Probabilistic Parameterized Polynomial Time^††thanks: Research for this paper has been funded through NWO EW TOP grant 612.001.601.

The simple yet powerful idea which lies at the heart of the theory of parameterized complexity is that the hardness of computational problems may be better studied by analyzing the effects of particular aspects of its instances, treating these as a distinguished problem parameter and allowing the time (or other measures such as space) required to find a solution to depend on this parameter by an unbounded factor. This leads to an account of fixed-parameter tractability (FPT) and a hardness theory based on classes belonging to the W-hierarchy which together mirror the parts played by P and NP in classical complexity theory.

In the two decades since the appearance of [6], the book by Downey and Fellows which largely formed the foundations of the field, research in parameterized complexity theory has gone far beyond this initial outlook and revealed a rich structure and many interesting questions to pursue, a lot of which covered in the updated [7]. Yet with a few notable exceptions, little attention has been paid to probabilistic computation in the parameterized setting. The most encompassing effort thus far has been made by Montaya and Müller in [18], where they show amongst other things that the natural analogue BPFPT relates to other complexity classes in much the same way as does BPP in the classical setting.¹¹1We also mention [3] which studies PFPT, the parameterized counterpart to PP.

Our aim with the present paper is to improve on this situation by demonstrating that studying parameterized probabilistic computation amounts to more than simply reconstructing results from the classical setting (which can already be a non-trivial task, as evidenced by the work done in [18]), and that results obtained in this way can have broader theoretical and practical significance. In particular we study a complexity class intended to capture probabilistic parameterized polynomial time computability, which we shall refer to as PPPT for this reason.²²2In [15, 16] this class was proposed under the name FERT for fixed-error randomized tractability, intended to be reminiscent of FPT. We believe that the name PPPT is more appropriate as it calls into mind the class PP as well as ppt-reductions.

This class PPPT is informally defined by considering probabilistic algorithms for parameterized problems, except allowing not the runtime but instead only the error bound to depend on the parameter by more than a polynomial factor. As such, PPPT can be thought of as containing those problems in PP which are nevertheless close to being in BPP and hence randomized tractable in a certain sense. This perspective, which we will explore more rigorously later on, formed much of the motivation of the class’s original proposal in [15].

In what follows, we assume the reader to be familiar with the basics of classical and parameterized complexity theory. However, we repeat the definitions of the complexity classes used here, mostly to facilitate the comparison with the class PPPT which we formally define in the next section. First recall the probabilistic complexity classes BPP and PP:

We mostly follow the original definition presented in [10] in that we consider a probabilistic Turing machine $\mathcal{M}$ to be a Turing machine with access to random bits which it may query at every step of its execution, and whose transition function may depend on the values read off in this way. However, we include the generalization that a probabilistic Turing machine $\mathcal{M}$ may query not one but $r_{\mathcal{M}}$ many random bits at each step, where as usual we drop the subscript when it can be inferred from the context. Before continuing we note that $r_{\mathcal{M}}\leq\log|\mathcal{M}|$.

The method of converting classical complexity classes C to parameterized classes paraC illustrated above originates from [9], and indeed $\mathsf{FPT}=\mathsf{paraP}$. One should keep in mind though that this construction does not yield the usual parameterized classes including BPFPT, as these are furthermore characterized by using at most $f(k)\log|x|$ random bits. (cf. [4]). In fact, as observed in [18], $\mathsf{BPFPT}=\mathsf{paraBPP}$ if and only if $\mathsf{P}=\mathsf{BPP}$. As we shall see in the next section, a similar statement remains true when we replace paraBPP by the class PPPT.

We now provide a formal definition of the class PPPT:

Based on the convention that the parameter value $k$ is given as a unary string along with the rest of the input $x$, from this definition it is immediate that PPPT is a subclass of PP. Moreover, it can be shown that this definition is robust in two ways, which makes it easy to see that PPPT is also a subclass of paraBPP.

One can extend the probabilistic amplification used in the proof of Proposition 1(i) to (instead) remove the term $f(k)$ from the error bound, at the cost of introducing this factor into the runtime. This is generally not allowed for PPPT, but permissible for paraBPP, hence PPPT is a subclass of paraBPP. In fact, this inclusion is easily shown to be strict.

Note that what the proof above actually shows is that there are problems in FPT which are not in PPPT. This tells us that the inclusion in the statement remains strict even if $\mathsf{BPP}=\mathsf{P}$, in which case we have that $\mathsf{paraBPP}=\mathsf{FPT}$: see Proposition 5.1 of [18]. Similar arguments can be used to show that $\mathsf{PPPT}\subseteq\mathsf{FPT}$ implies $\mathsf{BPP}=\mathsf{P}$. Most importantly however, we can extend these results by providing an exact characterization of PPPT, the proof of which relies on essentially the same strategy used to show that every problem in FPT is kernelizable.

For context it may be valuable to note that the idea of a complexity class being the intersection of a parameterized and a classical one has been considered before: while [2] looked into $\mathsf{W[P]}\cap\mathsf{NP}$, [18] briefly discussed $\mathsf{BPFPT}\cap\mathsf{BPP}$. However, we believe it is important to note first of all that the class PPPT arose from a reasonably natural definition intended to capture a slightly weaker kind of parameterization, instead of its definition being explicitly constructed to ensure a correspondence to the intersection of paraBPP and PP. Furthermore, and perhaps most importantly, we can actually exhibit natural problems which are complete for (the promise version of) the class PPPT, something which has not yet been done for BPP. In the remainder of this section we shall describe these problems, which originate from the domain of approximate Bayesian inference. We subsequently prove completeness for these problems in Section 3, which sets us up to explore the main implications of our results in Section 4.

Below we provide a definition of a Bayesian network, so that we may introduce the problem of inference within such networks; for the reader interested in a more detailed treatment, we refer to a standard textbook such as [14].

As reflected by the notation, one typically blurs the distinction between the node $V$ and the statistical variable which it represents, so that one may use $\mathrm{\Omega}(V)$ to refer directly to the set of possible outcomes of this variable.

One of the main computational problems associated with Bayesian networks is that of inference, which is to determine what the likelihood is of some given combination of outcomes, possibly conditioned on certain specified outcomes for another set of variables. Below we describe the corresponding decision problem.

Bayesian Inference
Input: A Bayesian network $\mathcal{B}=(G,\text{Pr})$, two sets of variables $H,E\subseteq V(G)$, joint value assignments $h\in\mathrm{\Omega}(H)$ and $e\in\mathrm{\Omega}(E)$, a rational threshold value $0\leq q\leq 1$.
Question: Is $\text{Pr}(h\mid e)>q$?

Based on whether $E=\emptyset$, we shall refer to this problem as Inference or Conditional Inference respectively, both of which are PP-complete (see also [19]). Despite their equivalence from a classical perspective, in terms of parameterized approximability the latter is more difficult: [16] has an overview of the main results known thus far. We consider the following specific parameterizations:

$\epsilon$-Inference
Input: A Bayesian network $\mathcal{B}=(G,\text{Pr})$, a set of variables $H\subseteq V(G)$, a joint value assignment $h\in\mathrm{\Omega}(H)$, and rational values $0\leq q\leq 1$ and $0<\epsilon\leq\frac{1}{2}$.
Parameter: $\lceil-\log\epsilon\rceil$.
Promise: $\text{Pr}(h)\not\in(q-\epsilon,q+\epsilon)$.
Question: Is $\text{Pr}(h)>q$?

$\{\epsilon,\text{Pr}(h),\text{Pr}(e)\}$-Conditional Inference
Input: A Bayesian network $\mathcal{B}=(G,\text{Pr})$, two sets of variables $H,E\subseteq V(G)$, joint value assignments $h\in\mathrm{\Omega}(H)$, $e\in\mathrm{\Omega}(E)$, rational values $0\leq q\leq 1$, $\epsilon>0$.
Parameters: $\epsilon$, $\text{Pr}(h)$, and $\text{Pr}(e)$.
Promise: $\text{Pr}(h)\not\in(q(1+\epsilon)^{-1},q(1+\epsilon))$.
Question: Is $\text{Pr}(h\mid e)>q$?

In what follows we use $\text{Pr}(h)\pm\epsilon\geq q$ as a shorthand for $\text{Pr}(h)>q$ with the promise that $\text{Pr}(h)\not\in(q-\epsilon,q+\epsilon)$, and similarly for the relative approximation; for more on this approach, see [17]. For completeness’ sake, we remind the reader of the definition of a promise problem, based on [11].

We can separate promise problems into complexity classes just as we do for decision problems by converting the familiar definitions in the following way.

We now show for both of the approximate inference problems given above that they are in the promise version of PPPT, the former by an explicit argument, the latter by an appeal to Theorem 2.1.

At this point we wish to show that the two parameterized problems considered in the previous section are actually complete for the class pPPPT. In order to do this, we first determine which notion of reduction is the most suitable with respect to the class PPPT, after which we identify the machine acceptance problem for pPPPT and demonstrate its completeness for the class under these reductions. We then construct an explicit reduction from this problem to $\epsilon$-Inference, and in turn reduce the latter to $\{\epsilon,\text{Pr}(h),\text{Pr}(e)\}$-Conditional Inference, thereby establishing completeness for both of these problems.

First of all, it is evident that while some form of parameterized reduction is required, the usual fpt-reductions are unsuitable because they allow the runtime to contain a factor superpolynomial in the parameter value and so PPPT is not closed under these. Furthermore, the reductions cannot be probabilistic either: while this is possible for BPP since the error can be reduced to constant using probabilistic amplification, mitigating the parameterized error bound is generally impossible without parameterized runtime (unless $\mathsf{P}=\mathsf{PP}$). Thus in this context it makes sense to consider the notion of a ppt-reduction⁴⁴4As [7] observes, the acronym ‘ppt’ can be read equally well as either “polynomial parameter transformation” or “parameterized polynomial transformation”., which was formally introduced in [1]:

However, we can remove the constraint that $k^{\prime}$ is bounded by a polynomial in $k$, and instead simply demand that its value depends only on $k$.⁵⁵5I hereby express my gratitude to the anonymous reviewer who raised this point. The resulting class of reductions is a slightly broader one for which we introduce the name pppt-reduction, for which it is easy to see that PPPT is closed under pppt-reductions. Thus we would like to exhibit a parameterized problem which is complete for PPPT under pppt-reductions; yet here we run into the same issue as with the class BPP, namely that it may be impossible to effectively decide whether a given probabilistic Turing machine has a suitably lower-bounded probability to be correct on all possible inputs. Hence we have to add this explicit requirement in the form of a promise, which means we arrive at the machine acceptance problem stated below, and instead study completeness for the promise class pPPPT.

Error PTM Acceptance
Input: Two strings $x$ and $1^{n}$ and a probabilistic Turing machine $\mathcal{M}$.
Parameter: A positive integer $k$.
Promise: For any valid input to $\mathcal{M}$ and after any number of steps, the probability of acceptance does not lie between $\frac{1}{2}-2^{-k}$ and $\frac{1}{2}+2^{-k}$.
Question: After $n$ steps, does $\mathcal{M}$ accept $x$ more often than it rejects?

We can now show the problem $\epsilon$-Inference to be complete for pPPPT as well, by providing what is essentially a Cook-style construction of a Bayesian network from the probabilistic Turing machine specification and number of steps and the input which together make up an instance of Error PTM Acceptance. In contrast to the previous result, the reduction resulting from this construction is a ppt-reduction rather than a pppt-reduction.

In turn, we can reduce $\epsilon$-Inference to $\{\epsilon,\text{Pr}(h),\text{Pr}(e)\}$-Conditional Inference, thereby extending the hardness and hence completeness to the latter.

To conclude this section, we discuss a question which may have occurred to the reader, namely whether one could simplify this approach by avoiding the inference problems altogether and working instead with the following variant of MajSat, which is the satisfiability problem complete for PP.

Gap-MajSat
Input: A propositional formula $\varphi$.
Parameter: A positive integer $k$.
Promise: The ratio of satisfying truth assignments of $\varphi$ does not lie between $\frac{1}{2}-2^{-k}$ and $\frac{1}{2}+2^{-k}$.
Question: Is $\varphi$ satisfied by more than half of its possible truth assignments?

The issue here is that the canonical reduction from Error PTM Acceptance to Gap-MajSat requires a number of variables proportional to both $n$ and the size of the machine $\mathcal{M}$, hence the original margin of $\pm 2^{-k}$ will shrink by a factor in the input size. The resulting parameter for the Gap-MajSat instance will thus depend on the input size, which means this reduction is not even an fpt-reduction. This points to a phenomenon also observed in the W-hierarchy, where W[SAT] (which is defined in terms of a parameterized satisfiability problem) is believed to a proper subclass of W[P] (which is defined in terms of a parameterized circuit satisfiability or machine acceptance problem). That the reduction does work for the inference problems suggests that Bayesian networks do have the direct expressive power of Turing machines lacked by propositional formulas.

Ultimately, one of the main open questions in the area of probabilistic computation is whether $\mathsf{P}=\mathsf{BPP}$. In contrast to the more famous open question whether $\mathsf{P}=\mathsf{NP}$, the generally accepted view is that BPP is likely to equal P, based on works such as [13]. However, due to the lack of natural problems which are known to be BPP-complete, it has not been possible to focus efforts on proving a particular problem to lie in P in order to demonstrate the collapse of BPP. We believe that our work makes an important contribution in that it indirectly provides a problem which can play this part, namely $\epsilon$-Inference. This relies in part on the following proposition adapted from [18] which we hinted at earlier.

Combined with Theorem 3.1, we arrive at the following result:

At the same time, we can provide some indication as to the hardness of $\epsilon$-Inference by means of the framework of kernelization lower bounds, which is where the notion of a ppt-reduction originated. Here we consider the following reformulation, inspired by [5], of a theorem by Drucker found in [8].

We can use this Theorem to prove that $\epsilon$-Inference has no randomized polynomial-sized kernels unless the polynomial hierarchy collapses.

While perhaps unsurprising, this result serves in particular as a reminder that hard problems in PPPT such as $\epsilon$-Inference are not solvable by polynomial kernelization followed by a probabilistic (PP) algorithm.

In this paper we have explored the proposal made in [15, 16] of an alternative parameterized randomized complexity class, which we have called PPPT and of which we have shown that it is identical to the intersection of PP and paraBPP. In the preceding sections we showed that the problem $\epsilon$-Inference is a natural fit for this class, as it is not only a member of the class in a straightforward way (Proposition 3), it is moreover complete for the corresponding problem class (Theorem 3.1). Because of the close relation between classical and parameterized probabilistic computation (Proposition 6), the class PPPT turns out to have unexpected broader relevance, as finding an efficient deterministic absolute approximation algorithm for Inference is necessary and sufficient for the derandomization of BPP to P (Theorem 4.1).

In other words, we are in the fortunate circumstances where efforts to address a long-standing open question originating in theory can actually coincide with the search for a novel algorithm capable of solving a practical problem, and most importantly the existence of such an algorithm actually follows from a conjecture supported by other considerations. With this paper we wish to call attention to this opportunity for researchers with theoretical and practical motivations alike to engage with a challenge which is broadly relevant to multiple research communities at once. It is our hope that such focused efforts on the $\epsilon$-Inference problem may lead to a valuable breakthrough in both the fields of structural complexity theory and of probabilistic graphical models.