On the Approximability of NP-complete Optimization Problems

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40 Chapter 4. Approximability classes If we now consider the new instance Inew = I1∪I2 and define Snew = S1∪S2, we have that opt 3DM (Inew) =n but max S |{w |∃yϕ(Inew,S,w, y)}| ≥ |{w |∃yϕ(Inew,Snew, w, y)}| ≥ n +1 because |{x1,...,xn} ∪ {z1,...,zn}| ≥ n +1. Here we have used the general principle A|= ∃xφ(x) ∧A⊆B⇒B|= ∃xφ(x) where φ(x) is a quantifier-free first-order formula and A⊆Bmeans that A is a submodel of B. ✷ The same proof may be applied to the corresponding bounded problems as well, though this was not noticed by the original authors. Theorem 4.19 a) Max Clique −B (Maximum clique in a graph with bounded degree) is not in Syntactic Max NP. b) Max 3SP −B (maximum bounded three-set packing) is not in Syntactic Max NP. c) Max 3DM −B (maximum bounded three dimensional matching) is not in Syntactic Max NP. Corollary 4.20 Syntactic Max NP does not contain PO PB. Proof The problems Max Clique −B and Max 2DM are not in Syntactic Max NP but are in PO PB. ✷ Thus Syntactic Max NP is a class that contains optimization problems which can be defined by the same type of formulas and can be approximated within a constant, but it is not entirely satisfactory that there are problems which can be solved in polynomial time which are not included in the class. Later we will see that the closure of Syntactic Max NP under P-reductions does contain PO PB. 4.10.2 Syntactic Max SNP Papadimitriou and Yannakakis also defined a subclass Syntactic Max SNP (strict NP) ofSyntactic Max NP using the same definition without the existence quantifier. Definition 4.16 Syntactic Max SNP is the class of NPO problems F which can be written in the form optF (I) =max S |{x :ΦF (I,S,x)}| where ΦF is a quantifier free formula, I an instance of F described as a finite structure and S a solution described as a finite structure.
Section 4.10. Classes defined by logical formulas 41 Example 4.3 Show that the problem Max 3Sat, which is the same problem as Max Sat with at most three literals in each clause, is in Syntactic Max SNP. Suppose that each clause is unique. Encode the input instance as four relations C0, C1, C2, C3 where Ci contains all clauses with exactly i negative literals. Let c =(x1,x2,x3) ∈ Ci mean that the i first variables x1,...,xi occur negated in clause c and the 3 − i remaining variables occur positive in clause c. If a clause contains less than three literals we duplicate the last variable in the encoding to fill up the three places in the clause. Now Max 3Sat can be defined as follows. opt(〈C0,C1,C2,C3〉) =max| (x1,x2,x3) : T ((x1,x2,x3) ∈ C0 ∧ (x1 ∈ T ∨ x2 ∈ T ∨ x3 ∈ T )) ∨ ∨ ((x1,x2,x3) ∈ C1 ∧ (x1 /∈ T ∨ x2 ∈ T ∨ x3 ∈ T )) ∨ ∨ ((x1,x2,x3) ∈ C2 ∧ (x1 /∈ T ∨ x2 /∈ T ∨ x3 ∈ T )) ∨ ∨ ((x1,x2,x3) ∈ C3 ∧ (x1 /∈ T ∨ x2 /∈ T ∨ x3 /∈ T )) | If we allow several copies of the same clause in the input the above definition will not work, but we can make it work by making the 3-ary relations Ci (0 ≤ i ≤ 3) 4-ary by including a clause number. For example the clause c17 = x2 ∨ x4 ∨ x11 is encoded by adding (x4,x2,x11,c17) torelationC1. Example 4.4 Show that the problem Max Ind Set −B, whichisthesame problem as Max Ind Set on a graph of degree at most B, isinSyntactic Max SNP. We encode the graph as a (B + 1)-ary relation A. There is one tuple (u, v1,...,vB) for every node u listing its neighbours v1,...,vB. If u has less than B neighbours, just repeat the last one to fill up the tuple. Now we can define Max Ind Set −B as follows, where S shall be interpreted as a set of independent nodes. opt(A) =max S |{(u, v1,...,vB) ∈ A : u ∈ S ∧ v1 /∈ S ∧ ...∧ vB /∈ S}| Since Syntactic Max SNP ⊆ Syntactic Max NP we know that every problem in Syntactic Max SNP can be approximated within a constant. Several natural optimization problems are contained in Syntactic Max SNP and have been shown to be complete in the class with respect to P-reductions. Proposition 4.21 (Papadimitriou and Yannakakis [88]) Max 3Sat, Max 3Sat −B (B ≥ 6) and Max Ind Set −B (B ≥ 7) are Syntactic Max SNP-complete under P-reductions. Max 3Sat −B is the same problem as Max 3Sat where the total number of occurrences of each variable is bounded by the constant B. Max Ind Set −B is the maximum independent set problem on a graph with degree bounded by B. Many more problems were shown to be Syntactic Max SNP-complete in [88] and further more problems will be shown to be complete in the following chapters of this thesis. In Chapter 8 there is a list of all problems currently known to be Max SNP-complete.
Page 1 and 2: On the Approximability of NP-comple
Page 3 and 4: Abstract This thesis deals with the
Page 5 and 6: Contents 1 Introduction 1 1.1 Backg
Page 7 and 8: Contents v 7 The travelling salespe
Page 9 and 10: Chapter 1 Introduction Is there any
Page 11 and 12: Section 1.3. Degrees of approximabi
Page 13 and 14: Section 1.5. Approximability of oth
Page 15 and 16: Chapter 2 Definitions 2.1 Basic not
Page 17 and 18: Section 2.3. How approximation is m
Page 19 and 20: Section 2.4. Approximation schemes
Page 21 and 22: Section 2.6. Why do NP-complete pro
Page 23 and 24: IF A ′ Section 3.2. Reencoding of
Page 25 and 26: Section 3.4. Relative error preserv
Page 27 and 28: iii) opt G(t1(x)) ≤ α · opt F (
Page 29 and 30: Section 3.5. Ratio preserving reduc
Page 31 and 32: Section 3.5. Ratio preserving reduc
Page 33 and 34: Section 3.7. Non-constructive reduc
Page 35 and 36: Section 3.8. Structure preserving r
Page 37 and 38: Chapter 4 Approximability classes 4
Page 39 and 40: Section 4.2. Fptas 31 Many of the p
Page 41 and 42: Section 4.4. Fptas ∞ and Ptas ∞
Page 43 and 44: 4.7 NPO Section 4.7. NPO 35 The cla
Page 45 and 46: Section 4.9. The existence of inter
Page 47: Section 4.10. Classes defined by lo
Page 51 and 52: Section 4.10. Classes defined by lo
Page 61 and 62: Section 4.11. The Max Ind Set class
Page 63 and 64: Section 5.2. Maximum three dimensio
Page 65 and 66: w T x y T T y w w x Section 5.2. Ma
Page 67 and 68: Section 5.2. Maximum three dimensio
Page 69 and 70: Section 5.3. Maximum three-set pack
Page 71 and 72: Section 5.5. Maximum k-dimensional
Page 73 and 74: Section 5.7. Maximum H-matching 65
Page 75 and 76: a b c b Section 5.7. Maximum H-matc
Page 79 and 80: a c b A B Section 5.7. Maximum H-ma
Page 83 and 84: Section 6.2. Maximum common induced
Page 85 and 86: x 1 − x 1 x 2 − x 2 x 3 − x 3
Page 87 and 88: x - x b Section 6.2. Maximum common
Page 89 and 90: A B 1 2 3 C D Section 6.2. Maximum
Page 91 and 92: Section 6.3. Maximum common edge su
Page 93 and 94: −→ a ←−a ←− c −→ c
Page 95 and 96: Chapter 7 The travellingsalesperson
Page 97 and 98: Section 7.3. Tsp with triangle ineq
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2−r1 Section 7.3. Tsp with triang
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Section 7.3. Tsp with triangle ineq
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Section 8.2. Problems in the plane
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Section 8.3. Summary of how all pro
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109 set of G of the same size. Once
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111 x0 may take the values [1..m],
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Section B.1. Graph theory 113 Algor
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Section B.1. Graph theory 115 I = {
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Section B.1. Graph theory 117 graph
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Section B.1. Graph theory 119 S(〈
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Section B.1. Graph theory 121 opt =
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Section B.1. Graph theory 123 Ai is
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Section B.1. Graph theory 125 n Al
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Section B.2. Network design 127 CES
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Section B.2. Network design 129 I =
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Section B.3. Sets and partitions 13
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Section B.3. Sets and partitions 13
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Section B.5. Sequencing and schedul
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Section B.6. Mathematical programmi
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m(〈A, b, c〉 ,x)=cT x = ncixi i
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Section B.7. Logic 141 I = {〈U, F
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Section B.8. Automata and language
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Section B.8. Automata and language
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Max PB 0 − 1 Programming, 36, 107
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feasible solution, 8 Fptas, 11, 30,
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graph colouring, 47, 100, 114 heigh
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Bibliography [1] A. Aho, J. E. Hopc
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Bibliography 155 [26] D. Z. Du and
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Bibliography 157 [55] V. Kann. On t
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Bibliography 159 [83] P. Orponen an
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On the Approximability of NP-complete Optimization Problems

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