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arXiv:2305.19104v1 [cs.CC] 30 May 2023
Hardness of Approximation in PSPACE and
Separation Results for Pebble Games∗
Siu Man Chan, Massimo Lauria1, Jakob Nordstr¨om2, and Marc Vinyals3
1Sapienza — Universit`a di Roma, Italy
2University of Copenhagen, Denmark, and Lund University, Sweden
3University of Auckland, New Zealand
May 31, 2023
Abstract
We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt ’70]
and the reversible version of this game in [Bennett ’89], and study the question of how hard it is to decide
exactly or approximately the number of pebbles needed for a given DAG in these games.
We prove that the problem of deciding whether spebbles suffice to reversibly pebble a DAG Gis
PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and
Tarjan ’80]. Via two different graph product constructions we then strengthen these results to establish
that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive
constant. To the best of our knowledge, these are the first hardness of approximation results for pebble
games in an unrestricted setting (even for polynomial time). Also, since [Chan ’13] proved that reversible
pebbling is equivalent to the games in [Dymond and Tompa ’85] and [Raz and McKenzie ’99], our results
apply to the Dymond–Tompa and Raz–McKenzie games as well, and from the same paper it follows that
resolution depth is PSPACE-hard to determine up to any additive constant.
We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space.
This improves on the additive logarithmic separation previously known and could plausibly be tight, al-
though we are not able to prove this.
We leave as an interesting open problem whether our additive hardness of approximation result could
be strengthened to a multiplicative bound if the computational resources are decreased from polynomial
space to the more common setting of polynomial time.
1 Introduction
In the pebble game first studied by Paterson and Hewitt [PH70], one starts with an empty directed acyclic
graph (DAG) Gwith bounded fan-in (and which in this paper in addition will always have a single sink) and
places pebbles on the vertices according to the following rules:
• If all (immediate) predecessors of an empty vertex vcontain pebbles, a pebble may be placed on v.
• A pebble may be removed from any vertex at any time.
∗This is the full-length version of the paper with the same title that appeared in Proceedings of the 56th IEEE Symposium on
Foundations of Computer Science (FOCS ’15).
HARDNESS OF APPROXIMATION IN PSPACE AND SEPARATION FOR PEBBLE GAMES
The goal is to get a pebble on the sink vertex of Gwith all other vertices being empty, and to do so while
minimizing the total number of pebbles on Gat any given time (the pebbling price of G). This game mod-
els computations with execution independent of the actual input. A pebble on a vertex indicates that the
corresponding value is currently kept in memory and the objective is to perform the computation with the
minimum amount of memory.
The pebble game has been used to study flowcharts and recursive schemata [PH70], register alloca-
tion [Set75], time and space as Turing-machine resources [Coo74,HPV77], and algorithmic time-space
trade-offs [Cha73,SS77,SS79,SS83,Tom78]. In the last 10–15 years, there has been a renewed interest
in pebbling in the context of proof complexity as discussed in the survey [Nor13] (although in this context
one is often interested also in the slightly more general black-white pebble game introduced in [CS76]), and
pebbling has also turned out to be useful for applications in cryptography [DNW05,AS15]. An excellent
overview of pebbling up to ca. 1980 is given in [Pip80] and some more recent developments are covered in
the upcoming survey [Nor20].
Bennett [Ben89] introduced the reversible pebble game as part of a broader program [Ben73] to investigate
possibilities to eliminate (or significantly reduce) energy dissipation in logical computation. Another reason
reversible computation is of interest is that observation-free quantum computation is inherently reversible.
In the reversible pebble game, the moves performed in reverse order should also constitute a legal pebbling,
which means that the rules for pebble placement and removal become symmetric as follows:
• If all predecessors of an empty vertex vcontain pebbles, a pebble may be placed on v.
• If all predecessors of a pebbled vertex vcontain pebbles, the pebble on vmay be removed.
Reversible pebblings of DAGs have been studied in [LV96,Kr´a04] and have been employed to shed light
on time-space trade-offs in reversible simulation of irreversible computation in [LTV98,LMT00,Wil00,
BTV01]. In a different line of work Potechin [Pot10] implicitly used the reversible pebble game for proving
lower bounds on monotone space complexity, with the connection made explicit in the follow-up works [CP14,
FPRC13].
Another pebble game on DAGs that will be of interest in this paper is the Dymond–Tompa game [DT85]
played on a DAG Gby a Pebbler and a Challenger. This game is played in rounds, with both players starting
at the sink in the first round. In the following rounds, Pebbler places a pebble on some vertex of Gafter
which Challenger either stays at the current vertex or moves to the newly pebbled vertex. This repeats until
at the end of a round Challenger is standing on a vertex with all (immediate) predecessors pebbled (or on a
source, in which case the condition vacuously holds), at which point the game ends. Intuitively, Challenger
is challenging Pebbler to “catch me if you can” and wants to play for as many rounds as possible, whereas
Pebbler wants to “surround” Challenger as quickly as possible. The Dymond–Tompa price of Gis the smallest
number rsuch that Pebbler can always finish the game in at most rrounds. The Dymond–Tompa game has
been used to establish that for parallel time a speed-up by a logarithmic factor is always possible [DT85],
and in [VT89] it was shown that a slightly modified variant of this game exactly characterizes parallelism
in complexity classes like ACi,NC, and P, and can be used to re-derive, for instance, Savitch’s theorem.
Furthermore, collapses or separations of these classes can in principle be recast (or discovered) as bounds on
Dymond–Tompa price. Interestingly, this characterization of parallelism extends to proof complexity as well
as discussed in [Cha13a].
A final game with pebbles that we want to just mention without going into any details is the Raz–McKenzie
game introduced in [RM99] to study the depth complexity of decision trees solving search problems. The
reason for bringing up the Dymond–Tompa and Raz–McKenzie games is that it was shown in [Cha13a] that
both games are actually equivalent to the reversible pebble game. Hence, any bounds derived for the reversible
pebble game also hold for Dymond–Tompa price and Raz–McKenzie price.
The main focus of this paper is to study how hard it is to decide exactly or approximately the pebbling
price of a DAG. For the standard pebble game Gilbert et al. [GLT80] showed that given a DAG Gand a
2
1 INTRODUCTION
positive integer sit is PSPACE-complete to determine whether space sis sufficient to pebble Gor not. It
would seem natural to suspect that reversible pebbling price should be PSPACE-complete as well, but the
construction in [GLT80] cannot be used to show this.
Given that pebbling price is hard to determine exactly, an even more interesting question is if anything
can be said regarding the hardness of approximating pebbling price. Although this seems like a very natural
and appealing question, apparently next to nothing has been known about this.
Wu et al. [WAPL14] showed that “one-shot” standard pebbling price is hard to approximate to within any
multiplicative constant assuming the so-called Small Set Expansion (SSE) hypothesis. In a one-shot pebbling
one is only allowed to pebble each vertex once, however, and this is a major restriction since the complexity
now drops from PSPACE-complete to NP-complete [Set75]. Note that containment in NP is easy to see
since any one-shot pebbling can be described concisely just by listing the order in which the vertices should
be pebbled (and it is easy to compute when a pebble is no longer needed and can be removed). In contrast,
in the general case pebbling strategies that are optimal with respect to space can sometimes provably require
exponential time.
One can also go in the other direction and study more general pebble games, such as the AND/OR pebble
game introduced by Lingas [Lin78] in one of the works leading up to [GLT80]. Here every vertex is labelled
AND or OR. For AND-vertices we have the usual pebbling rule, but for OR-vertices it is sufficient to just
have one pebble on some predecessor in order to be allowed to pebble the vertex. This game has a relatively
straightforward reduction from hitting set [FNPW10], which shows that it is hard to approximate to within a
logarithmic factor, but the reduction crucially depends on the OR-nodes.
We remark that hardness of approximation in PSPACE for other problems has been studied in [CFLS95],
but those techniques seem hard to adapt to pebble games since the reduction from QBF to pebbling is inher-
ently unable to preserve gaps.
Another problem that we study in the current paper is the relation between standard pebbling price and
reversible pebbling price. Clearly, the space needed to reversibly pebble a graph is at least the space required
in the standard pebble game. It is also not hard to see that there are graphs that require strictly more pebbles in
a reversible setting: for a directed path on nvertices only 2pebbles are needed in the standard game, while it is
relatively straightforward to show that the reversible pebbling space is Θ(log n)[Ben89,LV96]. However, for
“classic” graphs studied in the pebbling literature, such as binary trees, pyramids, certain superconcentrators,
and the worst-case graphs in [PTC77], the reversible and standard pebbling prices coincide asymptotically,
and are sometimes markedly closer than an additive logarithm apart.
This raises the question whether reversible and standard pebbling can be asymptotically separated with
respect to space. It might be worth pointing out in this context that for Turing machines it was proven
in [LMT00] that any computation can be simulated reversibly in exactly the same space. In the more re-
stricted pebbling model, it was shown in [Kr´a04] that if the standard pebbling price of a DAG Gon nvertices
is s, then Gcan be reversibly pebbled with at most s2log npebbles. Thus, if there is not only an additive but
also a multiplicative separation between standard and reversible pebbling price, such a separation cannot be
too large.
1.1 Our Results
We obtain the following results:
1. We establish an asymptotic separation between standard and reversible pebbling by exhibiting families
of graphs {Gn}∞
n=1 of size Θ(n)with a single sink and fan-in 2which have standard pebbling price s(n)
and reversible pebbling price Ω(s(n) log n). This construction works for any s(n) = On1/2−ǫwith
ǫ > 0constant, where the constant hidden in the asymptotic notation in the lower bound has a (mild)
dependence on ǫ.
3
HARDNESS OF APPROXIMATION IN PSPACE AND SEPARATION FOR PEBBLE GAMES
2. We prove that determining reversible pebbling price is PSPACE-complete. That is, given a single-sink
DAG Gof fan-in 2and a parameter s, it is PSPACE-complete to decide whether Gcan be reversibly
pebbled in space sor not.
3. Finally, we present two different graph products (for standard and reversible pebbling, respectively) that
take DAGs Giof size niwith pebbling price sifor i= 1,2and yield a DAG of size O(n1+n2)2with
pebbling price s1+s2+Kp(for Kp=±1depending on the flavour of the pebble game). Combining
these graph products with the PSPACE-completeness results for standard pebbling in [GLT80] and
reversible pebbling in item 2, we deduce that for any fixed Kthe promise problem of deciding for a
DAG G(with a single sink and fan-in 2) whether it can be pebbled in space sor requires space s+K
is PSPACE-hard in both the standard and the reversible pebble game.
We need to provide more formal definitions before going into a detailed discussion of techniques, but
want to stress right away that a key feature of the above results is the bounded fan-in condition. This is the
standard setting for pebble games in the literature and is also crucial in most of the applications mentioned
above. Without this constraint it would be much easier, but also much less interesting, to prove our results.1
Another aspect worth pointing out is that although the reversible pebble game is weaker than the standard
pebble game, it is technically much more challenging to analyze. The reason for this is that a standard pebbling
will always progress in a “forward sweep” through the graph in topological order, and so one can often assume
without loss of generality that once one has pebbled through some subgraph the pebbling will never touch this
subgraph again. For a reversible pebbling this is not so, since any pebble placed on any descendant of vertices
in the subgraph will also have to be removed at some later time, and this has to be done in reverse topological
order. Therefore, in any reversible pebbling there will be alternating phases of “forward sweeps” and “reverse
sweeps,” and these phases can also be interleaved at various levels. For this reason, controlling the progress of
a reversible pebbling is substantially more complicated. Despite the additional technical difficulties, however,
we consider the reversible pebble game to be at least as interesting to study as the standard and black-white
pebble games in view of its tight connection with parallelism in circuit and proof complexity as described
in [Cha13a].
1.2 Follow-up Work
Our hardness of approximation result for the standard pebble game was improved by Demaine and Liu [DL17],
who proved that it is PSPACE-hard to approximate the standard space of a graph of size nwithin an additive
n1/3−ǫterm.
1.3 Organization of This Paper
We present the necessary preliminaries in Section 2and then give a detailed overview of our results in Sec-
tion 3. We prove an asymptotic separation between standard and reversible pebbling in Section 4. In Section 5
we compute the exact price of some classic graphs, trees and pyramids, that we use in Section 6to construct
technical gadgets. These play a key role in Section 7, where we show that reversible pebbling is PSPACE-
complete. We detail the graph product for reversible pebbling in Section 8and its counterpart for standard
pebbling in Section 9. Some concluding remarks are presented in Section 10.
1The reason to emphasize this is that for unbounded fan-in the first author proved a PSPACE-completeness result for reversible
pebbling in [Cha13b], but this result uses simpler constructions and techniques that do not transfer to the bounded fan-in setting.
Another, somewhat related, example is that deciding space in the black-white pebble game has also been shown to be PSPACE-
complete for unbounded indegree in [HP10], but there the unbounded fan-in can be used to eliminate the white pebbles completely,
and again the techniques fail to transfer to the bounded indegree case.
4
2 PRELIMINARIES
2 Preliminaries
All logarithms in this paper are base 2unless otherwise specified. For a positive integer nwe write [n]to
denote the set of integers {1,2,...,n}. We use Iverson bracket notation
JBK=(1if the Boolean expression Bis true;
0otherwise; (2.1)
to convert Boolean values to integer values.
2.1 Boolean Formula Notation and Terminology
Aliteral aover a Boolean variable xis either the variable xitself or its negation x(a positive or negative literal,
respectively). A clause C=a1∨ · · · ∨ akis a disjunction of literals. A k-clause is a clause that contains at
most kliterals. A formula Fin conjunctive normal form (CNF) is a conjunction of clauses F=C1∧ · · ·∧Cm.
Ak-CNF formula is a CNF formula consisting of k-clauses. We think of clauses and CNF formulas as sets,
so that the order of elements is irrelevant and there are no repetitions.
Aquantified Boolean formula (QBF) is a formula φ=Q1x1Q2x2. . . QnxnF, where Fis a CNF
formula over variables x1,...,xnand Qi∈ {∀,∃} are universal or existential quantifiers (i.e., the formula is
in prenex normal form with all variables bound by quantifiers). It was shown in [SM73] that it is PSPACE-
complete to decide whether a QBF is true or not (where we can assume without loss of generality that Fis a
3-CNF formula).
2.2 Graph Notation and Terminology
We write G= (V, E)to denote a graph with vertices V(G) = Vand edges E(G) = E. All graphs in this
paper are directed acyclic graphs (DAGs). An edge (u, v)∈E(G)is an outgoing edge of uand an incoming
edge of v, and we say that uis a predecessor of vand that vis a successor of u. We write pred G(v)to denote
the set of all predecessors of vin Gand succG(v)to denote all its successors. Vertices with no incoming
edges are called sources and vertices with no outgoing edges are called sinks. For brevity, we will sometimes
refer to a DAG with a unique sink as a single-sink DAG, and this sink will usually be denoted z.
Taking the transitive closures of the predecessor and successor relations, we define the ancestors ancG(v)
of vto be the set of vertices that have a path to vand the descendants descG(v)to be the set of vertices on some
path from v. By convention, vis an ancestor and descendant of itself. We write anc∗
G(v) = ancG(v)\ {v}
and desc∗
G(v) = descG(v)\ {v}to denote the proper ancestors and proper descendants of v, respectively.
These concepts are extended to sets of pairwise incomparable vertices by taking unions so that ancG(U) =
Su∈UancG(u),anc∗
G(U) = Su∈Uanc∗
G(u), et cetera, where we say that the vertices in Uare pairwise
incomparable when no vertex in the set is an ancestor of any other vertex in the set. When the graph Gis
clear from context we will sometimes drop it from the notation.
2.3 Standard and Reversible Pebble Games
Apebble configuration on a DAG G= (V, E)is a subset of vertices P⊆V. We consider the following three
rules for manipulating pebble configurations:
1. P′=P∪ {v}for v /∈Psuch that pred G(v)⊆P(a pebble placement on v).
2. P′=P\ {v}for v∈P(a pebble removal from v).
3. P′=P\ {v}for v∈Psuch that pred G(v)⊆P(a reversible pebble removal from v).
5
HARDNESS OF APPROXIMATION IN PSPACE AND SEPARATION FOR PEBBLE GAMES
Astandard pebbling from P0to Pτis a sequence of pebble configurations P= (P0,P1,...,Pτ)where
each configuration is obtained from the preceding one by the rules 1and 2while in a reversible pebbling
rules 1and 3should be used. The time of a pebbling P= (P0,...,Pτ)is time(P) = τ, and the space is
space(P) = max0≤t≤τ{|Pt|}.
We say that a pebbling is unconditional if P0=∅and conditional otherwise. The pebbling price
PebG(P)of a pebble configuration Pis the minimum space of any unconditional standard pebbling on G
ending in Pτ=P, and we define the reversible pebbling price RPebG(P)by taking the minimum over all
unconditional reversible pebblings reaching P. The pebbling price of a single-sink DAG Gwith sink zis
Peb(G) = PebG({z}), and the reversible pebbling price of Gis RPeb(G) = RPebG({z}). We refer to such
pebblings as (complete) pebblings of Gor pebbling strategies for G. Again, when Gis clear from context we
can drop it from the notation, and from now on we will usually abuse notation by omitting the curly brackets
around singleton vertex sets.
For technical reasons, we will often be interested in distinguishing particular flavours of reversible peb-
blings. Suppose that vis a vertex in Gand that P= (P0=∅,P1,...,Pτ)is a reversible pebbling. We will
use the following terminology and notation:
•Pis a visiting pebbling of vif v∈Pτ. The visiting price RPebV(v)of vis the minimal space of any
such pebbling.
•Pis a surrounding pebbling of vif pred (v)⊆Pτand the surrounding price RPebS(v)is the minimal
space of any such pebbling.
•Pis a persistent pebbling of vif it is a reversible pebbling of vin the sense defined before, i.e., such
that Pτ={v}. We will sometimes refer to RPeb(v)as the persistent price of vto distinguish it from
the visiting and surrounding prices.
We also define the visiting price for a single-sink DAG Gwith sink zas RPebV(G) = RPebV
G(z)and the
surrounding price as RPebS(G) = RPebS
G(z).
Note that because of reversibility we could obtain exactly the same visiting space measure by defining a
visiting pebbling of vto be a pebbling P= (P0,P1,...,Pτ)such that P0=Pτ=∅and v∈S0≤t≤τPt,
and let the visiting price be the minimal space of any such pebbling. This is because once we have reached
a configuration containing vwe can simply run the pebbling backwards (because of reversibility) until we
reach the empty configuration again. We can therefore think of a pebbling as visiting vif there is a pebble on
vat some point but this pebble does not stay on vuntil the end of the pebbling. In a persistent pebbling the
pebble remains on vuntil all other pebbles have been removed. A surrounding pebbling, finally, is a pebbling
that reaches exactly the point where a pebble could be placed on v, since all its predecessors are covered by
pebbles (i.e., vis “surrounded” by pebbles), but where vis not necessarily pebbled.
It is not hard to see that for a single-sink DAG Gwe have the inequalities
Peb(G)≤RPebV(G)(2.2)
and
RPebS(G)≤RPebV(G)≤RPeb(G).(2.3)
Perhaps slightly less obviously, we also have the following useful equality.
Proposition 2.1. For any vertex vin a DAG Git holds that RPebS(v) = RPeb(v)−1.
Proof. To see that RPeb(v)≤RPebS(v)+1 consider a surrounding pebbling PSof space RPebS(v). Let P∗
be the pebbling which first runs PSto surround v, then puts a pebble on v, and finally runs the reverse of PS
to “unsurround” v(while keeping the pebble on v). Since P∗is a persistent pebbling of space RPebS(v) + 1,
the inequality follows.
6
2 PRELIMINARIES
We now prove that RPebS(v)≤RPeb(v)−1. Consider a persistent pebbling Pfor vof space RPeb(v).
Let tbe the last time that a pebble is put on v. Then vertex vis surrounded at time t, and there is a pebble on
vsince time t. Let P≥tbe the conditional pebbling obtained from Pafter time t, with the modification that
vertex vhas no pebble throughout P≥t, and let PR
≥tbe this pebbling run in reverse. Then PR
≥tis a surrounding
pebbling in space at most RPeb(v)−1, and the inequality follows.
2.4 The Dymond–Tompa and Raz–McKenzie Games
As described above, the Dymond–Tompa game on a single-sink DAG Gis played in rounds by two players
Pebbler and Challenger. In the first round Pebbler places a pebble on the sink zand Challenger challenges
this vertex. In all subsequent rounds, Pebbler places a pebble on an arbitrary empty vertex and Challenger
chooses to either challenge this new vertex (which we refer to as jumping) or to re-challenge the previously
challenged vertex (referred to as staying). The game ends when at the end of a round all the (immediate)
predecessors of the currently challenged vertex are covered by pebbles.2The Dymond–Tompa price DT(G)
of Gis the maximal number of pebbles rneeded for Pebbler to finish the game, or expressed differently the
smallest number rsuch that Pebbler has a strategy to make the game end in at most rrounds regardless of
how Challenger plays.
Let us also for completeness describe the Raz–McKenzie game, which is also played on a single-sink
DAG Gby two players Pebbler and Colourer. In the first round Pebbler places a pebble on the sink zand
Colourer colours it red. In all subsequent rounds, Pebbler places a pebble on an arbitrary empty vertex and
Colourer then colours this new pebble either red or blue. The game ends when there is a vertex with a red
pebble, while all its predecessors in the graph have blue pebbles. The Raz–McKenzie price RM(G)of Gis
the smallest number rsuch that Pebbler has a strategy to make the game end in at most rrounds regardless
of how Colourer plays.
The intuition for this game is that the vertices on the graphs have assigned values true (blue) or false
(red), with the condition that each vertex has value equal to the conjunction of the values of its predecessors.
Colourer claims that the sink is false, but the above condition vacuously implies that all source vertices must
be true. Colourer loses when Pebbler discovers a violation of the condition. Pebbler wants to find the violation
as soon as possible, while Colourer wants to fool Pebbler for as long as possible.
In [Cha13a] the first author proved that the equalities
DT(G) = RM(G) = RPeb(G)(2.4)
hold for any single-sink DAG G, i.e., that the reversible pebbling price, the Dymond–Tompa price and the
Raz–McKenzie price all coincide. Thus, any result we prove for one of these games is also guaranteed to hold
for the other games. The above equalities are very convenient in that they allow us to switch back and forth
between the reversible pebble game and the Dymond–Tompa game (or Raz–McKenzie game) when proving
upper and lower bounds, depending on which perspective is more suitable at any given time. In particular,
when proving lower bounds for reversible pebblings it is often helpful to do so by devising good Challenger
strategies in the Dymond–Tompa game. One final technical remark in this context is that in all such strategies
that we construct it holds that Challenger will either stay or jump to an ancestor of the currently challenged
vertex. Because of this we can assume without loss of generality that Pebbler only pebbles vertices in the
subgraph consisting of ancestors of the currently challenged vertex. If Pebbler pebbles some vertex outside
of this subgraph Challenger will just stay put on the current vertex, and so Pebbler just wastes a round.
2We remark that our description follows [Cha13a] and thus differs slightly from the original definition in [DT85], but the two
versions are equivalent for all practical purposes.
7
HARDNESS OF APPROXIMATION IN PSPACE AND SEPARATION FOR PEBBLE GAMES
(a) Path blown up to sequence of K3,3-graphs. (b) Road graph of length 9and width 3.
Figure 1: Modifications of path graphs to amplify difference between reversible and standard pebbling price.
3 Overview of Results and Sketches of Proofs
In this section we give a detailed overview of our results and also sketch some of the main ideas in the proofs.
In the rest of the paper, we then provide all the missing technical definitions and present the actual formal
proofs.
3.1 Separation Between Standard and Reversible Pebbling
As mentioned in Section 1, the strongest separation hitherto known between standard and reversible pebbling
is for the length-ℓpath on vertices {v1, v2,...,vℓ+1}with edges (vi, vi+1)for all i∈[ℓ], which has a standard
pebbling with 2pebbles whereas reversible pebblings require space Θ(log ℓ)[Ben89,LV96]. We give a
simple construction improving this to a multiplicative logarithmic separation.
Theorem 3.1. For any function s(n) = On1/2−ǫfor ǫ > 0constant there are DAGs {Gn}∞
n=1 of size Θ(n)
with a single sink and fan-in 2such that Peb(G) = O(s(n)) and RPeb(G) = Ω(s(n) log n)(where the
hidden constant depends linearly on ǫ).
A first observation is that if we did not have the bounded fan-in restriction, Theorem 3.1 would be very
easy. In such a case we could just take the path of length ℓ, blow up every vertex vito svertices v1
i,...,vs
i,
and add edges vj
i, vj′
i+1for all j, j′∈[s], so that we get a sequence of complete bipartite graphs Ks,s glued
together as shown in Figure 1a. It is not hard to show that any reversible pebbling of this DAG would have to
do sparallel, synchronized pebblings of the paths vj
1, vj
2,...,vj
ℓ+1for j∈[s], which would require space
Ω(slog ℓ), whereas a standard pebbling would clearly only need space O(s).
For bounded indegree it is not a priori clear what to do, however, or indeed whether there should even
be a multiplicative separation. But it turns out that one can actually simulate a lower bound proof along the
same lines as above by considering a layered graph as in Figure 1b, with sparallel paths of length up to ℓ
and with every path having an extra edge fanning out to its “neighbour path” above (or at the bottom for the
top row) at each level. We will refer to this construction as a road graph of length ℓand width s(where a
path is a maximally narrow road of width 1). It is easy to verify that the standard pebbling price of a road of
width s≥2is s+ 2. We claim that the reversible pebbling price is Ωslog(ℓ/s), from which Theorem 3.1
follows.
To prove the reversible pebbling lower bound it is convenient to think instead in terms of Challenger
strategies in the Dymond–Tompa game. The idea is that Challenger will stay put on the sink until Pebbler has
pebbled enough vertices so that there are no pebble-free paths from any source vertex to the sink. Intuitively,
the cheapest way for Pebbler to disconnect the graph is with a straight cut over some layer. When this happens,
Challenger looks at the latest pebbled vertex and compares the subgraph between the sources and the cut with
the subgraph between the cut and the sink. If more than half of the graph is before the cut, Challenger jumps
to the latest pebbled vertex. If not, Challenger stays on the sink. This strategy is then repeated on a graph of at
least half the length. Since every cut by Pebbler requires spebbles, Challenger can survive for roughly slog ℓ
8
3 OVERVIEW OF RESULTS AND SKETCHES OF PROOFS
rounds (except that the rigorous argument is not quite this simple, and the slightly smaller factor log(ℓ/s)in
the formal statement of the theorem is in fact inherent).
3.2 PSPACE-Completeness of Reversible Pebbling
Moving on to technically more challenging material, let us next discuss our PSPACE-completeness result for
reversible pebbling, which we restate here more formally for the record.
Theorem 3.2. Given a single-sink DAG Gof fan-in 2and a parameter s, it is PSPACE-complete to decide
whether Gcan be reversibly pebbled in space sor not. In more detail, given a QBF φ=Q1x1Q2x2. . . QnxnF,
where Fis a 3-CNF formula over variables x1,...,xn, there is a polynomial-time constructible single-
sink graph G(φ)of fan-in 2and a polynomial-time computable number γ(φ)such that RPebG(φ)=
γ(φ) + Jφis falseK.
At a high level, our proof is similar to that in [GLT80] for standard pebbling: we build gadgets for vari-
ables, clauses, and universal and existential quantifiers, and then glue them together in the right way so that
pebbling through the gadgets corresponds to verifying satisfying assignments for universally and existen-
tially quantified subformulas of the QBF φ. However, the execution of this simple idea is highly nontrivial
even in [GLT80], and we run into several additional technical difficulties when we want to do an analogous
reduction for reversible pebbling.
For starters, since the difference in pebbling price for graphs G(φ)obtained from true and false QBFs φ
is just an additive 1, we need exact control over the pebbling price of all components used in the reduction.
For standard pebbling there is no problem here—exact bounds on pebbling price are known for quite a wide
selection of graphs—but in the reversible setting this becomes an issue already for almost the simplest possible
graph: the complete binary tree of height h. An easy inductive argument shows that the standard pebbling
price of such a tree is exactly h+ 2. Since reversible pebblings find paths more challenging than do standard
pebblings, one could perhaps expect an extra additive log hor so in the reversible pebbling bound. However,
the asymptotically correct bound turns out to be h+ Θ(log∗h)as shown in [Kr´a04], and the upper and lower
bounds on the multiplicative constant obtained in that paper are far from tight.
The story is even worse for the workhorse of the construction in [GLT80] (and many other pebbling
results), namely pyramids of height h, which have ivertices at level ifor i= 1,...,h + 1, and where the
jth vertex at level ihas incoming edges from the jth and (j+ 1)st vertices at level i+ 1. There is a very
neat proof in [Coo74] that the standard pebbling price is again exactly h+ 2, but for reversible pebbling price
nothing has been known except that it has to be somewhere between h+ 2 and h+ O(log∗h)(where the
latter bound follows since any strategy for a complete binary tree of height hworks for any DAG of height h).
As a crucial first step towards establishing Theorem 3.2, we exactly determine the reversible pebbling price
of pyramids (and also binary trees).
Theorem 3.3. For ∆denoting a positive integer, let gbe the function defined recursively as
g(∆) = (0if ∆ = 1;
2g(∆−1)+∆−2+g(∆ −1) otherwise;
and let the inverse g−1of this function be defined as
g−1(h) = min{∆|g(∆) ≥h}.
Then the persistent pebbling price of a pyramid of height h, as well as of a complete binary tree of height h,
is h+g−1(h), where g−1is efficiently computable.
9
HARDNESS OF APPROXIMATION IN PSPACE AND SEPARATION FOR PEBBLE GAMES
Even though Theorem 3.3 is an important step, we immediately run into new problems when trying to
use it as a building block in our reduction for reversible pebbling. In the standard pebble game a complete
pebbling is any pebbling that reaches the sink. For the reversible game there is a subtle distinction in that
we can ask whether it is sufficient to just reach the sink or whether the rest of the graph must also be cleared
of pebbles. As discussed in Section 2, this leads to two different flavours of reversible pebblings, namely
persistent pebblings, which leave a pebble on the sink with the rest of the graph being empty, and visiting
pebblings, which just reach the sink (and can then be thought to run in reverse after having visited the sink
to clear the whole graph including the sink from pebbles). The pebblings we actually care about are the
persistent ones, but we cannot rule out the possibility that subpebblings of gadgets are visiting pebblings.
Clearly, the difference in pebbling space is at most 1, but this is exactly the additive 1of which we cannot
afford to lose control! To make things worse, for pyramids it turns out that persistent and visiting pebbling
prices actually do differ except in very rare cases.
Because of this, we have to build more involved graph gadgets for which we can guarantee that visiting
and persistent prices coincide. These gadgets are constructed in two steps. First, we take a pyramid and
append a path of suitable length, depending on the height of the pyramid, to the pyramid sink, resulting in
a graph that we call a teabag. Second, we take such teabags of smaller and smaller size and stack them on
top of one another, which yields a graph that looks a bit like a Christmas tree. These Christmas tree graphs
are guaranteed to have the same pebbling price regardless of whether a reversible pebbling is visiting or
persistent.
With this in hand we are almost ready to follow the approach in the PSPACE-completeness reduction
for standard pebbling in [GLT80]. The idea is that we want to build gadgets for the quantifiers in a formula
φ=∀x∃y···Qz F of specified pebbling price so that the only way to pebble the graph G(φ)without using
too much space is to first pebble the gadget for ∀x, then ∃y, et cetera, in the correct order until all quantifier
gadgets have been pebbled. Once we get to the clause gadgets, we would like that the pebbles in the quantifier
gadgets are locked in place encoding a truth value assignment to the variables, and that the only way to pebble
through the clause gadgets without exceeding the space budget is if every clause contains at least one literal
satisified by this truth value assignment.
In order to realize this plan, there remains one more significant technical obstacle to overcome, however.
To try to explain what the issue is, we need to discuss the PSPACE-completeness reduction in [GLT80] in
slightly more detail. The way this reduction imposes an order in which the quantifier gadgets have to be
pebbled is that pyramid graphs are included “at the bottom” of the gadgets (i.e., topologically first in order).
The source vertices of the quantifier gadgets all appear in such pyramids, and one has to pebble through these
pyramids to reach the rest of a gadget (where pebble placements encode variable assignments as mentioned
above).
In the first, outermost quantifier gadget the pyramids have large height. In the second gadget the pyramid
heights are slightly smaller, et cetera, down to the last, innermost quantifier gadget where the pyramids have
smallest height. In this way, the pyramids are used to “lock up” pebbles and force a strict order of pebbling of
the gadgets. It can be shown that in order not to exceed the pebbling space budget, any pebbling strategy has
to start by pebbling the highest pyramids in the first gadget. If the pebbling starts anywhere else in the graph,
this will mean that there are already pebbles elsewhere in the graph when the pebbling strategy reaches the
first, highest pyramids in the outermost quantifier, but if so the overall pebbling has to use up too much space
to pebble through this pyramid. One can also show that once the pyramids in the outermost quantifier gadget
have been pebbled, the pebbling cannot proceed until the next quantifier gadget is pebbled. The pyramids in
this gadget have smaller height, but there are also pebbles stuck in place in the outermost gadget, meaning
that pyramids must again be pebbled in exactly the right order to stay within the space budget.
These properties can be used to normalize pebbling strategies in the standard pebble game. Without loss
of generality, one can assume that any strategy that starts pebbling a pyramid in a gadget will complete this
local pebbling in one go, leaving a pebble at the sink of the pyramid, and will not place pebbles anywhere else
10
3 OVERVIEW OF RESULTS AND SKETCHES OF PROOFS
r
(a) Christmas tree.
a
b
r
(b) Turnpike.
Figure 2: Legend for technical gadget building blocks.
x′
i
xi
ri
¯x′
i
¯xi
ri
(a) Variable gadget.
x′
i
xi
ri
¯x′
i
¯xi
ri
(b) false position.
x′
i
xi
ri
¯x′
i
¯xi
ri
(c) true position.
Figure 3: Gadget for variable xiand pebble positions corresponding to truth value assignments.
until the pebbling of the pyramid has been completed. Also, once a pyramid in a quantifier gadget has been
pebbled in this way, one can prove that it will never be pebbled again since there is now at least one additional
pebble at some vertex later in the topological order in the graph, and a repeated pebbling of the pyramid in
question would therefore exceed the space budget. Thus, not only do the pyramids enforce that the gadgets
are pebbled in the right order—they also serve as single-entry access points to the gadgets, making sure that
each gadget is pebbled exactly once.
There is no hope of building gadgets with such properties in a reversible pebbling setting. It is simply
not true that a reversible pebbling will pebble through a subgraph and then never return. Instead, as already
discussed reversible pebblings will proceed in alternating phases of interleaved “forward sweeps” and “reverse
sweeps,” and subgraphs will be entered also in reverse topological order. Therefore, it is not sufficient to add
“space-locking” subgraphs at the source vertices of the gadgets. Rather, we have to insert “single-passage
points” inside and in between the gadgets for quantifiers and clauses. We obtain such subgadgets by further
tweaking our Christmas tree construction so that it can also connect two vertices in such a way that any
pebbling has to “pay a toll” to go through this subgraph. We cannot describe these gadgets, which we call
turnpikes, in detail here, but mention that the “space-locking” property that they have is that when the entrance
vertex is eliminated by having a pebble placed on that vertex, then the cost of pebbling through the rest of the
turnpike drops by 1. This is critically used in the subgraph compositions described next.
Assuming the existence of the necessary technical subgraph constructions sketched above, we can now
describe the overall structure of our reduction from quantified Boolean formulas to reversible pebbling (where
all parameters shown in the figures are fixed appropriately in the formal proofs). In the following figures we
denote a Christmas tree of (visiting and persistent) pebbling price rby the symbol in Figure 2a, where we
only display the sink vertex. We denote the turnpike gadget just discussed by the symbol in Figure 2b. We
write rto denote the toll parameter of the turnpike, where a turnpike with toll rhas persistent price r+ 2,
but only r+ 1 if we do not count the source aas part of the turnpike.
For every variable xiwe have a variable gadget as shown in Figure 3a, where we think of a truth value
assignment ρas represented by pebbles on vertices {¯xi, x′
i}when ρ(xi) = false and on {xi,¯x′
i}when
ρ(xi) = true, as shown in Figures 3b and 3c, respectively.
For every clause Cjwe have a clause gadget as depicted in Figure 4a. The vertices labelled ℓ′
j,k and ℓj,k in
Figure 4a are identified with the corresponding vertices for the positive or negative literal ℓj,k in the variable
11
HARDNESS OF APPROXIMATION IN PSPACE AND SEPARATION FOR PEBBLE GAMES
aj
bj
cj
uj
vj
pj
ℓ′
j,1
ℓ′
j,2
ℓ′
j,3
ℓj,1
ℓj,2
ℓj,3
βj
βj
βj
(a) Clause gadget.
z1
z2
d1
d2
d3d4e
r
r−1
r−2
(b) Conjunction gadget.
Figure 4: Gadgets for clauses and CNF formulas.
gadget in Figure 3a. If ρsatisfies a literal, then there is a pebble on the entrance vertex of the corresponding
turnpike, meaning that we can pebble through the gadget for a clause containing that literal with one less
pebble than if ρdoes not satisfy the clause.
To build the subgraph corresponding to a 3-CNF formula F=Vm
j=1 Cjwe join clause gadgets sequen-
tially using the conjunction gadget in Figure 4b. For technical reasons we start by joining a dummy graph
with the first clause gadget, then we join the result to the second clause gadget, and so on up to the mth clause
of F. The resulting graph has the property that if pebbles are placed on the variable gadgets according to an
assignment ρthat satisfies F, then the number of additional pebbles needed to pebble the graph is one less
than if the assignment is falsifying.
Finally we have one quantifier gadget for each variable. To describe this part of the construction, we
sort the variables indices in reverse order from the innermost to the outermost quantifier and denote by φithe
subformula with just the iinnermost quantifiers, so that φ0=F=Vm
j=1 Cj,φi=Qixiφi−1for Qi∈ {∀,∃},
and φ=φn. We construct graphs G(i):= G(φi), starting with G(0) which is just the subgraph corresponding
to the CNF formula F. To construct G(i+1) from G(i)we add an existential gadget as in Figure 5a if xiis
existentially quantified and a universal gadget as in Figure 5b if xiis universally quantified. An example of
the full construction can be found in Figure 6.
Given this construction we argue along the same lines as in in [GLT80], although as mentioned above
there are numerous additional technical complications that we cannot elaborate on in this brief overview of
the proof. We show that given an assignment ρito {xn,...,xi+1}, the number of additional pebbles needed
to pebble G(i)differs by 1depending on whether φiis true under the assignment ρior not. An existential
gadget can be optimally pebbled by setting xito any value that satisfies φi−1. To pebble a universal gadget one
needs to assign xito some value, pebble through the gadget, unset xiand assign it to the opposite value, and
finally pebble through the gadget again, and both assignments to ximust yield satisfying assignments to φi−1
in order for the pebbling not to go over budget. Proceeding by induction, we establish that the complete graph
G(n)can be pebbled within the specified space budget only if φ=φnis true, which yields Theorem 3.2.
12
3 OVERVIEW OF RESULTS AND SKETCHES OF PROOFS
x′
i
xi
ri
¯x′
i
¯xi
ri
figi
qi
qi−1
γi−5
(a) Existential quantifier gadget.
x′
i
xi
ri
¯x′
i
¯xi
ri
f′
i¯
f′
i
fi¯
fi
gi¯gi
hi¯
hi
qi
γi−6γi−6
qi−1
γi−7γi−7
(b) Universal quantifier gadget.
Figure 5: Quantifier gadgets for variable xi.
3.3 PSPACE-Inapproximability up to Additive Constants
Let us conclude the detailed overview of our contributions by describing what is arguably the strongest result
in this paper, namely a strengthening of the PSPACE-completeness of standard pebbling in [GLT80] and of
reversible pebbling in Theorem 3.2 to PSPACE-hardness results for approximating standard and reversible
pebbling price to within any additive constant K.
Theorem 3.4. For any fixed positive integer Kit is PSPACE-complete to decide whether a single-sink DAG G
with fan-in 2has (standard or reversible) pebbling price at most sor at least s+K.
We remark that it would of course have been even nicer to prove multiplicative hardness results. We
want to stress again, though, that to the best of our knowledge these are the first results ever for hardness of
approximation of pebble games in a general setting. The fact that these results hold even for PSPACE could
perhaps be taken both as an indication that it should be possible to prove much stronger hardness results for
algorithms limited to polynomial time, and as a challenge to do so.
We obtain Theorem 3.4 by defining and analyzing two graph product constructions, one for standard and
one for reversible pebbling, which take two graphs and output product graphs with pebbling price equal to the
sum of the pebbling prices of the two input graphs (except for an additive adjustment). These graph products
can then be applied iteratively K−1times to the graphs obtained by the reductions from QBFs. In the next
theorem we state the formal properties of these graph products.
Theorem 3.5. Given single-sink DAGs Giof fan-in 2 and size nifor i= 1,2, there are polynomial-time
constructible single-sink DAGs S(G1, G2)and R(G1, G2)of fan-in 2 and size O(n1+n2)2such that
•For standard pebbling price it holds that Peb(S(G1, G2)) = Peb(G1) + Peb(G2)−1.
•For reversible pebbling price it holds that RPebR(G1, G2)=RPeb(G1) + RPeb(G2)