Outline of this Chapter

Transcription

1 Outline of this Chapter Problems in DP Structure and Properties of the Boolean Hierarchy over NP Exact Graph Colorability J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 1 / 45

2 Problems in DP Reminder: Directed Hamilton Circuit Definition Directed Hamilton Circuit (DHC): Given: A directed graph G =(V(G),E(G)). Question: Does there exist a Hamilton cycle in G, i.e., a sequence (v 1,v 2,...,v n ), v i V(G), n= V(G), such that (v n,v 1 ) E(G) and (v i,v i+1 ) E(G) for 1 i <n? Theorem DHC is NP-complete. Proof: Has been presented in Kryptokomplexität I. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 2 / 45

3 Problems in DP Reminder: Hamilton Circuit Definition Hamilton Circuit (HC): Given: An undirected graph G =(V(G),E(G)). Question: Does there exist a Hamilton cycle in G, i.e., a sequence (v 1,v 2,...,v n ), v i V(G), n= V(G), such that {v n,v 1 } E(G) and {v i,v i+1 } E(G) for 1 i <n? Theorem HC is NP-complete. Proof: (Kryptokomplexität I.) Hint: Reduction from DHC. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 3 / 45

4 Problems in DP Reminder: Traveling Salesperson Problem Definition Traveling Salesperson Problem (TSP): Given: A complete undirected graph K n =(V,E), a cost function c :E N, and k N. Question: Does there exist a Hamilton cycle in K n such that the sum of the edge costs is at most k? Theorem TSP is NP-complete. Proof: TSP NP is easy to see. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 4 / 45

5 Problems in DP Reminder: TSP is NP-complete TSP is NP-hard: We show HC p mtsp. Given an undirected graph G =(V(G),E(G)) with V(G)={v 1,v 2,...,v n }, define f(g)=(k n,c,n), where K n =(V,E), V ={1,2,...,n}, and for each edge e ={i,j} of K n : 1 if {v i,v j } E(G) c({i,j})= 2 otherwise. Clearly, G HC if and only if f(g) TSP. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 5 / 45

6 Problems in DP Optimization and Search Variants of TSP Definition MinTSP: Given: A complete undirected graph K n =(V,E) and a cost function c :E N. Output: The minimum cost of a Hamilton cycle in K n with respect to c. SearchTSP: Given: A complete undirected graph K n =(V,E) and a cost function c :E N. Output: A Hamilton cycle in K n having minimum cost with respect to c. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 6 / 45

7 Problems in DP Exact Variant of the Traveling Salesperson Problem Definition Exact Traveling Salesperson Problem (Exact-TSP): Given: A complete undirected graph K n =(V,E), a cost function c :E N, and k N. Question: Is it true that tsp(k n,c)=k, where tsp(k n,c) denotes the length of an optimal tour in (K n,c)? J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 7 / 45

8 Problems in DP Exact Variant of the Traveling Salesperson Problem Fact Exact-TSP is NP-hard. Proof: HC p mexact-tsp can be shown with the same reduction f(g)=(k n,c,n), where K n =(V,E) and V ={1,2,...,n}, as in the previous proof: G HC if and only if tsp(k n,c)=n, which proves the fact. Question: Is it true that Exact-TSP NP? J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 8 / 45

9 Problems in DP Exact Variant of the Traveling Salesperson Problem Observation: Exact-TSP can be written as: Exact-TSP = {(K n,c,k) and tsp(kn,c)=k} = {(K n,c,k) and tsp(kn,c) k} {(K n,c,k) and tsp(kn,c) k} Note that: {(K n,c,k) and tsp(k n,c) k} is in NP and {(K n,c,k) and tsp(kn,c) k} is in conp. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 9 / 45

10 Problems in DP DP: Difference NP Definition DP = {L L=A B and A NP and B conp} = {L L=A B and A,B NP}. Lemma DP = {L L=L 1 L 2 and L 1,L 2 NP and L 2 L 1 }. Proof: See blackboard. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 10 / 45

11 Problems in DP Exact Variants of Other NP-complete Problems Analogously to Exact-TSP, exact variants can be defined for many NP-complete problems: Exact Vertex Cover, Exact Independent Set, Exact Clique, Exact Dominating Set, Exact-k-Color, J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 11 / 45

12 Problems in DP SAT-UNSAT Definition SAT-UNSAT: Given: A pair (H,H ) of boolean formulas in 3-CNF. Question: Is it true that H SAT and H SAT? Theorem SAT-UNSAT is DP-complete. Proof: See blackboard. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 12 / 45

13 Problems in DP Exact Independent Set Definition Exact Independent Set (XIS): Given: An undirected graph G =(V,E) and k N. Question: Is it true that mis(g)=k, where mis(g) denotes the size of a maximum independent set in G? Theorem Exact Independent Set is DP-complete. Proof: See blackboard. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 13 / 45

14 Problems in DP Critical Problems in DP Minimal-3-UNSAT: Given: A boolean formula ϕ in 3-CNF. Question: Is it true that ϕ is unsatisfiable, yet removing any clause from ϕ makes it satisfiable? Minimal-3-Uncolorability: Given: An undirected graph G. Question: Is it true that G is not 3-colorable, yet removing any one of its vertices makes it 3-colorable? Maximal Non-Hamilton Circuit: Given: An undirected graph G. Question: Is it true that G has no Hamilton cycle, yet adding any one edge to G creates one? J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 14 / 45

15 Problems in DP Critical Problems in DP Papadimitriou & Yannakakis (JCSS 1984) introduced DP to capture the complexity of problems that are known to be NP-hard or conp-hard, but not known to be in NP or conp. Critical problems tend to be extremely elusive and very hard to tackle. Papadimitriou & Yannakakis (JCSS 1984) write: This difficulty seems to reflect the extremely delicate and deep structure of critical problems too delicate to sustain any of the known reduction methods. One way to understand this is that critical graphs is usually the object of hard theorems. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 15 / 45

16 Problems in DP Critical Problems in DP Theorem (Papadimitriou & Wolfe (JCSS 1988)) Minimal-3-UNSAT is DP-complete. without proof Theorem (Cai & Meyer (SICOMP 1987)) Minimal-3-Uncolorability is DP-complete. without proof Theorem (Papadimitriou & Wolfe (JCSS 1988)) Maximal Non-Hamilton Circuit is DP-complete. without proof J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 16 / 45

17 Problems in DP Unique Solution Problems in DP Definition Unique SAT: Given: A boolean formula ϕ. Question: Does there exist a unique satisfying assignment for ϕ? Theorem Unique SAT is conp-hard and in DP. Proof: Excercise. Remark: It is still open whether Unique SAT is DP-complete. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 17 / 45

18 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP Definition (Boolean Hierarchy over NP) For classes C and D of sets, define: C D C D = {A B A C and B D}; = {A B A C and B D}. The boolean hierarchy over NP is inductively defined by: BH 0 (NP) = P, BH 1 (NP) = NP, BH 2 (NP) = NP conp, BH k (NP) = BH k 2 (NP) BH 2 (NP) for each k 3, and BH(NP) = BH k (NP). k 0 J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 18 / 45

19 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP Theorem (Cai et al. (SICOMP 1988)) For each k 1, BH k (NP)= A 1 A 2 A k A 1,A 2,...,A k NP and A k A k 1 A 1, where we agree by convention that parentheses may be omitted in such set differences: A 1 A 2 A k 1 A k =A 1 (A 2 ( (A k 1 A k ) )). without proof J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 19 / 45

20 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP (Hasse Diagram) BH(NP).. cobh 3 (NP) BH 3 (NP) codp=cobh 2 (NP) BH 2 (NP)=DP conp=cobh 1 (NP) BH 1 (NP)=NP P=BH 0 (NP) J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 20 / 45

21 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP (Venn Diagram). cobh 2 BH(NP) cobh 1 DP NP P BH 1 BH 2 conp codp. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 21 / 45

22 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP: Inclusion Structure Theorem For each k 0, BH k (NP) BH k+1 (NP) and cobh k (NP) cobh k+1 (NP) and, hence, BH k (NP) cobh k (NP) BH k+1 (NP). Proof: See blackboard. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 22 / 45

23 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP: Upward Collapse Theorem 1 For each k 0, if BH k (NP)=BH k+1 (NP), then BH k (NP)=coBH k (NP)=BH k+1 (NP)=coBH k+1 (NP)= =BH(NP). 2 For each k 1, if BH k (NP)=coBH k (NP), then BH k (NP)=coBH k (NP)=BH k+1 (NP)=coBH k+1 (NP)= =BH(NP). Proof: See blackboard. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 23 / 45

24 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP: Complete Problems Definition D k -SAT: Given: A tuple (H 1,H 2,...,H k ) of boolean formulas in CNF. Question: Does there exist an i such that H 1,...,H 2i 1 SAT and H 2i,...,H k SAT? Remark: Special cases: k =1: D 1 -SAT=SAT is NP-complete. k =2: D 2 -SAT=SAT-UNSAT is DP-complete. Theorem For each k 1, D k -SAT is BH k (NP)-complete. without proof J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 24 / 45

25 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP: Complete Problems Definition D k -Independent Set (D k -IS): Given: An undirected graph G =(V,E) and a tuple (m 1,m 2,...,m k ) of positive integers. Question: Is it true that: if k is even then 0<m 1 m 2 <m 3 m 4 < <m k 1 m k < V and mis(g) [m 1,m 2 ] [m 3,m 4 ] [m k 1,m k ]? if k is odd then 0<m 1 m 2 <m 3 m 4 < m k 1 <m k V and mis(g) [m 1,m 2 ] [m 3,m 4 ] [m k, V ]? J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 25 / 45

26 Structure and Properties of the Boolean Hierarchy over NP Boolean Hierarchy over NP: Complete Problems Remark: Special cases: k =1: D 1 -IS = IS is NP-complete. k =2 and m 1 =m 2 : D 2 -IS = XIS is DP-complete. Theorem For each k 1, D k -IS is BH k (NP)-complete. without proof J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 26 / 45

27 Structure and Properties of the Boolean Hierarchy over NP Wagner s Tool for Proving DP-Hardness Lemma (Wagner, TCS 1987) Let A be some NP-complete problem and let B be an arbitrary problem. If there exists a polynomial-time computable function f such that, for all input strings x 1 and x 2 for which x 2 A implies x 1 A, we have that (x 1 A x 2 A) f(x 1,x 2 ) B, then B is DP-hard. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 27 / 45

28 Structure and Properties of the Boolean Hierarchy over NP Wagner s Tool for Proving DP-Hardness Lemma (Wagner, TCS 1987) Let A be some NP-complete problem and let B be an arbitrary problem. If there exists a polynomial-time computable function f such that, for all input strings x 1 and x 2 for which x 2 A implies x 1 A, we have that then B is DP-hard. (x 1 A x 2 A) f(x 1,x 2 ) B, Has been applied to prove DP-completeness of, e.g., XIS, Exact-7-Color, etc. (Wagner, TCS 1987), Exact-4-Color (Rothe, IPL 2006). J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 27 / 45

29 Structure and Properties of the Boolean Hierarchy over NP Wagner s Tool for Proving DP-Hardness Lemma (Wagner, TCS 1987) Let A be some NP-complete problem and let B be an arbitrary problem. If there exists a polynomial-time computable function f such that, for all input strings x 1 and x 2 for which x 2 A implies x 1 A, we have that then B is DP-hard. (x 1 A x 2 A) f(x 1,x 2 ) B, Has been applied to prove DP-completeness of, e.g., XIS, Exact-7-Color, etc. (Wagner, TCS 1987), Exact-4-Color (Rothe, IPL 2006). Analogs for all levels of the boolean hierarchy and P NP, e.g., Dodgson Winner is P NP -complete (Hemaspaandra et al., J.ACM 1997) J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 27 / 45

30 Structure and Properties of the Boolean Hierarchy over NP Wagner s Tool for Proving BH 2k (NP)-Hardness Lemma (Wagner, TCS 1987) Let A be some NP-complete problem, B be an arbitrary problem, and k 1 a fixed integer. If there exists a polynomial-time computable function f such that, for all input strings x 1,x 2,...,x 2k for which ( j :1 j <2k) [x j+1 A x j A], we have that {i xi A} is odd f(x 1,x 2,...,x 2k ) B, then B is BH 2k (NP)-hard. Proof: See blackboard. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 28 / 45

31 Structure and Properties of the Boolean Hierarchy over NP Wagner s Tool for Proving BH 2k 1 (NP)/coBH 2k (NP)/coBH 2k 1 (NP)-Hardness Remark: Replacing 2k by 2k 1 in the lemma above yields an analogous criterium for BH 2k 1 (NP)-Hardness. Replacing odd by even yields an analogous criterium for cobh 2k (NP)- and cobh 2k 1 (NP)-Hardness, respectively. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 29 / 45

32 Exact Graph Colorability Reminder: Graph Colorability Definition Let G =(V(G),E(G)) be an undirected graph. A k-coloring of G is a mapping V(G) {1,2,...,k}. A k-coloring ψ of G is called legal if for any two vertices x and y in V(G), if {x,y} E(G) then ψ(x) ψ(y). The chromatic number of G, denoted by χ(g), is the smallest number k such that G is legally k-colorable. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 30 / 45

33 Exact Graph Colorability Reminder: Graph Colorability Definition For fixed k 1, define k-color={g G is a graph with χ(g) k}. Example: Figure: A 3-colorable graph J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 31 / 45

34 Exact Graph Colorability Exact Graph Colorability Definition For fixed k 1, define Exact-k-Color={G G is a graph with χ(g)=k}. Example: Figure: A graph in Exact-3-Color (left) and in Exact-4-Color (right) J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 32 / 45

35 Exact Graph Colorability Generalized Exact Graph Colorability Definition For fixed j 1, let M j N be a given set of j non-consecutive integers and define Exact-M j -Color={G G is a graph with χ(g) Mj }. Remark: For j =1 with M 1 ={k}, we write Exact-k-Color instead of Exact-M 1 -Color=Exact-{k}-Color. Theorem (Wagner, TCS 1987) For fixed k 1, let M k ={6k+1,6k+3,...,8k 1}. Exact-M k -Color is BH 2k (NP)-complete. In particular, for k =1 (i.e., M 1 ={7}), Exact-7-Color is DP-complete. without proof J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 33 / 45

36 Exact Graph Colorability Generalized Exact Graph Colorability Lemma There exists a polynomial-time computable function σ that p m-reduces 3-SAT to 3-Color and satisfies the following two properties: ϕ 3-SAT χ(σ(ϕ))=3; (1) ϕ 3-SAT χ(σ(ϕ))=4. (2) Proof: Use standard reduction 3-SAT p m3-color (next slides). Lemma (Guruswami and Khanna, CCC-2000) There exists a polynomial-time computable function ρ that p m-reduces 3-SAT to 3-Color and satisfies the following two properties: ϕ 3-SAT χ(ρ(ϕ))=3; (3) ϕ 3-SAT χ(ρ(ϕ))=5. (4) J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 34 / 45

37 Exact Graph Colorability Reminder: 3-Color is NP-complete Fact 2-Color is in P. without proof Theorem 3-Color is NP-complete. Proof: 1 3-Color NP is easy to see. 2 3-Color is NP-hard: We show 3-SAT p m3-color. Let ϕ(x 1,x 2,...,x n )=C 1 C 2 C m be a given 3-SAT instance with exactly three literals per clause. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 35 / 45

38 Exact Graph Colorability Reminder: 3-Color is NP-complete Define a reduction σ mapping ϕ to the graph G constructed as follows. The vertex set of G is defined by V(G) = {v 1,v 2,v 3 } {x i, x i 1 i n} {y j,k 1 j m and 1 k 6}, where the x i and x i are vertices representing the literals x i and their negations x i, respectively. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 36 / 45

39 Exact Graph Colorability Reminder: 3-Color is NP-complete The edge set of G is defined by E(G) = {{v 1,v 2 },{v 2,v 3 },{v 1,v 3 }} {{x i, x i } 1 i n} {{v 3,x i },{v 3, x i } 1 i n} {{a j,y j,1 },{b j,y j,2 },{c j,y j,3 } 1 j m} {{v 2,y j,6 },{v 3,y j,6 } 1 j m} {{y j,1,y j,2 },{y j,1,y j,4 },{y j,2,y j,4 } 1 j m} {{y j,3,y j,5 },{y j,3,y j,6 },{y j,5,y j,6 } 1 j m} {{y j,4,y j,5 } 1 j m}, where a j,b j,c j 1 i n{x i, x i } are vertices representing the literals occurring in clause C j =(a j b j c j ). J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 37 / 45

40 Exact Graph Colorability Reminder: Clause Graph in 3-SAT p m3-color a y 1 y 4 y 5 b y 2 y 6 c y 3 Figure: Graph H for clause C =(a b c) in 3-SAT p m3-color J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 38 / 45

41 Exact Graph Colorability Reminder: Properties of H in 3-SAT p m3-color 1 Vertices x i and x i corresponding to the literals x i and x i are legally colored 1 ( true ) or 2 ( false ). 2 Any coloring of the vertices a, b, and c that assigns color 1 to one of a, b, and c can be extended to a legal 3-coloring of H that assigns color 1 to y 6. Thus, if ϕ 3-SAT then G 3-Color. 3 If ψ is a legal 3-coloring of H with ψ(a)=ψ(b)=ψ(c)=i, then ψ(y 6 )=i. Thus, if ϕ 3-SAT then G 3-Color. It follows that ϕ 3-SAT σ(ϕ)=g 3-Color Clearly, reduction σ is polynomial-time computable. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 39 / 45

42 Exact Graph Colorability Guruswami Khanna Reduction: Sketch r i t i,1 s i t i,2 t i,3 Figure: Tree-like structure S i in the Guruswami Khanna reduction J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 40 / 45

43 Exact Graph Colorability Guruswami Khanna Reduction: Sketch Figure: Basic template in the Guruswami Khanna reduction J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 41 / 45

44 Exact Graph Colorability Guruswami Khanna Reduction: Sketch (111) (223) (332) (222) (331) (113) r i (333) (112) (221) (123) (132) (312) (231) (213) (321) (111) (233) (322) (222) (311) (133) t i,1 (111) (323) (232) (222) (131) (313) s i (333) (122) (211) (333) (212) (121) Figure: Connection pattern between the templates of a tree-like structure J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 42 / 45

45 Exact Graph Colorability Guruswami Khanna Reduction: Sketch Figure: Gadget connecting two leaves of the same row kind J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 43 / 45

46 Exact Graph Colorability Guruswami Khanna Reduction: Sketch Figure: Gadget connecting two leaves of the different rows kind J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 44 / 45

47 Exact Graph Colorability Exact-4-Color is DP-complete Theorem (Rothe, IPL 2001) For fixed k 1, let M k ={3k+1,3k+3,...,5k 1}. Exact-M k -Color is BH 2k (NP)-complete. In particular, for k =1 (i.e., M 1 ={4}), Exact-4-Color is DP-complete. Proof: See blackboard. J. Rothe (HHU Düsseldorf) Komplexitätstheorie II 45 / 45