independently by Lewis, Dobkin and Lipton in [LDL] and [L]. However our results are ..... [uig,Ujg] U [efg h I 1 ~ h ~ s-l} and the addition of some interconnecting ...
% NODE-AND
EDGE-DELETION NP-COMPLETE
PROBLEMS
Mihalis Y a n n a k a k i s PRINCETON UNIVERSITY Princeton, N e w Jersey
ABSTRACT If ~ is a graph property, the general node(edge) deletion p r o b l e m can be stated as follows: Find the m i n i m u m n u m b e r of nodes(edges), whose d e l e t i o n results in a subgraph s a t i s f y i n g property ~. In this paper we show that if ~ belongs to a rather b r o a d class of p r o p e r t i e s (the class of p r o p e r t i e s that are h e r e d i t a r y on induced subgraphs) then the n o d e - d e l e t i o n p r o b l e m is NP-complete, and the same is true for several r e s t r i c t i o n s of it. For the same class of properties, r e q u i ~ i n g the rem a i n i n g graph to be c o n n e c t e d does not change the N P - c o m p l e t e status of the problem; m o r e o v e r for a certain subclass, finding any "reasonable" a p p r o x i m a t i o n is also NP-complete. E d g e - d e l e t i o n problems seem to be less amenable to such generalizations. We show h o w e v e r that for several common p r o p e r t i e s (e.g. planar, outer-planar, line-graph, transitive digraph) the e d g e - d e l e t i o n p r o b l e m is NPcomplete.
the m a x clique, the feedback-node set, the f e e d b a c k - a r c set [K], and the simple m a x - c u t p r o b l e m [GJS]) can be formulated in an obvious w a y as node or edge deletion problems, s p e c i f y i n g a p p r o p r i a t e l y the p r o p e r t y ~. Furthermore, K r i s h n a m o o rthy and Deo showed in a recent paper [KD] that the n o d e - d e l e t i o n p r o b l e m for several o t h e r p r o p e r t i e s is also NPcomplete. (For an e x p o s i t i o n of NPcompleteness, see [GJ]). Since Cook's i n t r o d u c t i o n of the concept of N P - c o m p l e t e n e s s , the list of N P - c o m p l e t e problems has expanded rapidly, with more and more individual problems from various areas b e i n g added to it [GJ] Sections 2 and 3 are c o n c e r n e d with nodedeletion problems. O u r aim is to show, h o w this set of similar problems (with p r o p e r t i e s , drawn from a certain large class of properties) can be treated in a s y s t e m a t i c fashion in o r d e r to prove the N P - c o m p l e t e n e s s of all the members of the set. (A similar approach was taken also i n d e p e n d e n t l y by Lewis, Dobkin and Lipton in [LDL] and [L]. H o w e v e r our results are s i g n i f i c a n t l y more general than theirs.). In this paper we consider p r o p e r t i e s that are h e r e d i t a r y on induced subgraphs, i,e. if G is a graph (or digraph) with p r o p e r t y ~, then deletion of any node does not produce a graph v i o l a t i n g ~. We call a p r o p e r t y n o n t r i v i a l if it is true for a single node and is not satisfied by all the graphs in a given input domain. c l e a r l y the n o d e - d e l e t i o n p r o b l e m makes sense o n l y for n o n t r i v i a l properties. W e will require ~ to be easy (i.e. in p ) at least to recognize (although o u r results are valid even if , cannot be r e c o g n i z e d in n o n d e t e r m i n i s t i c p o l y n o m i a l time. In this case, the 'NPcomplete' clause should be replaced by 'NP-hard').
KEYWORDS¢ approximation, c o m p u t a t i o n a l complexity, edge-deletion, graph, graphproperty, hereditary, m a x i m u m subgraph, node-deletion, N P - c o m p l e t e , p o l y n o m i a l hierarchy. i.
INTRODUCTION
The general node(edge) deletion p r o b l e m can be stated as follows: Given a graph or digraph G find a set of nodes(edges) of m i n i m u m cardinality, whose deletion results in a subgraph or subdigraph s a t i s f y i n g the p r o p e r t y ~. (For the standard graph theory t e r m i n o l o g y the reader is referred to [H] or [Bg. Several of the w e l l - s t u d i e d polynomial g r a p h - p r o b l e m s ~ u c h as the connectivity of a graph [EV, TI], the a r c - d e l e t ion [K], the m a x i m u m m a t c h i n g and the bm a t c h i n g problems led]), as well as NPcomplete problems (such as the node cover, %work D a r t i a { l y
Mcs-76-15255
N o w suppose that ~ is such a p r o p e r t y and that there is an upper bound k on the order of graphs s a t i s f y i n g ~. Then the
s u p p o r t e d by N S F ' g r a n t
- 253 -
c o r r e s p o n d i n g n o d e - d e l e t i o n p r o b l e m is p o l y n o m i a l in a t r i v i a l way: Given a g r a p h G, e x a m i n e all the (induced) subg r a p h s of G of o r d e r up to k and find the one w i t h the l a r g e s t o r d e r that s a t i s f i e s ~. We call a p r o p e r t y i n t e r e s t i n g (on some input domain) if t h e r e e x i s t s no such u p p e r b o u n d (for the g r a p h s o f the input domain), i.e. if t h e r e are a r b i t r a r l y large g r a p h s s a t i s f y i n g n.
tion (with w o r s t - c a s e ratio 0 ( n l - e ) , for any e>0, w i t h n the n u m b e r of nodes) of the c o n n e c t e d n o d e - d e l e t i o n p r o b l e m is also N P - c o m p l e t e * , and (2) D e t e r m i n i n g w h e t h e r the l a r g e s t s u b g r a p h s s a t i s f y i n g are c o n n e c t e d or n o t is in 4 P2 [ N P U co-NP], p r o v i d e d NP M c o - N P ~ ~
that
The a d d i t i o n a l a s s u m p t i o n , that n is d e t e r m i n e d by the b l o c k s , is e s s e n t i a l as e x e m p l i f i e d by the p r o p e r t y n = 'comp l e t e graph' (or ~ = 'star'). This noded e l e t i o n p r o b l e m is e q u i v a l e n t to the node-cover problem, which has a polynomial 2-approximation.
In S e c t i o n 2 we s h o w that for any n o n trivial and i n t e r e s t i n g g r a p h - or d i g r a p h - p r o p e r t y that is h e r e d i t a r y on i n d u c e d s u b g r a p h s , the n o d e - d e l e t i o n p r o b l e m is N P - c o m p l e t e . M o r e o v e r the r e s t r i c t i o n to p l a n a r g r a p h s (or digraphs) and to a c y c l i c d i g r a p h s (in case of digraphp r o p e r t i e s ) is also N P - c o m p l e t e . For the r e s t r i c t i o n to b i p a r t i t e g r a p h s t h e r e are few (in a w a y t h a t is d e f i n e d m o r e p r e c i s e ly in S e c t i o n 2) e x c e p t i o n s . F o r e x a m p l e the node c o v e r p r o b l e m is p o l y n o m f a l (by K o n i g ' s t h e o r e m and the fact that the m a x i m u m m a t c h i n g is p o l y n o m i a l [Ed]). H o w e v e r we s h o w that it is a u n i q u e e x c e p t i o n a m o n g p r o p e r t i e s t h a t are det e r m i n e d by the c o m p o n e n t s . (We say that a p r o p e r t y ~ is d e t e r m i n e d by th___eec o m p o n e n t s - resp. by the b l o c k s - of a graph, if w h e n e v e r the c o m p o n e n t s - resp. the b l o c k s - of a g r a p h G s a t i s f y n~ then G s a t i s f i e s also ,).
In S e c t i o n 4 we t u r n to ~ e d g e - d e l e t i o n p r o b l e m s ~ w h i c h tend to be e a s i e r to solve (or h a r d e r to s h o w N P - c o m p l e t e ) than t h e i r n o d e - d e l e t i o n versions. Note for e x a m p l e the d i f f e r e n c e for ~ = 'acyclic graph' (tree) or 'degree constrained'. W e s h o w the f o l l o w l n g p r o b l e m s to be N P - c o m p l e t e : (i) w i t h o u t c y c l e s of s p e c i f i e d l e n g t h ~ or of any l e n g t h < ~, (2) d e g r e e - c o n s t r a i n e d w i t h m a x i m u m d e g r e e r > 2 and c o n n e c t e d , (3) outerplanar, (4) planar~ (5) l i n e - i n v e r t ible~ (6) c o m p a r a b i l i t y graph, (7) b i p a r t i t e (max-cut p r o b l e m ) , (8) t r a n s itive digraph. For p r o b l e m s (5), (6), (7) we d e t e r m i n e the b e s t p o s s i b l e b o u n d s on the n o d e - d e g r e e s for w h i c h the problems remain NP-complete.
In these c o n s t r u c t i o n s the r e m a i n i n g g r a p h a f t e r the d e l e t i o n of a m i n i m u m n u m b e r of n o d e s is o f t e n h i g h l y d i s c o n n e c t ed. O n e m a y w i s h to r e q u i r e the r e m a i n i n g g r a p h to be c o n n e c t e d and m i g h t e v e n h o p e that this t a s k c o u l d be easier: For e x a m p l e K r i s h n a m o o r t h y and D e o p r o v e d [KD] that the n o d e - d e l e t i o n p r o b l e m for = 'nonseparable' is N P - c o m p l e t e . (If the r e s u l t i n g g r a p h is d i s c o n n e c t e d t h e y r e q u i r e that each of its c o m p o n e n t s s a t i s f i e s ~). In this case the r e q u i r e m e n t for c o n n e c t i v i t y m a k e s the p r o b l e m very e a s y (linear): we can find the b l o c k s of the g r a p h IT2] and then d e t e r m i n e the b l o c k w i t h the m a x i m u m n u m b e r of nodes. In S e c t i o n 3 we s t u d y the e f f e c t of the i n c l u s i o n of a c o n n e c t i v i t y r e q u i r e m e n t . W e s h o w that for the same class of p r o p e r ties (hereditary~ n o n t r i v i a l and i n t e r e s t ing on c o n n e c t e d graphs) the N P - c o m p l e t e ness of the n o d e - d e l e t i o n p r o b l e m is n o t affected. (In case of d i g r a p h s we take c o n n e c t i v i t y to m e a n w h a t is u s u a l l y c a l l ed w e a k c o n n e c t i v i t y , i.e. c o n n e c t i v i t y of the u n d e r l y i n g u n d i r e c t e d graph). Moreo v e r for p r o p e r t i e s that are d e t e r m i n e d by the b l o c k s (i) Any nontrivial approxima-
of c o u r s e
A few w o r d s on n o t a t i o n : we use y (G) ~resp. vC(G)) to d e n o t e the m i n i m u m n u m b e r o~ n o d e s w h o s e d e l e t i o n r e s u l t s in a s u b g r a p h (resp. c o n n e c t e d subgraph) of G that s a t i s f i e s p r o p e r t y ~. Usually w h e n no a m b i g u i t y can arise, we d r o p the s u b s c r i p t ~. By S0(G) we d e n o t e the n o d e - c o v e r n u m b e r of G, i.e. S0(G ) = yw(G), w i t h ~ = ' i n d e p e n d e n t set of nodes' 2.
THE N O D E - D E L E T I O N P R O B L E M HEREDITARY PROPERTIES
FOR
T h e o r e m i. The n o d e - d e l e t i o n p r o b l e m for nontrivial, interesting graph-properties that are h e r e d i t a r y on i n d u c e d s u b g r a p h s is N P - c o m p l e t e . * **
- 254 -
For d e f i n i t i o n s c o n c e r n i n g a p p r o x i m a tion a l g o r i t h m s , see [J]. R e g a r d i n g the p o l y n o m i a l - t i m e h i e r a r chy and in p a r t i c u l a r ~ , see IS].
Proof:
B H { 8j s a t i s f i e s
For all m , n there is a n u m b e r r(m,n) (the s o - c a l l e d R a m s e y n u m b e r ) , such that e v e r y g r a p h w i t h no fewer than r(m,n) n o d e s contains K or K . W e c l a i m that e i t h e r all c l l•q u e sm or a~l i n d e p e n d e n t sets of n o d e s (or both) s a t i s f y ~. S u p p o s e , to the c o n t r a r y / that there are m, n such that K m and K n do not s a t i s f y W. Since is an i n t e r e s t i n g p r o p e r t y , t h e r e is a g r a p h s a t i s f y i n g ~, w i t h m o r e than r(m,n) nodes, and since ~ is h e r e d i t a r y on induced subgraphs either K or K has to m s a t i s f y ~. Define a complementary property ~ as follows: a_graph G satisfies iff its c o m p l e m e n t G s a t i s f i e s ~. Clearly s a t i s f i e s also the a s s u m p t i o n s of the theorem~ (since the c o m p l e m e n t of a subg r a p h is a s u b g r a p h of the c o m p l e m e n t ) , and the two n o d e - d e l e t i o n p r o b l e m s are e q u i v a l e n t (if the input d o m a i n of g r a p h s is u n r e s t r i c t e d ~ or at least c l o s e d u n d e r complementation).
c o m p l e t e p - p a r t i t e graph, then J c o n s i s t s of a s i n g l e edge and k = 2.) By nont r i v i a l l i t y of ~, there e x i s t s such a g r a p h J, and f u r t h e r m o r e , since all ind e p e n d e n t sets of n o d e s s a t i s f y ~, J has at least one c o m p o n e n t of o r d e r no less than 2.
S u p p o s e from n o w on w i t h o u t loss of g e n e r a l i t y that all i n d e p e n d e n t sets of n o d e s s a t i s f y n; o t h e r w i s e c o n s i d e r the e q u i v a l e n t p r o b l e m for ,. Let G be a g r a p h w i t h c o n n e c t e d compo n e n t s G I , G 2 ~ . . . ~ G t. For each G.I take a c u t p o i n t c. and sort the c o m p o n e n t s of G. r e l a t i v ~ to c. according to t h e i r i .i orders. T h i s glves a s e q u e n c e
i]i
(For e x a m p l e
if ~ =
Let Jl,...,J~ be the c o m p o n e n t s of J t s o r t e d a c c o r d i n g to t h e i r ~. s, c. the c u t p o i n t o f Jl that gave =_~ l J^u t~e largest c o m p o n e n t of J. r e l a t i v e to Cl, J. and , I I J the g r a p h s o b t a i n e d from Jl and J r e s p e c t i v e l y b y d e l e t i n g all n o d e s of J0 e x c e p t Cl, and d any• n o d e of J^u o t h e r than c I. (There e x l s t s such a node d since Jl' and c o n s e q u e n t l y J0 too, has at leas~ 2 nodes). Now, g i v e n a g r a p h G, input to the node c o v e r p r o b l e m , let G* c o n s i s t of n . k i n d e p e n d e n t c o p i e s of G, w h e r e n is the o r d e r of G. For e a c h node u of G* c r e a t e a c o p y of J' and a t t a c h it to u by i d e n t i f y i n g c I w i t h u. Replace every edge (u,v) of G* by a c o p y of Jo' a t t a c h e d to u and v by its n o d e s c] and d. (see Fig. i). (It does not m a t £ e r h o w we i d e n t i f y the n o d e s Cl,d w i t h the n o d e s u and v).
s
~.i =A < n i l ' n i 2 ' ' ' ' ' n . .>, .w i t .h nil . ~3 i > n.. ~ and a s s u m e that c. is the c u t -
~.
~
To
N
l
p o i n t of G. that gives the i c a l l y s m a l l e s t such ~..
lexicograph(If G. is
l
biconnected,
then c.
is any node of
it,
l
and a. = < n. >, w h e r e n. = iGil ) . Sort l l l . the s e q u e n c e s of e. 's a c c o r d l n g to the 1 l e x i c o g r a p h i c o r d e r i n g and let 8G = < ~i'
~2'''''~t
~i > ~2 > ~3
E
Fig.
i.
1
>' w h e r e
"'" > st"
E
The s e q u e n c e s 8 G induce a t o t a l ord e r i n g R a m o n g the graphs. (We m a y h o w e v e r ~ have~ 8 G = 8_ for two n o n i s o m o r p h i c g r a p h s G and ~). T a k e J to be a least g r a p h in this o r d e r i n g that c a n n o t be r e p e a t e d a r b i t r a r i l y m a n y t i m e s without v i o l a t i n g ~; i.e. there e x i s t s a n u m b e r k > i, such that k i n d e p e n d e n t c o p i e s of J (without any i n t e r c o n n e c t i n g edges) v i o l a t e ~, k - i i n d e p e n d e n t c o p i e s of J s a t i s f y ~, and any n u m b e r of ind e p e n d e n t c o p i e s of every H w i t h
Let G' be the r e s u l t i n g graph. We w i l l show that ~o(G) ~ ~ V (G') ~ nkA. i). Let V be a n o d e c o v e r for G, IVl < ~. D e l e t e V from e a c h copy of G. Every c o n n e c t e d c o m p o n e n t of the r e s u l t i n g subg r a p h of G' is e i t h e r (a) a c o m p o n e n t J. of J o t h e r than J1' or (b) a g r a p h formed by t a k i n g one cop9 of Ji and s e v e r a l c o p i e s of J0' d e l e t i n g e i t h e r c I or d from each copy of J0 and a t t a c h i n g it by the o t h e r n o d e (d or Cl) to n o d e c I of the copy of Ji (see Fig. 2 for an e x a m p l e ) , or (c) Jl - J0 (with c I d e l e t e d ) , or (d) -{Cl,d) (or ~ em, in case
the c o n n e c t e d c o m p o n e n t s of that the c o r r e s p o n d i n g d e l e t i o n s h a v e d i s c o n n e c t e d them. However this does not a f f e c t o u r a r g u m e n t s , since as it w i l l b e c o m e o b v i o u s in a minute, the w o r s t - c a s e is w h e n t h e y are all
- 255 -
connected). Thus the r e m a i n i n g g r a p h can be r e g a r d e d as a s u b g r a p h of r e p e t i tions of the f o l l o w i n g g r a p h J*: j* has t+ s-i c o m p o n e n t s , if s is the n u m b e r of g r a p h s of the form (b) for the p o s s i b l e c h o i c e s of the node (c I o r d) d e l e t e d from each copy of J0: £hese are J 2 , J 3 . . . , Jt and the s g r a p h s J[, i = i) ..., s of the form (b). For e x a m p l e , if J is the g r a p h of Fig. 2a) and the m a x i m u m d e g r e e of G is 3, then J* is as s h o w n in Fig. 2b.
be a s o l u t i o n to the n o d e - d e l e t i o n problem. Let m be the n u m b e r of c o p i e s of G) from w h i c h G ' - V c o n t a i n s J as an i n d u c e d subgraph. Since k independent c o p i e s of J v i o l a t e ~ and since , is h e r e d i t a r y on i n d u c e d s u b g r a p h s , m < k. T h a t is, from at least (n-l)k+l c o p i e s of G) G ' - V does not c o n t a i n J as an i n d u c e d subgraph. Let G:i' be such a c o p y of G and d e f i n e V~ = [ u 6 N i i V c o n t a i n s a l n o d e from the copy of J' a t t a c h e d to v ( p o s s i b l y v itself) or a n o d e from the copy of J0 that r e p l a c e d and e d g e (v,u) w i t h v < u (the o r d e r i n g of n o d e s is arbitrary)]. C l e a r l y IV~I S I V D N i S u p p o s e that there is an edge (v)u) of G i such that v) u ~ V~. T h e n V does not 1 c o n t a i n any n o d e from the c o p i e s of J' a t t a c h e d to v and u) or from the c o p y of J0 that r e p l a c e d (v,u) (since o t h e r w i s e the s m a l l e r of v)u w o u l d b e l o n g to V~) c o n s e q u e n t l y (see Fig. i) G: - I V N N.] c o n t a i n s J as an i n d u c e d l l s u b g r a p h ( r e g a r d l e s s of h o w the n o d e s c. and d w e r e i d e n t i f i e d to v and u). T ~ e r e f o r e V~ is a n o d e c o v e r for G. Thus i. V m u s t c o n t a l n at least Z+I n o d e s from e a c h of (n-l)k+l c o p i e s of G, i.e. IVI ~ [(n-l)k+l] (4+1) = n k 6 + Z + 2 + k ( n - l - Z ) ~- v(G') > nk~, since n > ~+i. O
A graph J Fi@. 2a. For all i) the c o m p o n e n t s of J~ r e l a t i v e to c I are (a) t h o s e of J1 e x c e p t jn and (b) j~ w i t h one of the n o d e s c ,d u deleted. S i n c e e a c h c o m p o n e n t l o f the s e c o n d k i n d has o r d e r less than Ij01 ) the c u t p o i n t c. g i v e s an ~ - s e q u e n c e for J~ w h i c h is l ~ x i c o g r a p h i c a l l y less than ~ h a t of Jl) and c o n c e q u e n t l y ~j~ { ~Jl' for all
C o r o l l a r y i. The node d e l e t i o n p r o b l e m r e s t r i c t e d to p l a n a r g r a p h s for g r a p h p r o p e r t i e s t h a t are h e r e d i t a r y on i n d u c e d s u b g r a p h s ) n o n t r i v i a l and intere s t i n g on p l a n a r g r a p h s is N P - c o m p l e t e .
i.
Proof:
D'j
~3"Z
The c o r r e s p o n d i n g Fig. 2b. Therefore
8j, ~ 8j.
For e v e r y n, t h e r e is an r(n) (may take for e x a m p l e r(n) = 4n)) such that all p l a n a r g r a p h s w i t h r(n) or m o r e n o d e s c o n t a i n an i n d e p e n d e n t set of n nodes. S~ince ~ is i n t e r e s t i n g on p l a n a r graphs, all i n d e p e n d e n t sets of n o d e s s a t i s f y ~. T h e node c o v e r p r o b l e m r e s t r i c t e d to p l a n a r g r a p h s is N P - c o m p l e t e [GJS]. N o w note, that if the o r i g i n a l g r a p h G and the g r a p h J d e f i n e d in the p r o o f of T h e o r e m 1 are p l a n a r , and in a d d i t i o n the two a t t a c h m e n t p o i n t s c.,d of JAu lie on a c o m m o n face in an e m b e d d i n g of it o n the plane, then the r e s u l t i n g g r a p h G' is also planar. S i n c e , is n o n t r i v i a l on p l a n a r graphs, we can c a r r y t h r o u g h the p r o o f of T h e o r e m 1 and find such a p l a n a r g r a p h J. M o r e o v e r we can c h o o s e n o d e d to lie on a c o m m o n face w i t h c I.
J--3 g r a p h J*
(In o u r e x a m p l e
8j = < < 5 , 3 > , < 4 , 1 > , < 4 > >
and
8 , = < , , < ~ , 4 , 4 , 3 > , , >.)
,
By o u r c h o i c e of J) any n u m b e r of ind e p e n d e n t c o p i e s of J* s a t i s f i e s ~) and by h e r e d i t a r i n e s s the r e m a i n i n g g r a p h does so too. T h e r e f o r e y(G') < n.k.i. 2).
Suppose
that
~o(G)
> I,+i, and
D T h e o r e m 2. The n o d e - d e l e t i o n p r o b l e m r e s t r i c t e d to b i p a r t i t e g r a p h s for g r a p h p r o p e r t i e s that are h e r e d i t a r y on i n d u c e d
let V
- "256 -
consider ~ = 'complete bipartite' If G = (N,E) is a bipartite graph with N = S U T a bipartition of the node set, let E' = [(u,v) I uES, vET, (u,v) ~E~. Then it is easy to see that 7,(G) = min
subgraphs, nontrivial on bipartite graphs, and are satisfied by any independent set of edges is NP-complete. Proof:
[~ (G), ~ (G')]. R o w e v e r it can be shown [y~] that°if property ~E has as its only forbidden subgraph k + l independent edges, with k > 2, then the corresponding node-deletion problem remains NP-complete even when restricted to bipartite graphs.
There are two cases to be considered. Case i. All graphs whose components are stars satisfy ~. Since ~ is nontrivial on bipartite graphs we can carry through the proof of T h e o r e m 1 using as J the Rleast bipartite graph satisfying the conditions stated there. Since all graphs whose components are stars satisfy n, Jl is not a star. (Note that the star SZ has ~-sequence ~S~ = < i,i~...,I>, and any connected graph H, with < ~ is itself a star Sr, with r ~ ~).
Co rpllary 3 [KD]. The node-deletion p E o b l e m for the following properties is NP-complete: n = i) planar, 2) outerplanar, 3) line-graph~ 4) chordal, 5) interval, 6) without cycles of specified length A, 7) without cycles of length < 4, 8) aegree-contrained with maximum degree r ~ I, 9) acyclic (forest), I0) bipartite, ii) comparability graph, 12) complete bipartite.
Thus there is at least one more node in the same set with c. in a bipartition of J0" 1 If we choose as d any such node, the graph G' constructed in the proof of T h e o r e m 1 is bipartite.
Furthermore the restriction to planar graphs for properties (2)-(12) and to bipartite graphs for properties (1)-(9) is also NP-complete.
Case 2. Some graph all of whose components are stars does not satisfy ~. Then J is connected~ i.e. has only one component Jl and this is a star S.. Since all independent sets of edges satisfy ~, i ~ 2. We distinguish two subcases depending on whether any number of independent copies of the graph shown in Fig. 3 satisfies or not. For these two subcases we apply two different reductions from the SAT-3 problem. (For a description of the reductions see [YI]) .
Regarding now digraph-properties note that the first argument used in the proof of T h e o r e m 1 does not hold in the case of digraphs, i.e. it may be the case that neither ~ n o r ~ is satisfied by an independent set of nodes. T h e o r e m 3. The node-deletion problem for nontrivial, interesting digraphproperties that are hereditary on induced subgraphs is NP-complete.
4-z
V
Proof: Using Ramsey's theorem we can show that for all PI~ P2' P3 there is a number r(Pl' P2' P3 ) such that all digraphs of Fig.
3. order at least r(Pl, P2' P3 ) contain as
c o r o l l a r y 2.. with the exception of the node cover problem, the node-deletion problem restricted to bipartite graphs for graphproperties that are h e r e d i t a r y on induced subgraphs, nontrivial on bipartite graphs and determined by the components is NPcomplete. If some independent set of edges does not satisfy ~, there are properties (besides ~O = 'independent set of nodes') for which the node-deletion problem becomes polynomial when restricted to bipartite graphs, and properties for which it remains NP-complete. For example
- 257 -
an induced subgraph either an independent set of P1 nodes, or a complete symmetric (abbreviated c.s.) digraph on P9 nodes, or a complete antisymmetric transitive (abbreviated c.a.t.) digraph on P3 nodes. since w is interesting and hereditary on induced subgraphs, it is satisfied either (i) by all independent sets of nodes, or (ii) by all c.s. digraphs, or (iii) by all c.a.t, digraphs. The proof of T h e o r e m 1 works for cases (i) and (ii) (in case (ii) the construction is c a r r i e d out for ~). It remains therefore tO show the result for case (iii).
Let s be the largest n u m b e r such that s i n d e p e n d e n t c.a.t, d i g r a p h s of any o r d e r s a t i s f y n, i.e. there exist n u m b e r s k. k2'" " " ' k s + l such that s + l independent' c.a.t, d i g r a p h s of o r d e r k l , . . . ,k +i violate ~. (There exists such a ~u~tber s if n is not s a t i s f i e d by all i n d e p e n d e n t sets of nodes, and s >__ i). Since ~ is h e r e d i t a r y on i n d u c e d s u b g r a p h s there exists a n u m b e r k such that s+l i n d e p e n d e n t c.a.t, d i g r a p h s of o r d e r k violate ~ (We can take for e x a m p l e k = m a x
v(G)
Given a graph G = (N,E), input to the node cover problem, w i t h N = [ul,... ,u n] E = [el,... ,e ], let r = m. (k-l).n. Form , m , , a digraph D = (N ,E ) as follows: N' = {uij I 1 < i < n, 1 < j ! r] U I 1
2+1, and let V be a s o l u t i o n to %he n ~ d e - d e l e t i o n problem. For an edge ef = (ui,u j) of G, let Kf = [g I Uig,
n > L+I.
Proof: (j = g and
that D' is formed by r copies of every edge ef = (ui,u j) r e p l a c e d
of some
since
C o r o l l a r y 4. The n o d e - d e l e t i o n p r o b l e m r e s t r i c t e d to a c y c l i c d i g r a p h s for digraph-properties that are h e r e d i t a r y on i n d u c e d subgraphs, n o n t r i v i a l and intere s t i n g on a c y c l i c d i g r a p h s is N P - c o m p l e t e .
1 < h _< s-l]
~re no eij k nodes)
(ui,Ug)
{ (eij h, efg h)
1 _< j _< r.
> r.Z,
The p r o o f of C o r o l l a r y 1 is v a l i d for digraph-properties too. A l s o the result of T h e o r e m 2 can be e x t e n d e d to d i g r a p h p r o p e r t i e s as well, a l t h o u g h there are m o r e s u b c a s e s to be c o n s i d e r e d in case 2 a c c o r d i n g to the o r i e n t a t i o n s of the edges.
{k l,k 2, .... k s + 1 }).
{eij h
h e r e d i t a r y on induced subgraphs, we m u s t have .IKfl < k-l, if D ' - V is to s a t i s f y ~. ~ h e ~ e f o r e from at least r-m. (k-l) = (n-l)m(k-l) copies of G, there is at least one of the s+l nodes that r e p l a c e d each edge of G deleted. Arguing as in the p r o o f of T h e o r e m 1 V m u s t c o n t a i n at least 6+1 nodes from each of these copies, i.e. IVl ~ (n-1)m(k-l) . (~+i) = m. (k-l) [n.t +n-(~+l)] - ~.
F u r t h e r m o r e the r e s t r i c t i o n of all p r o b l e m s to p l a n a r digraphs, and the r e s t r i c t i o n of p r o b l e m s 2,3,5,6,7 to a c y c l i c d i g r a p h s is also N P - c o m p l e t e .
efgl,...,efg,s_l
~V]. The nodes that r e p l a c e d edge e_ in the copies of G w i t h index in Kf, fo~m s+l i n d e p e n d e n t c o m p l e t e a n t i s y m m e t r i c t r a n s i t i v e d i g r a p h s of o r d e r I K f ! . By our choice of s and k and since n Is
3.
INCLUSION MENT.
Theorem
- 258 -
4.
OF A OONNECTIVITY
The
connected
REQUIRE-
node-deletion
p r o b l e m for g r a p h - p r o p e r t i e s that are h e r e d i t a r y on i n d u c e d (connected) graphs, n o n t r i v i a l and i n t e r e s t i n g on c o n n e c t e d g r a p h s is N P - c o m p l e t e .
c o n n e c t e d graphs). Given a planar graph G a m a x i m u m (induced) s u b g r a p h of G w i t h c o n s i s t s of a n o d e v and a m a x i m u m ind e p e n d e n t set of its neighborhood F(v). S i n c e G is planar, F(v) is o u t e r p l a n a r for e a c h n o d e v. S i n c e the m a x i m u m ind e p e n d e n t set of an o u t e r p l a n a r g r a p h can be found in p o l y m o n i a l time, the same is true for y~(G).
Proof: It is e a s y to s h o w that for all Z,n,m, there is a n u m b e r r(t,n~m) such that all c o n n e c t e d g r a p h s of o r d e r at least r(t,n,m) c o n t a i n as an i n d u c e d s u b g r a p h e i t h e r a star Si or a c l i q u e K n or a p a t h P of l e n g t h m. S i n c e ~ is i n t e r e s t i n g m o n - c o n n e c t e d g r a p h s and h e r e d i t a r y e i t h e r all cliques, or all stars or all p a t h s s a t i s f y ~.
T h e o r e m 4 can be e x t e n d e d to d i g r a p h p r o p e r t i e s as well. To case 1 (all c l i q u e s s a t i s f y ~) there c o r r e s p o n d two cases:(li) all c o m p l e t e s y m m e t r i c dig r a p h s s a t i s f y ~, and (lii) all c o m p l e t e antisymmetric transitive digraphs satisfy ~. In case (li) the p r o o f is as in case 1 of T h e o r e m 4. In case (lii) the p r o o f is as in case 2. To case 2 there c o r r e s p o n d t h r e e cases a c c o r d i n g to the o r i e n t a t i o n s Qf the e d g e s o f the stars(Fig. 4 on!ast~ In all 3 cases the p r o o f goes as in case 2 of T h e o r e m 4. F i n a l l y if n o n e of the p r e v i o u s c a s e s is true then an i n f i n i t e n u m b e r of s e m i p a t h s s a t i s f i e s ~. (A s e m i ~ t h is a d i g r a p h w h o s e u n d e r lying g r a p h is a path.) In this last case we n e e d to k n o w for e a c h n at least one g r a p h (for e x a m p l e a semipath) of o r d e r n s a t i s f y i n g ~ (or at least be able to g e n e r a t e such a g r a p h in p o l y n o m i a l time). U n d e r this last a s s u m p t i o n we have :
Case I. A l l c l i q u e s s a t i s f y ~. T h e n the c o n s t r u c t i o n of G' in T h e o r e m 1 is c a r r i e d o u t for ~. If V* is a node c o v e r for G*, G ' - V * is d i s c o n n e c t e d and c o n s e q u e n t l y G ' - V * is connected. C a s e 2. A l l stars s a t i s f y ~. Define a p r o p e r t y ~' as follows: A (not n e c e s s a r i l y connected) g r a p h G s a t i s f i e s ~' iff the g r a p h G 1 formed by t a k i n g a n e w node and c o n n e c t i n g it to all n o d e s of G s a t i s f i e s ~. C l e a r l y ~' is nontrivial, i n t e r e s t i n g , h e r e d i t a r y on i n d u c e d s u b g r a p h s and is s a t i s f i e d b y any i n d e p e n d e n t set of nodes. A p p l y the c o n s t r u c t i o n of T h e o r e m 1 for ~'. From the r e s u l t i n g g r a p h G' c o n s t r u c t G" by t a k i n g a n e w n o d e v and c o n n e c t i n g it w i t h an edge to all n o d e s of G'. We c l a i m that yC(G") = ¥_.(G'). Obviously the g r a p h f o ~ m e d by n~de v and any subg r a p h of G' s a t i s f y i n g ~' is c o n n e c t e d and s a t i s f i e s ~, thus yC(G") < V~ (G').
T h e o r e m 5. The connected node-deletion p r o b l e m for d i g r a p h - p r o p e r t i e s that are h e r e d i t a r y on i n d u c e d (connected) subgraphs, n o n t r i v i a l and i n t e r e s t i n g on c o n n e c t e d d i g r a p h s is N P - c o m p l e t e . C o r o l l a r y 6. The c o n n e c t e d n o d e - d e l e t i o n p r o b l e m r e s t r i c t e d to acyclic d i g r a p h s for d i g r a p h - p r o p e r t i e s that are h e r e d i t a r y on i n d u c e d (connected) subgraphs, nont r i v i a l and i n t e r e s t i n g on c o n n e c t e d a c y c l i c d i g r a p h s is N P - c o m p l e t e .
For the o t h e r d i r e c t i o n s u p p o s e that the o p t i m a l s o l u t i o n V to the (connected) n o d e - d e l e t i o n ~ p r o b l e m c o n t a i n s node v. T h e n V m u s t c o n t a i n all the n o d e s from n k - i c o p i e s of G, if G " - V is to be connected. T h u s IVI k (nk-l) n+l. Since y , (G') < n k ~ 0(G) < n k ( n - l ) t h i s is impossible. T h e r e f o r e V does not c o n t a i n node v and from the d e f i n i t i o n of ~', Y~, (G') ~ Y~CG").
F r o m n o w on we w i l l c o n c e n t r a t e on p r o p e r t i e s that are d e t e r m i n e d by the b l o c k s of a g r a p h and w i l l not d i s t i n g u i s h between graph-and digraph-properties. For such a p r o p e r t y W that is h e r e d i t a r y on i n d u c e d s u b g r a p h s , the f o l l o w i n g are equivalent: (i) ~ is i n t e r e s t i n g on c o n n e c t e d g r a p h s (2) a s i n g l e e d g e s a t i s f i e s ~.
C a s e 3. S o m e c l i q u e and some star do n o t s a t i s f y ~. T h e n all p a t h s h a v e to s a t i s y ~. In this case we use a r e d u c t i o n from the S A T - 3 problem. (see [Y2]), [] C o r o l l a r y 1 ( r e g a r d i n g the p l a n a r restriction) is no m o r e true if we include a c o n n e c t i v i t y r e q u i r e m e n t . As an e x a m p l e c o n s i d e r ~ = 'star' (~ is c l e a r l y h e r e d i t a r y on c o n n e c t e d i n d u c e d subgraphs~ n o n t r i v i a l and i n t e r e s t i n g on p l a n a r
-
Lemma 1 If ~ is d e t e r m i n e d by the b l o c k s and is h e r e d i t a r y on i n d u c e d s u b g r a p h s and there e x i s t s a f o r b i d d e n b i c o n n e c t e d subg r a p h H 1 for ~ w i t h an e d g e e, w h o s e d e l e t i o n r e s u l t s in a s i n g l y c o n n e c t e d g r a p h that s a t i s f i e s w~ t h e n (i) The
2 S 9
-
lying truth a s s i g h m e n t for S' satisfies at m o s t 2 literals in each clause. F u r t h e r m o r e there is such a truth assignment that satisfies all n o n c o m p l e m e n t e d variables. (4) If S is not satisfiable, then every truth a s s i g n m e n t for S' that satisfies all n o n c o m p l e m e n t e d variables, satisfies 3 literals in some clause.
a p p r o x i m a t i o n of the c o n n e c t e d n o d e - d e letion p r o b l e m n w i t h w o r s t - c a s e ratio 0(nl-6), for any ~ > 0, w i t h n the n u m b e r of nodes is N P - c o m p l e t e , (2) It is NPh a r d to decide w h e t h e r all largest ind u c e d subgraphs w i t h property ~, of a given graph are connected. Proof:
Lemma 3: If ~ is n o n t r i v i a l on c o n n e c t e d graphs, d e t e r m i n e d by the blocks and h e r e d i t a r y on induced subgraphs and there exists a graph H 2 w i t h at least 5 nodes, three of w h i c h are d i s t i n g u i s h e d , such that (i) d e l e t i o n of any d i s t i n g u i s h e d node results in a g r a p h s a t i s f y i n g ~, (ii) deletion of at m o s t 2 d i s t i n g u i s h e d nodes does not d i s c o n n e c t the graph, (iii) deletion of all 3 d i s t i n g u i s h e d nodes d i s c o n n e c t s the graph, then the a p p r o x i m a t i o n of the c o n n e c t e d noded e l e t i o n p r o b l e m ~ w i t h w o r s t - c a s e ratio 0(n l-t) is NP-complete.
(i) The r e d u c t i o n is from the SAT-3 ( s a t i s f i a b i l i t y w i t h 3 literals per clause) problem. For each clause we form a part of the graph w i t h one node c o r r e s p o n d i n g to each literal, so that for each clause at m o s t one such node can remain if the graph is to s a t i s f y ~. T h e n we take a cutpoint c of H.-e and 1 connect it to all nodes c o r r e s p o n d i n g to v a r i a b l e s through copies o f the one c o m p o n e n t of Hl-e relative to c and to the nodes c o r r e s p o n d i n g to n e g a t i o n s o f variables through copies of the o t h e r c o m p o n e n t of Hl-e relative to c. W e add an edge b e t w e e n any two nodes that c o r r e s p o n d to c o m p l e m e n t e d literals. Let G 1 be a graph with m nodes s a t i s f y i n g p r o p e r t y ~. A t t a c h a copy of G 1 to all nodes of the p r e v i o u s c o n s t r u c t i o n that do not c o r r e s p o n d to literals. For each clause connect a copy o f G 1 to the rest of the graph by the 3 nodes c o r r e s p o n d i n g to the literals of the clause and let G be the r e s u l t i n g graph. If the input set of clauses is s a t i s f i a b l e vC(G) = 2P (with P the n u m b e r of clauses) w~ereas if it is not satisfiable 7 c > m. The result follows by taking m ~ an a p p r o p r i a t e l y high (but constant for fixed 4) p o w e r of P. (2) If the set of clauses is s a t i s f i a b l e yC(G) = Y (G) = 2P and all s u b g r a p h s w i t h p ~ o p e r t y ~ of G o b t a i n e d by d e l e t i n g that few nodes are connected. If the set is not s a t i s f i a b l e y~(G) < y~(G). []
T h e o r e m 6. If ~ is n o n t r i v i a l and int e r e s t i n g on c o n n e c t e d graphs, d e t e r m i n e d by the blocks and h e r e d i t a r y on induced subgraphs, then the a p p r o x i m a t i o n of the connected node-deletion problem ~ with w o r s t - c a s e ratio 0(n I-E) is NP-complete. Proof: By showing that for each p r o p e r t y e i t h e r Lemma 1 or L e m m a 3 can be applied. O L e m m a 4: If , is d e t e r m i n e d by the b l o c k s of a graph and h e r e d i t a r y on induced subgraphs and there exists a graph H 3 w i t h at least 4 nodes, three of w h i c h are distinguished, s a t i s f y i n g a s s u m p t i o n s (i) and (ii) of L e m m a 3 and (iii) H 3 does not satisfy ~, then it is NP- and co-NPh a r d to d e c i d e w h e t h e r all largest induced s u b g r a p h s w i t h p r o p e r t y ~ are connected.
If a f o r b i d d e n s u b g r a p h H 1 as above does not exist, then we w i l l have to connect c o m p l e m e n t e d literals b y a forbidden subgraph. But then to get c o n n e c t i v i t y we will have to k e e p e x a c t l y one of these two nodes.
proof:
Lemma 2. G i v e n a set of clauses S = [C1,..,Cp] w i t h v a r i a b l e s X l , . . . , x n and e x a c t l y 3 literals per clause, there is a n o t h e r set of clauses S' = [C~,...,C' with variables Xl,...,Xn,Xn+l,.. ,x r p' with r and p' linear in p~ and at m o s t 3 literals per clause, such that: (i) S is s a t i s f i a b l e iff S' is, (2) Each ~ variable, whose c o m p l e m e n t appears in S', o c c u r s as m a n y times as its complement~ (3) If S is satisfiable, then e v e r y satis-
W e use r e d u c t i o n s from the SAT-3 problem, w h e r e we assume that the input set of clauses consists of a clause c o n t a i n i n g a single v a r i a b l e x and a set S of clauses that is satisfiable. (Note that cook's o r i g i n a l r e d u c t i o n for the s a t i s f i a b i l i t y p r o b l e m has e x a c t l y this form [C], if we take for example as x the variable that asserts that at the final time the T u r i n g m a c h i n e is in an a c c e p t i n g state). We a p p l y the t r a n s f o r m a t i o n of Lemma 2 to S and c o n s t r u c t a g r a p h J from the res u l t i n g set S' such that all m a x i m u m subgraphs w i t h ~ ~f J are connected, corresp o n d to a s a t i s f y i n g truth a s s i g h m e n t for
- 260 -
S' and k e e p all nodes literals.
corresponding
to true
Proposition
Relative
to any set X,
p,X For the NP-hardness proof we form a graph as in Fig. 5 where c is a node con tained in all m a x i m u m subgraphs with of J and the node b is connected to all nodes of the 2 copies of J corresponding to x by copies of H 3.
u
p,X
Thus T h e o r e m 7 gives a class of natural graph problems that testify to ~ ~ ~ NP U co-NP, in case that NP- ~ co-NP. A n o t h e r such p r o b l e m is reported in [Le] . C o r o l l a r y 7: The conclusions of Theorems 6,7 hold for the following n o d e - d e l e t i o n problems: planar, outerplanar, bipartite, chordal, acyclic graph (tree)~ cactus, acyclic digraph, symmetric, antisymmetric digraph~ without cycles of specified length £, of gmy length ~ 2..
C
d Fi~.
5.
For the co-NP-hardness graph of Fig. 6, where distinguished nodes of connected to all nodes to ~ by copies of H 3.
Fig.
4. proof we form the a, and b are H_, and b is o~ J corresponding
EDGE-DELETION
PROBLEMS
T h e o r e m 8. The following edge-deletion problems are NP-complete. (i) "without cycles of specified length ~", for any fixed £ ~ 3, (ii)
for even £, the same problem restricted to bipartite graphs,
(iii)
"without any cycles of length 6" restricted to bipartite graphs, for fixed £ ~ 4.
T h e o r e m 9. The e d g e - d e l e t i o n "connected, with m a x i m u m degreer" p r o b l e m is NPcomplete, for any fixed r ~ 2.
6.
T h e o r e m 7. If ~ is nontrivial and interesting on connected graphs, determined by the blocks and hereditary on induced subgraphs then it is NP-and co-NP hard to decide whether all largest induced subgraphs satisfying property ~ are connected.
Theorem planar"
i0. The e d g e - d e l e t i o n "outerp r o b l e m is NP-complete.
Theorem problem
ii. The edge-deletion is NP-complete.
Proof~
The reductions for the two last theorems are from the H a m i l t o n i a n path problem (with m a x i m u m degree 3 [GJS] for the planar case) and are b a s e d on counting arguments. We take two copies of the original graph and two new nodes that we connect to all the nodes of the original graphs. We show that if the original graph has a Hamiltonian path then the new graph contains a maximal o u t e r p l a n a r (resp. maximal planar minus one edge) subgraph. Conversely if there is such a subgraph then e m b e d d i n g it on the plane andusing properties of maximal o u t e r p l a n a r (resp. planar) graphs we can exhibit a H a m i l t o n i a n path of the original graph.
Proofs:
We show that Lemma 4 is applicable, unless the triangle does not satisfy property ~. In this case the N P - h a r d part is covered by Lemma I, and we shall give a separate proof for the co-NP h a r d part. [] The above problem
is easily
"planar"
seen to be
in ~ M (the set of languages recognizable . in polynomial time d e t e r m i n i s t i c a l l y by a query machine with oracle from NP; for the polynomial-time h i e r a r c h y see [S]) provided that ~ is in NP. Thus T h e o r e m 7 shows that NP ~ co-NP ==-> ~ ~ NP U co-NP. H o w e v e r this is not a peculzarity of the u n r e l a t i v i s e d polynomial hierarchy, i.e.:
-
261-
Theorem ii has been independently by Geldmacher and Liu.
shown l
e3
In the next theorems we use a restricted version of the SAT-3 problem. Lemma 5: The SAT-3 problem is NP-complete even when each variable appears 3 times. The requirements of the lemma are in a sense the best possible, since if each variable appears at most twice then theclauses are trivially satisfiable (assuming that each clause contains 2 or more literals). Lemma 5 appears to be useful in proving the NP-completeness of restricted problems. For example from it (rather from the transformation used) and Karp's reduction of the SAT-3 problem to the node cover problem [K], follows a result of [GJS]: that the node cover p r o b l e m on graphs with m a x i m u m degree 3 is NP-complete. We use it to determine the best possible bounds on the node degrees for the next three theorems. Theorem 12: The edge-deletion 'line invertible' graph problem on graphs with maximum degree 4 is NP-complete.
Fig. 7
Proof:
Remarks:
Given a set of clauses CI,...,C with variables X I , . . . , X - as in LemmaP5 conS truct the ~ollowlng "n graph G = (V,E).
(i)
The restriction of the m a x i m u m degree to 4 in the T h e o r e m 12 is best possible, i.e. for graphs with m a x i m u m degree 3 the p r o b l e m can be solved in Dolynomial time. The algorithm consists in applying successive transformations to the input graph (keeping the m a x i m u m degree 3) until a graph without triagles results. Then the p r o b l e m is reducible to the line-cover problem.
(2)
If a connectivity requirement is included, then the best bound can be brought down to 3.
V = [ai,bill S i ~ n] U {dijiX i occurs in Cj} U. [eiji Xi occurs in Cj} U [d' i, e~i
1 ~ i ~ n] U [Cjll ~ j ~ p] U
{C'jlCj has 2 literals] E = [(ai,bi)il S i ~ n] U [(ai,dij), (ai,eik),
(d~, dij),
(e~, eik) i
dij~eik E V] U {(dij , Cj), (dij'die),
(eij, eie)
(eij,Cj) '
i j ~ e; dij , die ,
eij, eie E V~ U {(Cj, C~) 3
I C[ E V] 3 "
T h e o r e m 13. The e d g e - d e l e t i o n "bipartite" (simple max-cut) problem on cubic graphs is NP-complete. The proof uses Lemmas 2 and 5 (rather the transformations employed there). The 'NP-completeness of the simple max-cut problem (without the restriction on the degrees) was shown in [GJS], where there was also m e n t i o n e d the open status of the p r o b l e m on graphs with restricted m a x i m u m degree. The NP-completeness of the w e i g h t e d version (i.e. with the edges having weights) was first shown in Karp's paper [K].
For example
if C 1 = Xl V X 2 V X3' C 2 X2 V X3, C3 = X1 V X 2 V X 3. the graph G is an in Fig. 7. We show that G has a line-invertible subgraph obtained by deleting r edges, where r is the total number of literal-occurences iff the clauses are satisfiable. In our example the set of edges that are deleted corresponding to the truth assignment X 1 = i, X 2 = 0, X 3 = i, is shown with heavy lines in the figure.
T h e o r e m 14. The e d g e - d e l e t i o n 'comparability graph' p r o b l e m is NP-complete,
- 262 -
even on cubic graphs.
f e e d b a c k - n o d e set (or any o t h e r p r o b l e m of the class), w h e r e a s the node cover p r o b l e m can be e a s i l y a p p r o x i m a t e d within ratio 2, but also b e c a u s e it w o u l d shed m o r e light into the n a t u r e of N P - c o m p l e t e p r o b l e m s from the c o m b i n a t o r ial point of view and into their beh a v i o u r w i t h respect to a p p r o x i m a t i o n algorithms.
T h e o r e m 15. The e d g e - d e l e t i o n 'transitive-digraph' p r o b l e m is NP-complete. For the proof of the two last theorems we first m o d i t y the c o n s t r u c t i o n of T h e o r e m 13 to get a graph w i t h o u t t r i a n g l e s and then reduce the simple m a x - c u t p r o b l e m to them. F i n a l l y we note that for all the above p r o p e r t i e s the edge d e l e t i o n p r o b l e m rem a i n s N P - c o m p l e t e if we conclude a c o n n e c t i v i t y requirement. 5.
ACKNOWLEDGEMENT I w i s h to thank J. D. U l l m a n for his e n c o u r a n g e m e n t , support, the p a t i e n t and c a r e f u l reading of various forms of the m a n u s c r i p t , as well as m a n y s u g g e s t i o n s that improved the p r e s e n t a t i o n of the material.
CONCLUSIONS
In S e c t i o n s 2 and 3 we saw h o w a set of similar p r o b l e m s - the n o d e - d e l e t i o n p r o b l e m s - can be a t t a c k e d in a s y s t e m a t i c w a y to prove the N P - c o m p l e t e n e s s of all the m e m b e r s of the set. It w o u l d be intere s t i n g to find o t h e r classes of p r o b l e m s for w h i c h a similar result holds. In p a r t i c u l a r it w o u l d be nice if the same k i n d of t e c h n i q u e s could be a p p l i e d to the e d g e - d e l e t i o n p r o b l e m s (of course for an a p p r o p r i a t e l y r e s t r i c t e d class of properties). U n f o r t u n a t e l y we suspect that this is not the case-the r e d u c t i o n s we found for the p r o p e r t i e s c o n s i d e r e d in S e c t i o n 4 do not seem to fall into a pattern. A class of p r o b l e m s w h i c h seems more likely to be a m e n a b l e to such a t r e a t m e n t is the class of p o l y n o m i a l and integer d i v i s i b i l i t y p r o b l e m s [PI,P2], w h e r e m o s t of the N P - c o m p l e t e n e s s proofs e m p l o y s i m i l a r reductions. R e g a r d i n g the class of n o d e - d e l e t i o n p r o b l e m s two q u e s t i o n s suggest themselves: i) H o w m u c h can we e x p a n d the class of p r o p e r t i e s for w h i c h the p r o b l e m remains N P - c o m p l e t e , 2) the r e d u c t i o n schemes we d e s c r i b e d in S e c t i o n 2 show that the n o d e - d e l e t i o n p r o b l e m (without the c o n n e c t i v i t y requirement) has at least as rich a s t r u c t u r e (in the c o m b i n a t o r i a l sensesee also [ADP]) as the node cover problem. It is an immediate c o r o l l a r y of the proofs that any E - a p p r o x i m a t e a l g o r i t h m for any of the n o d e - d e l e t i o n p r o b l e m s could be u s e d to derive an E - a p p r o x i m a t e a l g o r i t h m for the node cover problem. W h a t can we say in the o t h e r direction, and what are the i n t e r r e l a t i o n s h i p s among the various p r o b l e m s in the class with respect to their c o m b i n a t o r i a l structure? This is very interesting, in v i e w of the fact that there is for example no k n o w n a p p r o x i m a t i o n a l g o r i t h m w i t h b o u n d e d w o r s t - c a s e ratio for the
- 263 -
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2 6 4
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