algorithms:
SOL ≤ OPT + c
I Constant Approximation Algorithms:
SOL ≤ c ·OPT
I Logarithmic Approximation Algorithms:
SOL = O(log n) ·OPT
I Polynomial Approximation Algorithms:
SOL = O(nc) ·OPT ,
where c ≤ 1
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
18 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Example: Vertex Cover Problem
Definition (Vertex Cover [Vazirani, 2001] )Given a graph G = (V ,E), a vertex cover is a subset C ⊂ Vsuch that every edge has at least one end point incident at C .
Definition (Minimum Vertex Cover Problem)Given a complete weighted graph G = (V ,E), find a minimumcardinality vertex cover C .
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
18 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Example: Vertex Cover Problem
Definition (Vertex Cover [Vazirani, 2001] )Given a graph G = (V ,E), a vertex cover is a subset C ⊂ Vsuch that every edge has at least one end point incident at C .
Definition (Minimum Vertex Cover Problem)Given a complete weighted graph G = (V ,E), find a minimumcardinality vertex cover C .
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
18 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Example: Vertex Cover Problem
Definition (Vertex Cover [Vazirani, 2001] )Given a graph G = (V ,E), a vertex cover is a subset C ⊂ Vsuch that every edge has at least one end point incident at C .
Definition (Minimum Vertex Cover Problem)Given a complete weighted graph G = (V ,E), find a minimumcardinality vertex cover C .
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
19 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
Maximal Matching
Definition (Matchings)Given a graph G = (V ,E), a subset M ⊂ E is called a matchingif no two edges in M are adjacent in G.
Figure : (a) Maximal matching (b) Perfect matching
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
20 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
2-Approximation Algorithm
1. Find a maximal matching in G, and output the set ofmatched vertices.
Algorithm 1: 2-Approximation Algorithm for minimum vectorcover problem
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
21 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
2-Approximation Algorithm
TheoremAlgorithm 1 is a 2-approximation algorithm.
Proof.Let M be a maximal matching in G = (V ,E) and OPT be theminimum vertex cover in G, then
|M | ≤ |OPT |
The cover picked by the algorithm has cardinality |SOL| ≤ 2 · |M |.
|SOL| ≤ 2 · |OPT |
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
21 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
2-Approximation Algorithm
TheoremAlgorithm 1 is a 2-approximation algorithm.
Proof.Let M be a maximal matching in G = (V ,E) and OPT be theminimum vertex cover in G, then
|M | ≤ |OPT |
The cover picked by the algorithm has cardinality |SOL| ≤ 2 · |M |.
|SOL| ≤ 2 · |OPT |
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
22 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
2-Approximation Algorithm
Example:
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithms
22 Vertex Cover
Traveling Salesman Problem
References
CS 397October 14, 2014
2-Approximation Algorithm
Example:
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
23 Traveling Salesman Problem
References
CS 397October 14, 2014
Solving Traveling Salesman Problem
INPUT: Edge weighted graph1. Compute a Minimum Spanning Tree for G2. Select a vertex r ∈ V (G) to be the root vertex3. Let L be the list of vertices visited in a preorder walk
OUTPUT: Hamiltonian Cycle that visitsA preorder tree walk recursively visits every vertex in thetree, listing a vertex when it is first encountered , beforeany of its children are visited.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
24 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
25 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
26 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
27 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
28 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
29 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
30 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
31 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
32 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
33 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
34 Traveling Salesman Problem
References
CS 397October 14, 2014
Computing the Minimum Spanning Tree
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
35 Traveling Salesman Problem
References
CS 397October 14, 2014
Preorder
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
36 Traveling Salesman Problem
References
CS 397October 14, 2014
Preorder
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
37 Traveling Salesman Problem
References
CS 397October 14, 2014
Preorder
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
38 Traveling Salesman Problem
References
CS 397October 14, 2014
Preorder
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
39 Traveling Salesman Problem
References
CS 397October 14, 2014
Preorder
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
40 Traveling Salesman Problem
References
CS 397October 14, 2014
Preorder
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
41 Traveling Salesman Problem
References
CS 397October 14, 2014
Preorder
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
42 Traveling Salesman Problem
References
CS 397October 14, 2014
Hamiltonian Cycle from the Preorder Walk
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
43 Traveling Salesman Problem
References
CS 397October 14, 2014
Hamiltonian Cycle from the Preorder Walk
4 + 4 + 10 + 15 + 10 + 16 = 59
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
44 Traveling Salesman Problem
References
CS 397October 14, 2014
Total tour cost vs. the optimal cost
Let H ∗ denote an optimal Hamiltonian tour in G. Let T be theminimum spanning tree of G.
c(T ) ≤ c(H ∗)
Let W be the full length cost of the preorder walk.
c(W ) = 2c(T )
Let H be the output Hamiltonian Cycle from the algorithm
c(W ) ≥ c(H )
Thenc(H ) ≤ 2c(H ∗)
Approximation ratio (ρ) is 2
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
44 Traveling Salesman Problem
References
CS 397October 14, 2014
Total tour cost vs. the optimal cost
Let H ∗ denote an optimal Hamiltonian tour in G. Let T be theminimum spanning tree of G.
c(T ) ≤ c(H ∗)
Let W be the full length cost of the preorder walk.
c(W ) = 2c(T )
Let H be the output Hamiltonian Cycle from the algorithm
c(W ) ≥ c(H )
Thenc(H ) ≤ 2c(H ∗)
Approximation ratio (ρ) is 2
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
45 Traveling Salesman Problem
References
CS 397October 14, 2014
Approximable Problems [Vazirani, 2001]
Definition (APX)I An abbreviation for “Approximable", is the set of NP
optimization problems that allow polynomial-timeapproximation algorithms with approximation ratio boundedby a constant.
I Problems in this class have efficient algorithms that can findan answer within some fixed percentage of the optimalanswer.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
45 Traveling Salesman Problem
References
CS 397October 14, 2014
Approximable Problems [Vazirani, 2001]
Definition (APX)I An abbreviation for “Approximable", is the set of NP
optimization problems that allow polynomial-timeapproximation algorithms with approximation ratio boundedby a constant.
I Problems in this class have efficient algorithms that can findan answer within some fixed percentage of the optimalanswer.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
46 Traveling Salesman Problem
References
CS 397October 14, 2014
Polynomial-time Approximation Schemes
Definition (PTAS [Aaronson et al., 2008])The subclass of NPO problems that admit an approximationscheme in the following sense.For any ε > 0, there is a polynomial-time algorithm that isguaranteed to find a solution whose cost is within a 1 + ε factorof the optimum cost. Contains FPTAS, and is contained in APX.
Definition (FPTAS [Aaronson et al., 2008] )The subclass of NPO problems that admit an approximationscheme in the following sense.For any ε > 0, there is an algorithm that is guaranteed to find asolution whose cost is within a 1 + ε factor of the optimum cost.Furthermore, the running time of the algorithm is polynomial in n(the size of the problem) and in 1/ε.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
46 Traveling Salesman Problem
References
CS 397October 14, 2014
Polynomial-time Approximation Schemes
Definition (PTAS [Aaronson et al., 2008])The subclass of NPO problems that admit an approximationscheme in the following sense.For any ε > 0, there is a polynomial-time algorithm that isguaranteed to find a solution whose cost is within a 1 + ε factorof the optimum cost. Contains FPTAS, and is contained in APX.
Definition (FPTAS [Aaronson et al., 2008] )The subclass of NPO problems that admit an approximationscheme in the following sense.For any ε > 0, there is an algorithm that is guaranteed to find asolution whose cost is within a 1 + ε factor of the optimum cost.Furthermore, the running time of the algorithm is polynomial in n(the size of the problem) and in 1/ε.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
47 Traveling Salesman Problem
References
CS 397October 14, 2014
Approximation Ratio of well known HardProblems
1. FPTAS: Bin Packing Problem2. PTAS: Makespan Scheduling Problem3. APX:
3.1 Min Steiner Tree Problem (1.55 Approximable) [Robins,2005]3.2 Min Metric TSP (3/2 Approximable) [Christofides,1977]3.3 Max SAT (0.77 Approximable) [Asano,1997]3.4 Vertex Cover (2 Approximable) [Vazirani,2001]
4. MAX SNP4.1 Independent Set Problem4.2 Clique Problem4.3 Travelling Salesman Problem
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
48 Traveling Salesman Problem
References
CS 397October 14, 2014
Complexity Classes [Ausiello et al., 2011]
FPTAS ( PTAS ( APX ( NPO
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
49 Traveling Salesman Problem
References
CS 397October 14, 2014
Inapproximable Problems
Many problems have polynomial-time approximation schemes.However, there exists a class of problems that is not so easy[Williamson and Shmoy, 2010]. This class is called MAX SNP.
TheoremFor any MAXSNP-hard problem, there does not exist apolynomial-time approximation scheme, unless P = NP[Williamson and Shmoy, 2010].
TheoremIf P 6= NP, then for any constant ρ ≥ 1, there is nopolynomial-time approximation algorithm with approximationratio ρ for the general travelling salesman problem.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
49 Traveling Salesman Problem
References
CS 397October 14, 2014
Inapproximable Problems
Many problems have polynomial-time approximation schemes.However, there exists a class of problems that is not so easy[Williamson and Shmoy, 2010]. This class is called MAX SNP.
TheoremFor any MAXSNP-hard problem, there does not exist apolynomial-time approximation scheme, unless P = NP[Williamson and Shmoy, 2010].
TheoremIf P 6= NP, then for any constant ρ ≥ 1, there is nopolynomial-time approximation algorithm with approximationratio ρ for the general travelling salesman problem.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
49 Traveling Salesman Problem
References
CS 397October 14, 2014
Inapproximable Problems
Many problems have polynomial-time approximation schemes.However, there exists a class of problems that is not so easy[Williamson and Shmoy, 2010]. This class is called MAX SNP.
TheoremFor any MAXSNP-hard problem, there does not exist apolynomial-time approximation scheme, unless P = NP[Williamson and Shmoy, 2010].
TheoremIf P 6= NP, then for any constant ρ ≥ 1, there is nopolynomial-time approximation algorithm with approximationratio ρ for the general travelling salesman problem.
50
ApproximationAlgorithms
Jhoirene B Clemente
Optimization Problems
Hard CombinatorialOptimization ProblemsClique
Independent Set Problem
Vertex Cover
Approaches in SolvingHard Problems
ApproximationAlgorithmsVertex Cover
Traveling Salesman Problem
50 References
CS 397October 14, 2014
References I
Aaronson, S., Kuperberg, G., and Granade, C. (2008).The complexity zoo.
Ausiello, G., Bonifaci, V., and Escoffier, B. (2011).Complexity and approximation in reoptimization.In Computability in Context: Computation and Logic in the Real World,volume 2, pages 101–129.
Garey, M. R. and Johnson, D. S. (1979).Computers and Intractability: A Guide to the Theory of NP-Completeness.
Papadimitriou, C. and Steiglitz, K. (1998).Combinatorial optimization: algorithms and complexity.
Vazirani, V. V. (2001).Approximation algorithms, volume 1997.
Williamson, D. and Shmoy, D. (2010).The Design of Approximation Algorithms.
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