TCC28102005
The techniques used in proving NP-Completeness results vary almost as widely as the
NP-Complete problems themselves, and therefore not all can be illustrated. However,
there are several general types of proofs that occur frequently and that can provide a
suggestive framework for deciding how to go about proving a new problem NPComplete. We call these a) restriction, b) local replacement, and c) component design.
Restriction
Proof by restriction is the simplest, and perhaps the most frequently applicable of the 3
types. It simply says the problem  that we want to prove NP-Complete does contain
 as a subproblem which is already known NP-complete. We have already seen the
example of EXACT COVER BY 3-SETS, that contains 3DM as a subproblem which
was already shown to be NP-Complete. Similarly DIRECTED HAMILTONIAN
CIRCUIT can be shown to be NP-Complete restricting to digraphs in which if
(u, v)  E so must (v, u )  E since then it becomes identical to HAMILTONIAN
CIRCUIT problem.
Here is a list of problems that can be proved NP-Complete by restriction.
1.MINIMUM COVER
INSTANCE: Collection C of subsets of a set S, positive integer K.
QUESTION: Does C contain a cover of S of size K or less, that is, a subset C  C
with | C  | K and such that c  S ?
Proof: Restrict X3C by allowing only instances having |c|=3 for all c  C and having
k=|S|/3.
2. HITTING SUBSET
INSTANCE: Collection C of subsets of S, positive integer K.
QUESTION: Does S contain a hitting set for C of size K or less, that is a subset
S   S with
| S  | K and such that S  contains at least one lement from each subset in C
Proof: Restrict VC by allowing only instances having |c|=2 for all c  C
3. SUBGRAPH ISOMORPHISM
INSTANCE: Two graphs, G  (V1 , E1 ) and H=(V2 , E2 )
QUESTION: Does G contain a subgraph isomorphic to H, that is, a subset
V  V1 and E  E1 such that | V || V1 |,| E || E1 | and that there exists a 1-1 function
f : V2  V satisfying {u, v}  E2 iff {f(u),f(v)}  E?
Proof: Restrict CLIQUE by allowing only instances for which H is a complete graph, that
is, E2 contains all possible edges joining 2 members of V2 ?
4. BOUNDED DEGREE SPANNING TREE
INSTANCE: A graph G=(V,E) and a positive integer K | V | 1
QUESTION: Is there a spanning tree for G in which no vertex has degree exceeding K,
E   E such that | E  || V | 1 ,
that
is,
a
subset
the
graph
G  (V , E )is connected, and no vertex is included in more than K edges from E  ?
Proof: restrict to HP by allowing only instances in which K=2.
5. KNAPSACK
INSTANCE:
A
finite
set
U,

a
“size”
s(u )  Z  and

a
value

v(u)  Z for each u U ,a size constraint B  Z , and a value goal K  Z
QUESTION: Is there a subset U   U such that  s(u )  B and  v(u)  K?
uU 
uU
Local Replacement
In proofs by local replacement, the transformations are sufficiently
nontrivial to warrant spelling out in the standard proof format, but they still
tend to be relatively uncomplicated. All we do is pick some aspect of the
known NP-complete problem instance to make up a collection of basic units,
an we obtain the corresponding instance of the target problem by replacing
each basic unit, in a uniform way, with a different structure. The
transformation from SAT to 3SAT was of this type. In that transformation,
basic units of an instance of SAT were the clauses, and each clause was
replaced by a collection of clauses according to the same general rule. The
key point is that each replacement constituted only local modification of
structure. The replacements were essentially independent of one another,
except in so far as the reflected parts of the original instance that were not
changed.
Example
PARTITION INTO TRIANGLES
INSTANCE: A graph G=(V,E), with | V | 3q for a positive integer q.
QUESTION: Is there a partition of V into q disjoint sets V1 ,V2 ,...,Vq of 3
vertices
each
such
that,
for
each
Vi  {vi[1] , vi[2] ., vi[3]}, the three edges {vi[1] , vi[2] },{vi[1] , vi[3] }{vi[2] .vi[3] } all belong to E?
Theorem 3.7 PARTITION INTO TRIANGLES is NP-Complete.
Proof: We transform X3C to PARTITION INTO TRAINGLES. Let the set
X with |X|=3q and the collection C of 3-element subsets of X be an arbitrary
instance of X3C. we shall construct a graph G=(V,E), with |V|=3q’, such that
the desired partition exists for G iff C contains an exact cover.
The basic units of the X3C instance are the 3-element subsets in C. The local
replacement
substitutes
for
each
subset
ci  {xi , yi , zi }  C the collection Ei of 18 edges as shown in the figure. Thus
G=(V,E) is defined by
VX
|C|
i 1
{ai [ j ]:1  j  9} E 
|C|
Ei
i 1
Notice that the only vertices that appear in edges belonging to more than a
single Ei are those that are in the set X. Notice also that
| V || X | 9 | C | 3q  9 | C | so that q  q  3 | C | . It is not hard to see that the
construction can be performed in polynomial time.
If c1,c2 ,..., cq are the 3-element subsets from C in any exact cover for X, then
the corresponding partition V  V1  V2  ...  Vq of V is given by taking
{ai [1], ai [2], xi },{ai [4], ai [5], yi },{ai [7], ai [8], zi },{ai [3], ai [6], ai [9]} from the vertices
meeting Ei whenever ci  {xi , yi , zi } is in the exact cover, and by taking
{ai [1], ai [2], ai [3]},{ai [4], ai [5], ai [6]},{ai [7], ai [8], ai [9]} from the vertices meeting
Ei whenever ci is not in the exact cover. This ensure that each element of X
is included in exactly one 3-vertex subset in the partition.
Conversely, if V  V1  V2  ...  Vq is any partition of G into triangles, the
corresponding exact cover is given by choosing
those
ci  C such that {ai [3], ai [6], ai [9]}  V j , for some j ,1  j  q .
Example
MINIMUM TEST COLLECTION
INSTANCE: A finite set A of “possible diagnoses”, a collection C of subsets
of A, representing binary “tests”, and a positive integer J | C |
QUESTION: Is there a subcollection C   C with | C  | J such that for every
pair ai , a j of possible diagnoses from A, there is some test c  C for which
|{ai , a j }  c | 1?
Proof: 3DM can be transformed to an instance of MINIMUM TEST
COLLECTION.
Component Design
This appears to be the most complicated of all our proof techniques. The
NP-Completeness proofs for 3DM, MATCHING, VC and HC all fall in this
category.
The basic idea is to use constituents of the target problem instance to design
certain “components” that can be combined to “realize” instances of known
NP-Complete problem.