table of Contents
Proof of NP-complete problems
First, limit law
Minimum coverage problems (VC)
- Examples of the problem set \ (S \) of a set of subsets \ (C \) , positive integer \ (K \) . Q \ (C \) if there are \ (S \) size of not more than \ (K \) coverage, i.e. contains a subset \ (C '\ subseteq C \ ) such that \ (| C' | = k \) and \ (\ bigcup C '= S \) .
- Proof: Limit \ (\ FORALL c \ in C \) , \ (| c | = 3 \) , \ (| S | = SK \) , compared X3C problem.
Subgraph isomorphism problem
- Problem instance
FIG \ (G = (V_1, E_1) \) , \ (H = (V_2, E_2) \) .
Q \ (G \) if there is isomorphic in \ (H \) submap, i.e., whether there is a subset \ (V \ subseteq V_1 \) , \ (E \ subseteq E_1 \) , so \ (| V | = | V_2 | \) , \ (| E | = | E_2 | \) , and there is a bijective function \ (F: V_2 \ to V \) , so \ ((u, v) \ in E \ Longleftrightarrow (f (U), f (v)) \ in E_2 \) . - Proof: limit \ (H \) is a complete graph, and \ (| V_2 | = K \) , then the problem is the group.
0-1 Backpack (Knapsack)
- Problem instance
a finite set \ (U \) , \ (\ FORALL U \ in U \) , the size \ (S (U) \ in the Z-^ + \) , the value of \ (v (u) \ in Z ^ + \ ) , size constraints \ (B \ in the Z-^ + \) , the target value \ (K \ in the Z-^ + \) .
Asked whether there subset \ (the U-'\ subseteq the U-\) , such that
\ [\ begin {equation} \ sum _ {\ boldsymbol {u} \ in U} \ boldsymbol {s} (\ boldsymbol {u}) \ leq \ boldsymbol {B}, \ quad \ sum _ {\ boldsymbol {u} \ in \ boldsymbol {U} ^ {\ prime}} \ boldsymbol {v} (\ boldsymbol {u}) \ geq \ boldsymbol {K} \ end { equation} \] - Proof: limit \ (\ FORALL V \ in the U-\) , \
[S (U) = V (U) \]
\ [B = \ left \ lfloor \ FRAC {. 1} {2} \ sum_ {U \ in U} s (u) \ right
\ rfloor, \ quad K = \ left \ lceil \ frac {1} {2} \ sum_ {u \ in U} v (u) \ right \ rceil \] becomes the equalization of .
Covering three yuan proper set (X3C)
- Problem instance
finite set \ (S \) , \ (| S | = 3q \) , \ (S \) ternary subset \ (C \) .
Q. Is there a \ (C '\ subseteq C \) , so that \ (S \) each element appears exactly \ (C' \) a member of. - Proof: Limit
- \(S=W \cup X \cup Y\)
- \(|W|=|X|=|Y|=q\)
- \ (C = \ {(w , x, y) | (w, x, y) \ in W \ times X \ times Y) \} \) then \ (| C '| = Q \) , and \ ( C '\) any two elements do not intersect, become 3DM problem.
Focus set
- Set \ (S \) subset of the set of \ (C \) , positive integer \ (K \) . Q \ (S \) contains \ (C \) is no larger than \ (k \) centralized collection, that is, whether there is \ (S '\ subseteq S \) , \ (| S' | \ Leq k \) , so that \ (S '\) comprises at least \ (C \) is a subset of each element.
- Proof: limit \ (\ FORALL C \ in C \) , \ (| C | = 2 \) , so \ (V = S \) , \ (E = C \) , the configuration of FIG \ (G = ( V, E) \) vertex cover problem.
There spanning tree of bounded degree
Problem instance
FIG \ (G = (V, E
) \) Q \ (G \) contains no more than one degree of vertex \ (K \) spanning tree, i.e., whether there is a subset \ (E '\ subseteq E \ ) , \ (| E '| = | V | -1 \) , FIG \ (G' = (V, E ') \) is connected, and \ (V \) in each vertex included in up \ ( E '\) a \ (K \) Article sides.Proof: limit \ (K = 2 \) , was the problem Hamiltonian path problem.
Multiprocessor Scheduling
- Problem instance a finite set of tasks \ (A \) , \ (\ FORALL A \ in A \) , the length \ (L (A) \ in the Z ^ + \) , the number of processor units \ (m \ in Z ^ + \ ) , the deadline \ (D \ in the Z-^ + \) .
To ask whether there is a set of disjoint \ (A_1, A_2, ..., A_m \) such that
\ [A = A_. 1 {} \ {2} A_ Cup \ Cup \ ldots \ A_ Cup {m} \]
\ [\ max \ left \ {\ sum_ { a \ in A_ {i}} l (a): 1 \ leq i \ leq m \ right \} \ leq D \] - Proof: limit \ (m = 2 \) , \
[D = \ left \ lfloor \ FRAC. 1} {2} {\ sum_ {A \ A} in L (A) \ right \ rfloor \]
becomes the average problem.
Second, the partial substitution method
Some elements selected completely known problem instance as a basic unit, the basic unit and the replacement to the specified configuration, so as to obtain a corresponding instance of the target problem.