The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

We show that the counting class LWPP [FFK94] remains unchanged even if one allows a polynomial number of gap values rather than one. On the other hand, we show that it is impossible to improve this from polynomially many gap values to a superpolynomial number of gap values by relativizable proof techniques. The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction) is in LWPP (and thus low for PP, i.e., $\rm PP^\mbox{Legitimate Deck} = PP$) if the weakened version of the Reconstruction Conjecture holds in which the number of nonisomorphic preimages is assumed merely to be polynomially bounded. This strengthens the 1992 result of K\"{o}bler, Sch\"{o}ning, and Tor\'{a}n [KST92] that the Legitimate Deck Problem is in LWPP if the Reconstruction Conjecture holds, and provides strengthened evidence that the Legitimate Deck Problem is not NP-hard.


Introduction
Nothing is more natural than wanting to better understand an object by knowing what it can and cannot do. Whether wondering how fast a (rental?) car can go in reverse or wondering if NP NP can without loss of generality be assumed to ask at most one question per nondeterministic path (as it indeed can, as is implicit in the quantifier characterization [SM73,Wra77] of NP NP ), we both in life and as theoreticians want to find how robust things are.
We are often particularly happy when a class proves to be quite robust under definitional perturbations. Such robustness on one hand suggests that perhaps there is something broadly natural about the class, and on the other hand such robustness often makes it easier to put the class to use. This paper shows that the counting classes LWPP and WPP, defined in 1994 in the seminal work of Fenner, Fortnow, and Kurtz [FFK94] on gap-based counting classes, are quite robust. Even though their definitions are in terms of having the gap function (the difference between the number of accepting and rejecting paths of a machine) hit a single target value, we prove (in Section 3) that one can allow a list of up to polynomially many target values without altering the descriptive richness of the class, i.e., without changing the class.
We then apply this to the question of whether the Legitimate Deck Problem is in LWPP. The Legitimate Deck Problem (a formal definition will be given in Section 4) is the decision problem of determining whether, given a multiset of (unlabeled) graphs, there exists a graph G such that that multiset is precisely (give or take isomorphisms) the multiset of one-node-deleted subgraphs of G (a.k.a. the deck of G) [KH94]. The Reconstruction Conjecture [Kel42,Ula60]which in the wake of the resolution of the Four-Color Conjecture was declared by the editorial board of the Journal of Graph Theory to be the foremost open problem in graph theory [Edi77]states that every graph with three or more nodes is uniquely determined (give or take isomorphisms) by its multiset of one-node-deleted subgraphs. The Legitimate Deck Problem was defined in 1978 by Nash-Williams [NW78], in his paper that framed the algorithmic/complexity issues related to reconstructing graphs-such as telling whether a given deck is legitimate (i.e., is the deck of some graph).
Our application of our LWPP robustness result to the question of whether the Legitimate Deck Problem is in LWPP is the following. The strongest previous evidence of the simplicity of the Legitimate Deck Problem is the 1992 result of Köbler, Schöning, and Torán [KST92] that the Legitimate Deck Problem is in LWPP (and thus is PP-low, i.e., PP Legitimate Deck = PP) if the Reconstruction Conjecture (i.e., that each deck whose elements all have at least two nodes has at most one preimage, give or take isomorphisms) holds. Using this paper's main robustness result as a tool, Section 4 proves that the Legitimate Deck Problem is in LWPP (and thus is PP-low) if a weakened version of the Reconstruction Conjecture holds, namely, that each deck has at most a polynomial number of nonisomorphic preimages. This weakened version is not known to be equivalent to the Reconstruction Conjecture itself. And so our result for the first time gives a path to proving that the Legitimate Deck Problem is PP-low that does not require one to, on the way, resolve the foremost open problem in graph theory [Edi77].
We started this section by noting that it is natural to want to know both flexibilities and limitations of classes. Our main result is about flexibility: going from one target gap to instead a polynomial number. But are we leaving money on the table? Could we extend our result to slightly superpolynomial numbers of target gaps, or even to exponential numbers of target gaps? In Section 5 we note that if the robustness of LWPP were to hold up to exponentially many target gaps, then NP would be in LWPP and so would be PP-low (i.e., PP NP = NP); yet NP is widely suspected not to be PP-low. We also, by encoding nondeterministic oracle Turing machines by low-degree multivariate polynomials so as to capture the gap functions of those machines, show an oracle relative to which robustness fails for all superpolynomial numbers of target gaps; thus, no extension beyond this paper's polynomial-number-of-target-gaps robustness result for LWPP can be proven by a relativizable proof.
To summarize: In this paper, we prove that LWPP and WPP are robust enough that they remain unchanged when their single target gap is allowed to be expanded to a polynomial-sized list; we apply this new robustness of LWPP to show that the PP-lowness of the Legitimate Deck Problem follows from a weaker hypothesis than was previously known; and we show that our polynomial robustness of LWPP is optimal with respect to relativizable proofs.

Preliminaries
We first present the definitions of many of the counting classes that we will be speaking of, taking the definitions from the seminal paper of Fenner, Fortnow, and Kurtz [FFK94].
The following class, WPP, is potentially larger than SPP. Instead of the "target value" 1 for the case x ∈ A, we allow a target value f (x), where f may be any polynomial-time computable function whose image does not contain 0. FP denotes the class of polynomial-time computable functions.
Definition 2.4 ( [FFK94]). WPP is the class of all sets A such that there exists a GapP function g and a function f ∈ FP that maps from Σ * to Z − {0} such that for all x ∈ Σ * , The class LWPP is the same as WPP except that the "target function" f may depend on only the length of the input.
Definition 2.5 ( [FFK94]). LWPP is the class of all sets A such that there exists a GapP function g and a function f ∈ FP that maps from 0 * to Z − {0} such that for all x ∈ Σ * , We now generalize the definition of LWPP to the case of having the target of the GapP function be, for members of the set, not a single value but a collection of values.
One might expect us to formalize this by simply having the polynomial-time computable "what is the target" function output a list of the nonzero-integer targets. That would work fine and be equivalent to what we are about to do, as long as we are dealing with lists having at most a polynomial number of elements. However, to be able to speak of even longer lists-as will be important in our negative results offsetting our main result-we use an indexing approach, as follows.
Definition 2.6. Let r be any function mapping from N to N. Then the class r-LWPP is the class of all sets A such that there exists a GapP function g and a function f ∈ FP that maps to Z − {0} such that for each x ∈ Σ * , The following class, Poly-LWPP, will be central to this paper: Our main result is that this class in fact equals LWPP. The "+ c" in Definition 2.7 may seem strange at first. But without it we would have a boundary-case pathology at n = 0, namely, the class could not contain any set that contains the empty string. 1 Definition 2.7.
It is easy to see that 1-LWPP = LWPP, and that, of course, more flexibility as to targets never removes sets from the class, i.e., speaking loosely for the moment as to notation (and the log case will not be defined or used again in this paper, but it is clear from context here what we mean by it; the exponential case will be defined only in Section 5), 1-LWPP ⊆ 2-LWPP ⊆ 3-LWPP ⊆ · · · ⊆ Log -LWPP ⊆ Poly-LWPP ⊆ Exp-LWPP. As mentioned above, in this paper we will prove that the first five of these "⊆"s are in fact all equalities. We will also prove that the sixth "⊆" cannot be an equality unless NP is PP-low.
We now show that for every function r, r-LWPP is contained in the co-class of the well-known counting class C = P. Wag86]). C = P is the class of all sets A such that there is a nondeterministic polynomial-time Turing machine N and a function f ∈ FP such that for each x ∈ Σ * , The "+ c" in Definition 2.7 also, on its surface, would seem to make a difference at length 1, by allowing lists of size greater than one; however, one could work around that issue. In contrast, the exclusion of the empty string would not be avoidable if our class of polynomials were to be a class-such as n c -such that all of its members evaluate to 0 at n = 0. In any case, our use of n c + c avoids any special worries at lengths 0 and 1. And since for every polynomial p there is a c such that (∀n ∈ N)[p(n) ≤ n c + c], using polynomials just of the form n c + c is in fact not a restriction.
More convenient for us is the following characterization of C = P using GapP functions.
Theorem 2.9 ( [FFK94]). For each A ⊆ Σ * , A ∈ C = P if and only if there exists a function g ∈ GapP such that for all x ∈ Σ * , We thus certainly have the following, which holds simply by taking the GapP function g required in Theorem 2.9 to be the same as the function g in Definition 2.6.

Main Result: LWPP Stays the Same If for Accepted Inputs We Allow Polynomially Many Gap Values Instead of One
We now state our main result: LWPP altered to allow even a polynomial number of target gap values is still LWPP (i.e., with just one target gap value).
For the proof, we need the following closure properties shown by Fenner, Fortnow, and Kurtz [FFK94]. 2 For function classes F 1 and F 2 , Closure Property 3.4 ( [FFK94]). If g ∈ GapP and q is a polynomial, then the function

Closure Property 3.5 ([FFK94]). GapP is closed under addition, subtraction, and multiplication.
Proof of Theorem 3.1. As mentioned previously, it is easy to see that LWPP ⊆ Poly-LWPP.
To show Poly-LWPP ⊆ LWPP, let A be a set in Poly-LWPP defined by g ∈ GapP, f ∈ FP, and polynomial r(n) = n c + c according to Definitions 2.6 and 2.7.
Let h 1 be a function such that for all x ∈ Σ * and i ∈ N + , We have h 1 ∈ GapP since f ∈ FP ⊆ GapP, g ∈ GapP, and GapP is closed under subtraction [FFK94]. We define h 2 such that for all x ∈ Σ * , It follows that for every x ∈ Σ * , (1) Now we define the function g such that for all x ∈ Σ * , Using the closure properties, it is easy to see that g ∈ GapP.
Note that by Eqn.
(1), we have that for all x ∈ Σ * , (2) Let f be a function such that for all ℓ ∈ N, It is easy to see that f ∈ FP. Keeping in mind Eqn. (2), it follows from the above that for every x ∈ Σ * , Since g ∈ GapP and f ∈ FP, this implies that A ∈ LWPP.
A theorem analogous to Theorem 3.1 also holds for the corresponding class that allows the target values to depend on the actual input instead on only the length of the input.
The proof is almost exactly the same as the proof of Theorem 3.1, simply taking into account the fact that for WPP the "gap" function can vary even among inputs of the same length.

Applying the Main Result to Graph Reconstruction
Definition 4.1. Let G 1 , G 2 , . . . , G n be a sequence of graphs and G = (V, E) a graph with V = {1, 2, . . . , n}. Suppose that there is a permutation π ∈ S n such that for each k ∈ {1, 2, . . . , n}, the The Reconstruction Conjecture (see, e.g., the surveys [BH77, Man88, Bon91] and the book [LS03]) says that each legitimate deck consisting of graphs with at least two vertices has exactly one preimage up to isomorphism. This conjecture is a very prominent conjecture in graph theory-as mentioned in Section 1 it is perhaps the most important conjecture in that area-and has been studied for many decades.
Nash-Williams [NW78], Mansfield [Man82], Kratsch and Hemachandra [KH94], and Hemaspaandra et al. [HHRT07] introduced various decision problems related to the Reconstruction Conjecture, as part of a stream of work studying the algorithmic and complexity issues of reconstruction. We here are interested mainly in the Legitimate Deck Problem, which is defined as the following decision problem.
Mansfield [Man82] showed that the Graph Isomorphism Problem, GI (given two graphs G 1 and G 2 , are they isomorphic?), is polynomial-time many-one reducible to the Legitimate Deck Problem. However, to this day it remains open whether there is a polynomial-time many-one reduction from the Legitimate Deck Problem to GI.
So how hard is the Legitimate Deck Problem? It is easy to see that the Legitimate Deck Problem is in NP. In the following, we will see that there is some evidence that the Legitimate Deck Problem is not NP-hard, and we will improve that evidence.
Let us define the following function problem.
is in GapP.
Theorem 4.2 has the following corollary.  Unfortunately, we do not know whether the Reconstruction Conjecture holds. However, perhaps we can prove the membership of the Legitimate Deck Problem in LWPP under a weaker assumption than the Reconstruction Conjecture, for instance, what if the number of nonisomorphic preimages is not always 0 or 1 (as holds under the Reconstruction Conjecture), but rather is merely relatively small, e.g., some constant or some polynomial in the number of vertices (as must hold for each graph class having bounded minimum degree)? We will now use the results of the previous section to prove that this indeed is the case. For any function r, we now define a complexity class r-LWPP. This class may not seem very natural, but we will see that it is very-well suited to helping us classify the problem Legitimate Deck. In some sense, it is a tool that we will use in our proof, and then will discard by noting that it in fact turns out to be a disguised version of r-LWPP.
Definition 4.6. Let r be any function mapping from N to N. Then the class r-LWPP is the class of all sets A such that there exists a GapP function g, and a function f ∈ FP that maps from 0 * to Z − {0}, such that for each x ∈ Σ * , Theorem 4.7. Let q be any nondecreasing function from N to N. Then the following holds. If the q-Reconstruction Conjecture holds, then the Legitimate Deck Problem is in q-LWPP.
This almost directly implies that Legitimate Deck ∈ q-LWPP. However, note that to satisfy Definition 4.6, function h must be a function that depends only on the length of the input G 1 , G 2 , . . . , G n . The problem here is that (depending on how exactly we decide to encode the graphs G 1 , G 2 , . . . , G n in the input G 1 , G 2 , . . . , G n ) even the value n (the number of graphs in the deck) may depend on the actual input G 1 , G 2 , . . . , G n and not only on the length of the input G 1 , G 2 , . . . , G n . Fortunately, there is a way to get around this problem. From the length of the input, we get at least an upper bound for n since we can certainly assume that n ≤ | G 1 , G 2 , . . . , G n |.
Define a function h such that for each m ∈ N, h(0 m ) is the product of all "h-values" up to length m. That is, define function h such that for all m ∈ N, Define function h ′ such that for all inputs G 1 , G 2 , . . . , G n , Now we can see that for all inputs G 1 , G 2 , . . . , G n , ..,Gn | ) for some i ∈ {1, 2, . . . , q(n)} if G 1 , G 2 , . . . , G n ∈ Legitimate Deck. (3) Note that h, h ′ ∈ FP and g ∈ GapP. By Closure Properties 3.2 and 3.5, the function is in GapP. Finally, note that since q is nondecreasing and n ≤ | G 1 , G 2 , . . . , G n |, we have that q(n) ≤ q(| G 1 , G 2 , . . . , G n |) in Eqn.
(3). Thus by Definition 4.6-with the GapP function g there being our g(x) · h ′ (x) and the function f there being our h-we have that Legitimate Deck ∈ q-LWPP.
We want to show that if there exists a polynomial q such that the q-Reconstruction Conjecture holds, then Legitimate Deck is in LWPP. To this end, we need the following inclusion. Proof. Let A be a set in r-LWPP via f 1 ∈ FP and g ∈ GapP. Let f 2 ∈ FP be the function defined such that for every n, i ∈ N + , f 2 ( 0 n , i ) = i · f 1 (0 n ). Then A is in r-LWPP via f 2 ∈ FP and g ∈ GapP.
Corollary 4.9. If the q-Reconstruction Conjecture holds for some polynomial q, then the Legitimate Deck Problem is in LWPP.
Proof. Suppose the q-Reconstruction Conjecture holds for nondecreasing polynomial q. (If q is not nondecreasing but the q-Reconstruction Conjecture holds, then obviously we can replace q with a nondecreasing polynomial q ′ that on each input n is greater than or equal to q(n) and the q ′ -Reconstruction Conjecture will hold. So we may w.l.o.g. take it that q is nondecreasing.) By Theorems 4.7 and 4.8, Legitimate Deck ∈ q-LWPP ⊆ q-LWPP and hence Legitimate Deck ∈ Poly-LWPP. With Theorem 3.1, it follows that Legitimate Deck ∈ LWPP. This gives us our new, more flexible-though still conditional-evidence that the Legitimate Deck Problem is not NP-hard.
Corollary 4.10. If the q-Reconstruction Conjecture holds for some polynomial q, then the Legitimate Deck Problem is not NP-hard (or even NP-Turing-hard) unless NP is PP-low.
The following is easy to see.
Theorem 4.11. Let H be any P-recognizable class of graphs such that decks of its graphs have a number of nonisomorphic preimages that is bounded polynomially in the number of vertices. Then the Legitimate Deck Problem restricted to H is in LWPP.
As usual, for each graph G, δ(G) denotes the degree of a minimum-degree vertex of G.
Theorem 4.12. For each k ∈ N + , let Then the Legitimate Deck Problem restricted to H k is in LWPP, i.e., Legitimate Deck∩H k ∈ LWPP.
Proof. In the proof of their Theorem 6.1, Kratsch and Hemaspaandra [KH94] showed that for each class of graphs with bounded minimum degree, the number of nonisomorphic preimages is polynomially bounded. Now the theorem follows from Theorem 4.11.
It is interesting to note that for each of the H k classes GI H k ≡ p m GI trivially holds (for example, via adding to each of the two graphs being tested for isomorphism an isolated node), notwithstanding the fact that intersection with H k pulls the Legitimate Deck Problem's complexity into LWPP.
In two of this section's corollaries we used the fact that all LWPP sets are PP-low. We mention that since all LWPP set are also C = P-low [KST92,FFK94], the altered versions of Corollaries 4.4 and 4.10 in which the conclusion is changed from "unless NP is PP-low" to "unless NP is C = P-low" both hold, respectively due to Köbler, Schöning, and Torán [KST92] and the present paper.

Optimality of the Main Result
It is easy to see that the proof of Theorem 3.1 breaks down if Poly-LWPP is replaced by the analogous class where the size of the set of allowed gap values can be larger than polynomial in the input length. This does not necessarily imply that the corresponding theorem does not hold. However, in this section, we establish that relativizable proof techniques are not sufficient to improve Theorem 3.1 from Poly-LWPP to r-LWPP for any superpolynomial function r. That is, we show that our main result is optimal with respect to what can be proven by relativizable proof techniques.
But first, let us briefly consider the "more extreme" case of the class Exp-LWPP, where the number of allowed gap values can be an exponential function.

Exp-LWPP
We show that the whole class NP is contained in in Exp-LWPP. This gives strong evidence that Exp-LWPP ⊆ LWPP because LWPP is known to be low for PP [FFK94], but NP is widely believed not to be low for PP.
Since NP ⊆ coC = P and all LWPP sets are PP-low, we obtain the following corollaries. All of the above, except the PP-lowness claims, analogously hold for Exp-WPP/WPP, since r-LWPP is a subset of r-WPP. So for example we have the following.
Proof of Theorem 5.2. Let A ∈ coC = P. By Theorem 2.9, there exists a function g ∈ GapP such that for every x ∈ Σ * , x ∈ A =⇒ g(x) = 0, and Let p be a polynomial such that for each x ∈ Σ * , −2 p(|x|) ≤ g(x) ≤ 2 p(|x|) . Let f ∈ FP be the function such that, for every n ∈ N + and every i ∈ N + , We can now see that according to Definition 2.6, A ∈ 2 p(n)+1 -LWPP and hence A ∈ Exp-LWPP.
Theorem 5.6. Let r be any function from N to N such that for every c ∈ N, r / ∈ O(n c ). Then there exists an oracle O such that r- To prove Theorem 5.6, we will encode nondeterministic oracle Turing machines by low-degree multivariate polynomials. This technique is apparently folklore and has been used, for example, by de Graaf and Valiant [dGV02] to construct a relativized world where the quantum complexity class EQP is not contained in the modularity-based complexity class MOD p k P. The general technique of replacing oracle machines by simpler combinatorial objects such as circuits, decision trees, or polynomials and then using properties of such combinatorial objects to show the existence of a desired oracle dates to the seminal work of Furst, Saxe, and Sipser [FSS84], who made the connection between circuit lower bounds and the relativization of the polynomial hierarchy-a connection that has led to the resolution of many previously long-open relativized questions, such as the achievement of an oracle making the polynomial hierarchy infinite [Yao85,Hås87] and of oracles making the polynomial hierarchy extend exactly k levels [Ko89].
As indicated above, in Definition 5.7 and Proposition 5.8 below, we will encode nondeterministic oracle Turing machines by low-degree multivariate polynomials, in order to obtain polynomials that compute the gap of polynomial-time nondeterministic oracle machines.
A polynomial encoding of N (·) (x) is a multilinear polynomial p ∈ Z[y 1 , y 2 , . . . , y m ] defined as follows: Call a computation path ρ valid if ρ is a computation path of N D (x) for some oracle D ⊆ Σ * . Let x i 1 , x i 2 , . . . , x i ℓ be the distinct queries along a valid computation path ρ. Create a monomial mono(ρ) that is the product of terms Here, sign(ρ) = 1 if ρ is an accepting path and sign(ρ) = −1 if ρ is a rejecting path.
The next proposition states that the multilinear polynomial p has low total degree, and contains all the necessary information about N (·) (x) to yield the value gap N B (x) for every oracle B ⊆ Σ * .
Lemma 5.10 states a variant of the prime number theorem. We will need it in our oracle construction to get a lower bound for the number of primes in a given set of natural numbers (see set S defined below).
Proof of Theorem 5.6. First, we need a test language. For every set B ⊆ Σ * , define L B as where B =n denotes B ∩ Σ n . If B satisfies the condition that for every length n it holds that B =n ≤ r(n), then L B ∈ r-LWPP. (To see this, note that the function d(0 n ) = B =n is a #P B function, where #P is Valiant's [Val79] class of functions that count the number of accepting paths of nondeterministic polynomial-time Turing machines. But #P ⊆ GapP [FFK94], and that fact itself relativizes, so d(0 n ) ∈ GapP B . Since our test language should reject when d(0 n ) = 0 and should accept when 1 ≤ d(0 n ) ≤ r(n), and by our "if B satisfies" those are the only possibilities, we have L B ∈ r-LWPP B .) We will construct an oracle B such that for each n, B =n ≤ r(n) and L B / ∈ LWPP B . Let (N j , M j , p j ) j≥1 be an enumeration of all triples such that N j is a nondeterministic polynomial-time oracle Turing machine, M j is a deterministic polynomial-time oracle Turing machine computing a function, and p j is a monotonically increasing polynomial such that the running time of both N j and M j is bounded by p j regardless of the oracle. We construct the oracle B in stages. In stage j, we decide the membership in B of strings of length n j and extend the initial segment B j−1 of B to B j . Initially, we set B 0 = ∅.
Stage j, where j ≥ 1: Let n j be large enough that: (a) n j > p j−1 (n j−1 ) (to ensure that the previous stages are not affected), (b) r(n j ) ≥ p j (n j ) 4 , (c) (2 n j − p j (n j ))/2 ≥ p j (n j ) 4 , and (d) p j (n j ) 3 −p j (n j ) ≥ p j (n j ) 2 . Such an n j exists because p j is a monotonically increasing polynomial and r is a superpolynomial function, i.e., for each c ∈ N, r / ∈ O(n c ). We diagonalize against nondeterministic polynomial-time oracle Turing machine N j and deterministic polynomial-time oracle Turing machine M j . That is, we make sure that L B is not decided according to the definition of LWPP by GapP function g computed by N j together with FP function f computed by M j (see Definition 2.5). Let val be the value computed by M Note that T ≤ p j (n j ) since the computation time of M B j−1 j (0 n j ) is bounded by p j (n j ). In the following, we will never add any string from T to the oracle. This ensures that the value val computed by M B j−1 j (0 n j ) is never changed when we replace oracle B j−1 by B j . ( * ) We choose a set C ⊆ Σ n j − T such that This construction guarantees that for each n, B =n ≤ r(n) and L B / ∈ LWPP B . Thus our proof is complete if we can show that it is always possible to find a set C satisfying ( * ). We state and prove that as the following claim and its proof.
To obtain a lower bound for val, we determine a lower bound for the number of primes in S. First, note that at the beginning of stage j, we have taken n j large enough such that N/2 ≥