Lecture Notes Discrete Optimization - Applied Mathematics

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9.4 Examples of NP-completeness Proofs . . . . . . . . . . . . . . . . . . . 72 9.5 More on Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . 76 9.5.1 NP-hard Problems . . . . . . . . . . . . . . . . . . . . . . . . . 76 9.5.2 Complexity Class co-NP . . . . . . . . . . . . . . . . . . . . . . 77 9.5.3 Pseudo-polynomiality and Strong NP-completeness . . . . . . . . 78 9.5.4 Complexity Class PSPACE . . . . . . . . . . . . . . . . . . . . . 79 10 Approximation Algorithms 80 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 10.2 Approximation Algorithm for Vertex Cover . . . . . . . . . . . . . . . . 81 10.3 Approximation Algorithms for TSP . . . . . . . . . . . . . . . . . . . . 82 10.4 Approximation Algorithm for Steiner Tree . . . . . . . . . . . . . . . . . 85 10.5 Approximation Scheme for Knapsack . . . . . . . . . . . . . . . . . . . 87 10.5.1 Dynamic Programming Approach . . . . . . . . . . . . . . . . . 88 10.5.2 Deriving a FPTAS for Knapsack . . . . . . . . . . . . . . . . . . 89 v
1. Preliminaries 1.1 <strong>Optimization</strong> Problems We first formally define what we mean by an optimization problem. The definition below focusses on minimization problems. Note that it extends naturally to maximization problems. Definition 1.1. A minimization problem Π is given by a set of instancesI. Each instance I ∈I specifies • a set F of feasible solutions for I; • a cost function c :F →R. Given an instance I = (F,c) ∈I, the goal is to find a feasible solution S ∈ F such that c(S) is minimum. We call such a solution an optimal solution of I. In discrete (or combinatorial) optimization we concentrate on optimization problems Π, where for every instance I = (F,c) the set F of feasible solutions is discrete, i.e., F is finite or countably infinite. We give some examples below. Minimum Spanning Tree Problem (MST): Given: Goal: We have An undirected graph G=(V,E) with edge costs c : E →R. Find a spanning tree of G of minimum total cost. F ={T ⊆ E | T is a spanning tree of G} and c(T)= ∑ c(e). e∈T Traveling Salesman Problem (TSP): Given: An undirected graph G=(V,E) with distances d : E →R. Goal: Find a tour of G of minimum total length. Here we have F ={T ⊆ E | T is a tour of G} and c(T)= ∑ d(e) e∈T Linear Programming (LP): Given: A set F of feasible solutions x=(x 1 ,...,x n ) defined by m linear constraints { F = (x 1 ,...,x n )∈R n n } ≥0 | ∑ i j x i ≥ b j ∀ j = 1,...,m i=1a Goal: and an objective function c(x)=∑ n i=1 c ix i . Find a feasible solution x∈F that minimizes c(x). 1
Page 1 and 2: Document last modified: January 10,
Page 3 and 4: Changes made with respect to Lectur
Page 5: 5.1 Introduction . . . . . . . . .
Page 9 and 10: whether a given edge(i, j) is part
Page 11 and 12: 1.6 Basics of Linear Programming Th
Page 13 and 14: 2. Minimum Spanning Trees 2.1 Intro
Page 15 and 16: X v ¯X u e e ′ T X v ¯X u e e
Page 17 and 18: Input: undirected graph G=(V,E) wit
Page 19 and 20: d(w) changes, then it can only decr
Page 21 and 22: Input: independent set system(S,I)
Page 23 and 24: Proof. We first show that the greed
Page 25 and 26: thus c(P ′ ) ≤ c(P). It therefo
Page 27 and 28: That is, d(v) decreases throughout
Page 29 and 30: Input: directed graph G=(V,E), nonn
Page 31 and 32: Input: directed graph G=(V,E), nonn
Page 33 and 34: 5. Maximum Flows 5.1 Introduction T
Page 35 and 36: p 12 q p 12 q 11 5 11 19 s 8 5 11 3
Page 37 and 38: Input: directed graph G=(V,E), capa
Page 39 and 40: (3)⇒(1): By Lemma 5.4, the value
Page 41 and 42: Note that the distance of u from s
Page 43 and 44: 2 p 1,1 −2 q p 1 q 2,1 1,1 x 2 2,
Page 45 and 46: flow by x·c(u,v) for every backwar
Page 47 and 48: Input: directed graph G=(V,E), capa
Page 49 and 50: V + x and Vx − , respectively, be
Page 51 and 52: 2,4 0,0 p 3,3 0,4 0,−2 p 2,3 4,0
Page 53 and 54: which is equivalent to c π (e)0 th
Page 55 and 56: 4,0 s 0,2 0,2 0,0 p 2,1 1,1 x 0,0 1
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M M ∗ M△ M ∗ Figure 11: Illus
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Input: undirected bipartite graph G
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Let G ′ be the resulting graph an
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Now, fix an arbitrary matching M an
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Proof. Let x∈conv-hull(S). Then w
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···+λ k x k of vectors x 1 ,...
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Theorem 8.6. Let A ∈ Z m×n be a
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The size of a maximum matching can
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9. Complexity Theory 9.1 Introducti
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[6]). This algorithm is an example
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Π 2 which correspond to easy insta
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Theorem 9.1. SAT is NP-complete. Th
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this mapping can be done in polynom
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other than explicitly listing L sub
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complete problem, unless P=NP. Proo
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solution S∈F whose cost c(S) is a
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We first show the following inappro
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Input: Complete graph G=(V,E) with
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Input: Complete graph G=(V,E) with
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Input: A set N ={1,...,n} of items
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References [1] R. K. Ahuja, T. L. M
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Lecture Notes Discrete Optimization - Applied Mathematics

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