- Email: [email protected]
Logic vs. complexity theoretic properties of the graph accessibility problem for directed graphs of bounded degree
Information Processing North-Holland
Letters 34 (1990) 143-146
9 April 1990
LOGIC VS. COMPLEXITY THEORETIC PROPERTIES OF THE GRAPH ACCESSIBILITY PROBLEM FOR DIRECTED GRAPHS OF BOUNDED DEGREE Christoph Sektion
MEINEL
Mathemotik,
Humboldt-Uniuersitiit
zu Berlin, Unter den Linden, PF 1297, DDR-1086
Berlin
Communicated by T. Lengauer Received 13 June 1989 Revised 5 December 1989
Recently, Ajtai and Fagin [l] succeeded in logically characterizing inherent differences between the graph accessibility problems for directed graphs (GAP) on one hand and that for undirected graphs (UGAP) and directed graphs of bounded ink E WI) on the other hand. Generalizing the approach of [7], we show that these results, at least with and outdegree (GAP,,,, k > 2, cannot be interpreted as a proof that GAP is strictly harder than GAP,., in a complexity theoretic respect to GAP,,,, sense.
Keywordr:
Logarithmic UGAP
space-bounded -
computation,
nonuniform
Introduction One of the most investigated
is a path from one distinguished directed logspace accepted
programs,
graph
accessibility
problems
GAP and
Aleliunas et al. [2] and those of [6,3] it is widely expected that UGAP is not. However, up to now problems
in com-
plexity theory is the graph accessibility problem GAP which consists in the decision whether there to another
branching
one. Savitch
node in a graph
[12] proved its version
for
graphs to be complete (with respect to reduction) for NL, the class of languages by nondeterministic Turing machines
running in O(log n) space. In order to characterize the power of nondeterminism in the case of loga-
no rigorous
proof
recently, Ajtai characterization
was given for this belief.
Very
and Fagin [l] presented a logical of the inherent differences be-
tween GAP and UGAP.
They discoved
the follow-
ing discrepancy: l
GAP
l
UGAP
is not monadic 2:. as well as GAP,,,, k E N, are monadic $, where GAP, , denotes the graph accessibility problem for direcied graphs with indegree bounded by i and outdegree bounded by j.
rithmically bounded space a great variety of restricted versions of the graph accessibility problem (e.g. for undirected graphs, for acyclic and monotone graphs, for graphs of bounded outdegree or
Ajtai and Fagin interpreted this result “as the first prove in a precise technical sense” that GAP “is harder than” UGAP and GAP,,,, k E IV. In the
restricted bandwidth, etc.) under different reductions (e.g. logspace reduction, one-way logspace reduction, p-projection reduction, etc.) was investigated (see e.g. [10,5,4,7&l). Interestingly, it is not known whether the graph accessibility problem UGAP for undirected graphs is complete in NL. According to the results of
k > 2, not true in a (nonuniform)
0020-0190/90/$3.50
0 1990 - Elsevier Science Publishers
following
we prove that this is, at least for GAPk.k, complexity theo-
retic sense. Even working with the stringent notion of p-projection reducibility [13] one can prove the following completeness results which show e.g. that GAP and GAP, k, k 2 2, are of the same complexity theoretic difficulty (modulo p-projection reduction).
B.V. (North-Holland)
143
Volume 34, Number 3
INFORMATION
PROCESSING
Theorem 1. (i) GAPk,k, k > 2 (as well as GAP [12,7]) is complete in nonuniform NL. (ii) GAP,,,, k 2 2 (as well as UGAP [2,11]) is complete in nonuniform L. In the following we will prove this theorem in the stronger settings of acyclic and monotone graphs. If GAP,:, denotes the graph accessibility problem for acyclic and monotone graphs with indegree i and outdegree j (i, j E N), then it holds: Theorem 2. (i) GAP,,, , k 2 2, is complete in nonumform NL. (ii) GAP ;,i, k >, 2, is complete in nonuniform L. Obviously, Theorem 1 is an easy consequence of Theorem 2. In order to prove Theorem 2 we make use of well-known branching program based descriptions of the nonuniform complexity classes L and NL which are given in detail e.g. in [9]. Simulating within polynomial size branching programs by certain funnel branching programs from these descriptions we can derive the hardness of the considered graph accessibility problems fully similar as that of the more general graph accessibility problems GAP’,,, (for acyclic and monotone graphs of unbounded indegree and outdegree bounded by j) considered in [7]. The completeness immediately follows from the completeness of GAP;, 2 and GAP’,,, in nonuniform NL and L, respectively [7].
1. (Disjunctive) funnel branching programs and the proof of the theorem Recall, a branching program is a directed acyclic graph where each node has outdegree 2 or 0. Nodes with outdegree 0 are called sinks and are labelled by Boolean constants. The remaining nodes are labelled by Boolean variables taken from the set X = {x,, . . . , xn} of input variables. There is a distinguished node, called the source, which has indegree 0. A branching program computes an n-argument Boolean function f E S?Z as follows: Starting at the source, the value of the
144
LE’ITERS
9 April 1990
variable labelling the current node is tested. If this is 0 (1) the next node which will be tested is the left (right) successor to the current node. The branching program computes f if, for any input w E (0, l}“, the path traced under w finishes at a sink labelled by f(w). Without loss of generality we may assume that a branching program has exactly two sinks, one O-sink and one l-sink. The complexity measure of a branching program P is the number of its nonsink nodes, the size of P. By gBP we denote the set of all languages acceptable by sequences of polynomial size branching programs. Beside of branching programs and decision trees we consider disjunctive branching programs [8] where some of the nonsink nodes are labelled by the 2-argument Boolean disjunction V instead of Boolean variables. Disjunctive branching programs accept an input w E (0, l}” if there exists a path traced under w which finishes in a sink labelled by 1. We denote the set of all languages acceptable by sequences of polynomial size disjunctive branching programs by .?F{v ) ~ Bp. It is well known that (sequences of) polynomial size branching programs and nonuniform logarithmic space-bounded deterministic Turing machines are of the same computational power and that the same relation holds between polynomial disjunctive branching programs and nonuniform logarithmic space-bounded nondeterministic Turing machines. Proposition 3. (i) 9,, = (nonuniform) L [12]. (ii) 6@‘tV ) _ Bp = (nonuniform) NL [7]. The key point in the proof of Theorem 2 is the following lemma which states that polynomial size (disjunctive) branching programs can be transformed within polynomial size into (disjunctive) branching programs whose nodes have (beside of small outdegree) small indegree, too. A (disjunctive) branching program P is called a (disjunctiue) funnel branching program if all its nodes u have either exactly one in-coming edge starting in an arbitrary node of P or at most two edges labelled by the same Boolean value which start in nodes labelled merely by Boolean variables.
Volume
34, Number
INFORMATION
3
PROCESSING
Lemma 4. Each (disjunctive)
LETTERS
9 April 1990
branching program P of size s can be simulated by a (disjunctive) funnel branching program P’ of size 0(s2). In particular, polynomial size (disjunctive) branching programs can be simulated by (disjunctive) funnel branching programs of polynomial size.
polynomial in n, and if for every f, of F there is a mapping
Proof. Let P be a (disjunctive) branching program of size s and let u be a node of P with indegree k labelled by z E { V, x, (1 < i < n)}. To v we consider the (disjunctive) funnel branching program component P, (see Fig. 1) where x denotes an arbitrary one of the input variables x,, l
In order to see that the graph accessibility problems GAPi,2 (GAP;,,) are p-projection hard in nonuniform NL (L) we take an arbitrary problem F = { f,} of nonuniform NL (L) which, due to Proposition 3 and Lemma 4, can be solved by a sequence {P, } of polynomial size disjunctive funnel branching programs (ordinary funnel branching programs) P,, n E N. Since P,, is based on an acyclic graph with p(n) nodes we can enumerate its nodes by 1, 2,. . , p(n) in such a way that _ the source is numbered by 1, _ the accepting l-sink is numbered by p(n), and _ each edge leads always from a node with a lower number to a node with a higher one. Now, to every input (x,, . . . , x,) E (0, l}” of a graph G(x,, . . . , x,) = f, we assign
such that
Proof of Theorem 2. Since the graph accessibility problems GAP’ (= GAP’,,,) and GAP’,,i are p-projection complete in nonuniform NL and nonuniform L [7] respectively, it suffices to prove Theorem 2 for the case k = 2. In order to do this we proceed similar as in [7]. Recall, a problem F = {f,} (f, : (0, l}” + (0, l}, n E IV) is p-projection reducible to a problem G = {g,} (g, : (0, l}” + (0, l}, n E N), abbreviated
F< G, if there is a function
p(n)
(~,(x,,),~,~,<~~~~) . .
of size p(n)
if vertex i is an V -node and if (i, j) is an edge of P,
1, I %(Xi,)
=
xk/~!i
with
)
if vertex i is labelled by xk and if vertex j is reached from i if xk is l/O,
i 0,
bounded above by a
otherwise,
for 1 < i
P”
Fig. 1.
14.5
Volume
34, Number
INFORMATION
3
PROCESSING
Obviously, G(x,, . . . , xn) is of outdegree < 2 (1). Since P,, was assumed to be funnel G(x,, _. . , xn) is of indegree < 2. Furthermore, xn) is monotone by means of the special G(x I,“‘, nature of the enumeration of the vertices of P,, and it holds
f”(X 1,“‘, x,)=1 iff P, accepts
(x1,. . _, xn)
iff GAP;,,(G(x,,...,x,))
=l
(GAP~,,(G(x,,...,x,))
=I)-
But this implies F G GAP;,,
(GAP;,,)
_
The completeness of GAP;,, and GAP;,, in the classes nonuniform NL and nonuniform L, respectively, immediately follows from the completeness of the more general graph accessibility problems GAP’,,, and GAP’,,, [7]. 0
References [l] M. Ajtai and R. Fagin, Reachibility than for undirected (1988).
146
finite graphs,
is harder for directed IBM Res. Rept. RJ 6240
LETTERS
9 April 1990
[2] R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz and C. Rackoff, Random walks, universal traversing sequences and the complexity of maze problems, in: Proc. 2Uth FOCS (1979) 218-223. [31 A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo and M. Tompa, Two applications of complementation via inductive counting (1987). und das Verhaltnis 141 L. Budach, Klassifizierungsprobleme van deterministischer und nichtdeterministischer Raumkomplexitlt, Sem. Ber. 68, Humbold-Univ., Berlin (1985). Languages which capture complexity [51 N. Immerman, classes, in: Proc. 15th STOC (1983) 347-354. Symmetric space]61 H.R. Lewis and C.H. Papadimitriou, bounded computation, Theoret. Comput. Sci. 19 (1982) 161-187. reducibility and the complexity ]71 C. Meinel, P-projection classes L (nonuniform) and NL (nonuniform), in: Proc. 12th MFCS (Bratislava) 1986, Lecture Notes in Computer Science 233 (Springer, Berlin, 1986) 527-535. size O-branching ]81 C. Meinel, The power of polynomial programs, in: Proc. STACS ‘88 (Bordeaux), Lecture Notes in Computer Science 294 (Springer, Berlin, 1988) 81-90. ]91 C. Meinel, Modified Branching Programs and Their Compufational Power, Lecture Notes in Computer Science 370 (Springer, Berlin, 1989). Time and space bounded 1101 B. Monien and I.H. Sudborough, complexity classes and bandwidth constrained problems, in: Proc. MFCS ‘81, Lecture Notes in Computer Science 118 (Springer, Berlin, 1981) 78-93. Ill1 P. Pudlak, The hierarchy of Boolean circuits, Tech. Rept. 20 Ceskoslovenska Akademie VED (1986). between nondeterministic and ]121 W. Savitch, Relationships deterministic tape complexities, J. Compuf. System Sci. 4 (2) (1970) 177-192. ]131 S. Skyum and L.G. Valiant, A complexity theory based on Boolean algebra, in: Proc. 22th FOCS (1981) 244253.
Letters 34 (1990) 143-146
9 April 1990
LOGIC VS. COMPLEXITY THEORETIC PROPERTIES OF THE GRAPH ACCESSIBILITY PROBLEM FOR DIRECTED GRAPHS OF BOUNDED DEGREE Christoph Sektion
MEINEL
Mathemotik,
Humboldt-Uniuersitiit
zu Berlin, Unter den Linden, PF 1297, DDR-1086
Berlin
Communicated by T. Lengauer Received 13 June 1989 Revised 5 December 1989
Recently, Ajtai and Fagin [l] succeeded in logically characterizing inherent differences between the graph accessibility problems for directed graphs (GAP) on one hand and that for undirected graphs (UGAP) and directed graphs of bounded ink E WI) on the other hand. Generalizing the approach of [7], we show that these results, at least with and outdegree (GAP,,,, k > 2, cannot be interpreted as a proof that GAP is strictly harder than GAP,., in a complexity theoretic respect to GAP,,,, sense.
Keywordr:
Logarithmic UGAP
space-bounded -
computation,
nonuniform
Introduction One of the most investigated
is a path from one distinguished directed logspace accepted
programs,
graph
accessibility
problems
GAP and
Aleliunas et al. [2] and those of [6,3] it is widely expected that UGAP is not. However, up to now problems
in com-
plexity theory is the graph accessibility problem GAP which consists in the decision whether there to another
branching
one. Savitch
node in a graph
[12] proved its version
for
graphs to be complete (with respect to reduction) for NL, the class of languages by nondeterministic Turing machines
running in O(log n) space. In order to characterize the power of nondeterminism in the case of loga-
no rigorous
proof
recently, Ajtai characterization
was given for this belief.
Very
and Fagin [l] presented a logical of the inherent differences be-
tween GAP and UGAP.
They discoved
the follow-
ing discrepancy: l
GAP
l
UGAP
is not monadic 2:. as well as GAP,,,, k E N, are monadic $, where GAP, , denotes the graph accessibility problem for direcied graphs with indegree bounded by i and outdegree bounded by j.
rithmically bounded space a great variety of restricted versions of the graph accessibility problem (e.g. for undirected graphs, for acyclic and monotone graphs, for graphs of bounded outdegree or
Ajtai and Fagin interpreted this result “as the first prove in a precise technical sense” that GAP “is harder than” UGAP and GAP,,,, k E IV. In the
restricted bandwidth, etc.) under different reductions (e.g. logspace reduction, one-way logspace reduction, p-projection reduction, etc.) was investigated (see e.g. [10,5,4,7&l). Interestingly, it is not known whether the graph accessibility problem UGAP for undirected graphs is complete in NL. According to the results of
k > 2, not true in a (nonuniform)
0020-0190/90/$3.50
0 1990 - Elsevier Science Publishers
following
we prove that this is, at least for GAPk.k, complexity theo-
retic sense. Even working with the stringent notion of p-projection reducibility [13] one can prove the following completeness results which show e.g. that GAP and GAP, k, k 2 2, are of the same complexity theoretic difficulty (modulo p-projection reduction).
B.V. (North-Holland)
143
Volume 34, Number 3
INFORMATION
PROCESSING
Theorem 1. (i) GAPk,k, k > 2 (as well as GAP [12,7]) is complete in nonuniform NL. (ii) GAP,,,, k 2 2 (as well as UGAP [2,11]) is complete in nonuniform L. In the following we will prove this theorem in the stronger settings of acyclic and monotone graphs. If GAP,:, denotes the graph accessibility problem for acyclic and monotone graphs with indegree i and outdegree j (i, j E N), then it holds: Theorem 2. (i) GAP,,, , k 2 2, is complete in nonumform NL. (ii) GAP ;,i, k >, 2, is complete in nonuniform L. Obviously, Theorem 1 is an easy consequence of Theorem 2. In order to prove Theorem 2 we make use of well-known branching program based descriptions of the nonuniform complexity classes L and NL which are given in detail e.g. in [9]. Simulating within polynomial size branching programs by certain funnel branching programs from these descriptions we can derive the hardness of the considered graph accessibility problems fully similar as that of the more general graph accessibility problems GAP’,,, (for acyclic and monotone graphs of unbounded indegree and outdegree bounded by j) considered in [7]. The completeness immediately follows from the completeness of GAP;, 2 and GAP’,,, in nonuniform NL and L, respectively [7].
1. (Disjunctive) funnel branching programs and the proof of the theorem Recall, a branching program is a directed acyclic graph where each node has outdegree 2 or 0. Nodes with outdegree 0 are called sinks and are labelled by Boolean constants. The remaining nodes are labelled by Boolean variables taken from the set X = {x,, . . . , xn} of input variables. There is a distinguished node, called the source, which has indegree 0. A branching program computes an n-argument Boolean function f E S?Z as follows: Starting at the source, the value of the
144
LE’ITERS
9 April 1990
variable labelling the current node is tested. If this is 0 (1) the next node which will be tested is the left (right) successor to the current node. The branching program computes f if, for any input w E (0, l}“, the path traced under w finishes at a sink labelled by f(w). Without loss of generality we may assume that a branching program has exactly two sinks, one O-sink and one l-sink. The complexity measure of a branching program P is the number of its nonsink nodes, the size of P. By gBP we denote the set of all languages acceptable by sequences of polynomial size branching programs. Beside of branching programs and decision trees we consider disjunctive branching programs [8] where some of the nonsink nodes are labelled by the 2-argument Boolean disjunction V instead of Boolean variables. Disjunctive branching programs accept an input w E (0, l}” if there exists a path traced under w which finishes in a sink labelled by 1. We denote the set of all languages acceptable by sequences of polynomial size disjunctive branching programs by .?F{v ) ~ Bp. It is well known that (sequences of) polynomial size branching programs and nonuniform logarithmic space-bounded deterministic Turing machines are of the same computational power and that the same relation holds between polynomial disjunctive branching programs and nonuniform logarithmic space-bounded nondeterministic Turing machines. Proposition 3. (i) 9,, = (nonuniform) L [12]. (ii) 6@‘tV ) _ Bp = (nonuniform) NL [7]. The key point in the proof of Theorem 2 is the following lemma which states that polynomial size (disjunctive) branching programs can be transformed within polynomial size into (disjunctive) branching programs whose nodes have (beside of small outdegree) small indegree, too. A (disjunctive) branching program P is called a (disjunctiue) funnel branching program if all its nodes u have either exactly one in-coming edge starting in an arbitrary node of P or at most two edges labelled by the same Boolean value which start in nodes labelled merely by Boolean variables.
Volume
34, Number
INFORMATION
3
PROCESSING
Lemma 4. Each (disjunctive)
LETTERS
9 April 1990
branching program P of size s can be simulated by a (disjunctive) funnel branching program P’ of size 0(s2). In particular, polynomial size (disjunctive) branching programs can be simulated by (disjunctive) funnel branching programs of polynomial size.
polynomial in n, and if for every f, of F there is a mapping
Proof. Let P be a (disjunctive) branching program of size s and let u be a node of P with indegree k labelled by z E { V, x, (1 < i < n)}. To v we consider the (disjunctive) funnel branching program component P, (see Fig. 1) where x denotes an arbitrary one of the input variables x,, l
In order to see that the graph accessibility problems GAPi,2 (GAP;,,) are p-projection hard in nonuniform NL (L) we take an arbitrary problem F = { f,} of nonuniform NL (L) which, due to Proposition 3 and Lemma 4, can be solved by a sequence {P, } of polynomial size disjunctive funnel branching programs (ordinary funnel branching programs) P,, n E N. Since P,, is based on an acyclic graph with p(n) nodes we can enumerate its nodes by 1, 2,. . , p(n) in such a way that _ the source is numbered by 1, _ the accepting l-sink is numbered by p(n), and _ each edge leads always from a node with a lower number to a node with a higher one. Now, to every input (x,, . . . , x,) E (0, l}” of a graph G(x,, . . . , x,) = f, we assign
such that
Proof of Theorem 2. Since the graph accessibility problems GAP’ (= GAP’,,,) and GAP’,,i are p-projection complete in nonuniform NL and nonuniform L [7] respectively, it suffices to prove Theorem 2 for the case k = 2. In order to do this we proceed similar as in [7]. Recall, a problem F = {f,} (f, : (0, l}” + (0, l}, n E IV) is p-projection reducible to a problem G = {g,} (g, : (0, l}” + (0, l}, n E N), abbreviated
F< G, if there is a function
p(n)
(~,(x,,),~,~,<~~~~) . .
of size p(n)
if vertex i is an V -node and if (i, j) is an edge of P,
1, I %(Xi,)
=
xk/~!i
with
)
if vertex i is labelled by xk and if vertex j is reached from i if xk is l/O,
i 0,
bounded above by a
otherwise,
for 1 < i
P”
Fig. 1.
14.5
Volume
34, Number
INFORMATION
3
PROCESSING
Obviously, G(x,, . . . , xn) is of outdegree < 2 (1). Since P,, was assumed to be funnel G(x,, _. . , xn) is of indegree < 2. Furthermore, xn) is monotone by means of the special G(x I,“‘, nature of the enumeration of the vertices of P,, and it holds
f”(X 1,“‘, x,)=1 iff P, accepts
(x1,. . _, xn)
iff GAP;,,(G(x,,...,x,))
=l
(GAP~,,(G(x,,...,x,))
=I)-
But this implies F G GAP;,,
(GAP;,,)
_
The completeness of GAP;,, and GAP;,, in the classes nonuniform NL and nonuniform L, respectively, immediately follows from the completeness of the more general graph accessibility problems GAP’,,, and GAP’,,, [7]. 0
References [l] M. Ajtai and R. Fagin, Reachibility than for undirected (1988).
146
finite graphs,
is harder for directed IBM Res. Rept. RJ 6240
LETTERS
9 April 1990
[2] R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz and C. Rackoff, Random walks, universal traversing sequences and the complexity of maze problems, in: Proc. 2Uth FOCS (1979) 218-223. [31 A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo and M. Tompa, Two applications of complementation via inductive counting (1987). und das Verhaltnis 141 L. Budach, Klassifizierungsprobleme van deterministischer und nichtdeterministischer Raumkomplexitlt, Sem. Ber. 68, Humbold-Univ., Berlin (1985). Languages which capture complexity [51 N. Immerman, classes, in: Proc. 15th STOC (1983) 347-354. Symmetric space]61 H.R. Lewis and C.H. Papadimitriou, bounded computation, Theoret. Comput. Sci. 19 (1982) 161-187. reducibility and the complexity ]71 C. Meinel, P-projection classes L (nonuniform) and NL (nonuniform), in: Proc. 12th MFCS (Bratislava) 1986, Lecture Notes in Computer Science 233 (Springer, Berlin, 1986) 527-535. size O-branching ]81 C. Meinel, The power of polynomial programs, in: Proc. STACS ‘88 (Bordeaux), Lecture Notes in Computer Science 294 (Springer, Berlin, 1988) 81-90. ]91 C. Meinel, Modified Branching Programs and Their Compufational Power, Lecture Notes in Computer Science 370 (Springer, Berlin, 1989). Time and space bounded 1101 B. Monien and I.H. Sudborough, complexity classes and bandwidth constrained problems, in: Proc. MFCS ‘81, Lecture Notes in Computer Science 118 (Springer, Berlin, 1981) 78-93. Ill1 P. Pudlak, The hierarchy of Boolean circuits, Tech. Rept. 20 Ceskoslovenska Akademie VED (1986). between nondeterministic and ]121 W. Savitch, Relationships deterministic tape complexities, J. Compuf. System Sci. 4 (2) (1970) 177-192. ]131 S. Skyum and L.G. Valiant, A complexity theory based on Boolean algebra, in: Proc. 22th FOCS (1981) 244253.