Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

The algebraic analogue of the $\mathsf{P}$ versus $\mathsf{NP}$ problem, famously referred to as the $\mathsf{VP}$ versus $\mathsf{VNP}$ question, is one of the most significant problem in algebraic complexity theory. Valiant [28] showed that the Permanent polynomial is $\mathsf{VNP}$-complete (over fields of char $\neq$ 2). A striking aspect of this polynomial is that the underlying decision problem, in fact even the search problem, is in $\mathsf{P}$. Given a graph, we can decide in polynomial time whether it has a perfect matching, and if so find a maximum matching in polynomial time [12]. Since the underlying problem is an easier problem, it helped in establishing $\mathsf{VNP}$-completeness of a host of other polynomials by a reduction from the Permanent polynomial (cf. [4]). Inspired from classical results in structural complexity theory, in particular [20], Bürgisser [5] proved that if Valiant’s hypothesis (i.e. $\mathsf{VP}\neq\mathsf{VNP}$) is true, then, over any field there is a $p$-family in $\mathsf{VNP}$ which is neither in $\mathsf{VP}$ nor $\mathsf{VNP}$-complete with respect to $c$-reductions. Let us call such polynomial families $\mathsf{VNP}$-intermediate (i.e. in $\mathsf{VNP}$, not $\mathsf{VNP}$-complete, not in $\mathsf{VP}$). Further, Bürgisser [5] showed that over finite fields, a specific family of polynomials is $\mathsf{VNP}$-intermediate, provided the polynomial hierarchy $\mathsf{PH}$ does not collapse to the second level. On an intuitive level these polynomials enumerate cuts in a graph. This is a remarkable result, when compared with the classical $\mathsf{P}$-$\mathsf{NP}$ setting or the BSS-model. Though the existence of problems with intermediate complexity has been established in the latter settings, due to the involved “diagonalization” arguments used to construct them, these problems seem highly unnatural. That is, their definitions are not motivated by an underlying combinatorial problem but guided by the needs of the proof and, hence, seem artificial. The question of whether there are other naturally-defined $\mathsf{VNP}$-intermediate polynomials was left open by Bürgisser [4]. We remark that to date the cut enumerator polynomial from [5] is the only known example of a natural polynomial family that is $\mathsf{VNP}$-intermediate.

The question of whether the classes $\mathsf{VP}$ and $\mathsf{VNP}$ are distinct is often phrased as whether $\mathsf{Perm}_{n}$ is not a quasi-polynomial-size projection of $\mathsf{Det}_{n}$. The importance of this reformulation stems from the fact that it is a purely algebraic statement, devoid of any dependence on circuits. While we have made very little progress on this question of determinantal complexity of the permanent, the progress in restricted settings has been considerable. One of the success stories in theoretical computer science is unconditional lower bound against monotone computations [24, 25, 1]. In particular, Razborov [25] proved that computing the permanent over the Boolean semiring requires monotone circuits of size at least $n^{\Omega(\log n)}$. Jukna [18] observed that if the Hamilton cycle polynomial is a monotone $p$-projection of the permanent, then, since the clique polynomial is a monotone projection of the Hamiltonian cycle [28] and the clique requires monotone circuits of exponential size [1], one would get a lower bound of $2^{n^{\Omega(1)}}$ for monotone circuits computing the permanent, thus improving on [25]. The importance of this observation is also highlighted by the fact that such a monotone $p$-projection, over the reals, would give an alternate proof of the result of Jerrum and Snir [17] that computing the permanent by monotone circuits over $\mathbb{R}$ requires size at least $2^{n^{\Omega(1)}}$. (Jerrum and Snir [17] proved that the permanent requires monotone circuits of size $2^{\Omega(n)}$ over $\mathbb{R}$ and the tropical semiring.) The first progress on this question raised in [18] was made recently by Grochow [15]. He showed that the Hamiltonian cycle polynomial is not a monotone sub-exponential-size projection of the permanent. This already answered Jukna’s question in its entirety, but Grochow [15] used his techniques to further establish that polynomials like the perfect matching polynomial, and even the $\mathsf{VNP}$-intermediate cut enumerator polynomial of Bürgisser [5], are not monotone polynomial-size projections of the permanent. This raises an intriguing question of whether there are other such non-negative polynomials which share this property.

While the $\mathsf{Perm}$ vs $\mathsf{Det}$ problem has become synonymous with the $\mathsf{VP}$ vs $\mathsf{VNP}$ question, there is a somewhat unsatisfactory feeling about it. This rises from two facts: one, that the $\mathsf{VP}$-hardness of the determinant is known only under the more powerful quasi-polynomial-size projections, and, second, the lack of natural $\mathsf{VP}$-complete polynomials (with respect to polynomial-size projections) in the literature. (In fact, with respect to $p$-projections, the determinant is complete for the possibly smaller class $\mathsf{VBP}$ of polynomial-sized algebraic branching programs.) To remedy this situation, it seems crucial to understand the computation in $\mathsf{VP}$. Bürgisser [4] showed that a generic polynomial family constructed using a topological sort of a generic $\mathsf{VP}$ circuit, while controlling the degree, is complete for $\mathsf{VP}$. Raz [23], using the depth reduction of [29], showed that a family of “universal circuits” is $\mathsf{VP}$-complete. Thus both families directly depend on the circuit definition or characterization of $\mathsf{VP}$. Last year, Durand et al. [11] made significant progress and provided a natural, first of its kind, $\mathsf{VP}$-complete polynomial. However, the natural polynomials studied by Durand et al. lacked a bit of punch because their completeness was established under polynomial-size constant depth c-reductions rather than projections.

In this paper, we make progress on all three fronts. First, we provide a list of new natural polynomial families, based on basic (combinatorial) $\mathsf{NP}$-complete problems [14] whose completeness is via parsimonious reductions [27], that are $\mathsf{VNP}$-intermediate over finite fields (Theorem 3.1). Then, we show that over reals, some of our intermediate polynomials are not monotone affine polynomial-size projections of the permanent (Theorem 4.1). As in [15], the lower bound results about monotone affine projections are unconditional. Finally, we improve upon [11] by characterizing $\mathsf{VP}$ and establishing a natural $\mathsf{VP}$-complete polynomial under polynomial-size projections (Theorem 5.3). A modification yields a family similarly complete for $\mathsf{VBP}$ (Theorems 5.4, 5.5).

In [5], Bürgisser showed that unless PH collapses to the second level, an explicit family of polynomials, called the cut enumerator polynomial, is $\mathsf{VNP}$-intermediate. He raised the question, recently highlighted again in [15], of whether there are other such natural $\mathsf{VNP}$-intermediate polynomials. In this section we show that in fact his proof strategy itself can be adapted to other polynomial families as well. The strategy can be described abstractly as follows: Find an explicit polynomial family $h=(h_{n})$ satisfying the following properties.

M: Membership.

The family is in $\mathsf{VNP}$.

E: Ease.

Over a field $\mathbb{F}_{q}$ of size $q$ and characteristic $p$, $h$ can be evaluated in $\mathsf{P}$. Thus if $h$ is $\mathsf{VNP}$-hard, then we can efficiently compute #$\mathsf{P}$-hard functions, modulo $p$.

H: Hardness.

The monomials of $h$ encode solutions to a problem that is #$\mathsf{P}$-hard via parsimonious reductions. Thus if $h$ is in $\mathsf{VP}$, then the number of solutions, modulo $p$, can be extracted using coefficient computation.

Then, unless $\mathsf{Mod}_{p}\mathsf{P}\subseteq\mathsf{P/poly}$ (which in turn implies that PH collapses to the second level, [19]), $h$ is $\mathsf{VNP}$-intermediate.

We provide a list of $p$-families that, under the same condition $\mathsf{Mod}_{p}\mathsf{P}\not\subseteq\mathsf{P/poly}$, are $\mathsf{VNP}$-intermediate. All these polynomials are based on basic combinatorial $\mathsf{NP}$-complete problems that are complete under parsimonious reduction.

For the next three polynomials, we consider the complete graph $G_{n}$ on $n$ nodes, and we have the set of variables $\tilde{X}=\{X_{e}\}_{e\in E_{n}}$ and $\tilde{Y}=\{Y_{v}\}_{v\in V_{n}}$.

We show that if $\mathsf{Mod}_{p}\mathsf{P}\not\subseteq\mathsf{P/poly}$, then all five polynomials defined above are $\mathsf{VNP}$-intermediate.

We now show that some of our intermediate polynomials are not monotone $p$-projections of the Permanent polynomial. The results here are motivated by the recent results of Grochow [15]. Recall that a polynomial $f(x_{1},\ldots,x_{n})$ is a projection of a polynomial $g(y_{1},\ldots,y_{m})$ if $f(x_{1},\ldots,x_{n})=g(a_{1},\ldots,a_{m})$, where $a_{i}$’s are either constants or $x_{j}$ for some $j$. The polynomial $f$ is an affine projection of $g$ if $f$ can be obtained from $g$ by replacing each $y_{i}$ with an affine linear function $\ell_{i}(\tilde{x})$. Over any subring of $\mathbb{R}$, or more generally any totally ordered semi-ring, a monotone projection is a projection in which all constants appearing in the projection are non-negative. We say that the family $(f_{n})$ is a (monotone affine) projection of the family $(g_{n})$ with blow-up $t(n)$ if for all sufficiently large $n$, $f_{n}$ is a (monotone affine) projection of $g_{t(n)}$.

Before giving the proof, we set up some notation. For more details, see [2, 26, 15]. For any polynomial $p$ in $n$ variables, let $\mathsf{Newt}(p)$ denote the polytope in $\mathbb{R}^{n}$ that is the convex hull of the vectors of exponents of monomials of $p$. For any Boolean formula $\phi$ on $n$ variables, let p-SAT($\phi$) denote the polytope in $\mathbb{R}^{n}$ that is the convex hull of all satisfying assignments of $\phi$. Let $K_{n}=(V_{n},E_{n})$ denote the $n$-vertex complete graph. The travelling salesperson (TSP) polytope is defined as the convex hull of the characteristic vectors of all subsets of $E_{n}$ that define a Hamiltonian cycle in $K_{n}$.

For a polytope $P$, let $\mathsf{c}(P)$ denote the minimal number of linear inequalities needed to define $P$. A polytope $Q\subseteq\mathbb{R}^{m}$ is an extension of $P\subseteq\mathbb{R}^{n}$ if there is an affine linear map $\pi\colon\mathbb{R}^{m}\to\mathbb{R}^{n}$ such that $\pi(Q)=P$. The extension complexity of $P$, denoted $\mathsf{xc}(P)$, is the minimum size $c(Q)$ of any extension $Q$ (of any dimension) of $P$.

The following are straightforward, see for instance [15, 13].

We use the following recent results.

The quest for a natural $\mathsf{VP}$-complete polynomial has generated a significant amount of research [4, 23, 22, 7, 11]. The first success story came from [11], where some naturally defined homomorphism polynomials were studied, and a host of them were shown to be complete for the class $\mathsf{VP}$. But the results came with minor caveats. When the completeness was established under projections, there were non-trivial restrictions on the set of homomorphisms $\mathcal{H},$ and sometimes even on the target graph $H$. On the other hand, when all homomorphisms were allowed, completeness could only be shown under seemingly more powerful reductions, namely, constant-depth $c$-reductions. Furthermore, the graphs were either directed or had weights on nodes. It is worth noting that the reductions in [11] actually do not use the full power of generic constant-depth $c$-reductions; a closer analysis reveals that they are in fact linear p-projection. That is, the reductions are linear combinations of polynomially many $p$-projections (see Chapter 3, [4]). Still, this falls short of $p$-projections.

In this work, we remove all such restrictions and show that there is a simple explicit homomorphism polynomial family that is complete for $\mathsf{VP}$ under $p$-projections. In this family, the source graphs $G$ are specific bounded-tree-width graphs, and the target graphs $H$ are complete graphs. We also show that a similar family with bounded-path-width source graphs is complete for $\mathsf{VBP}$ under $p$-projections. Thus, homomorphism polynomials are rich enough to characterise computations by circuits as well as algebraic branching programs.

The polynomials we consider are defined formally as follows.

Note that for every $G,H,\mathcal{H}$, $f_{G,H,\mathcal{H}}(\bar{Y})$ equals $\hat{f}_{G,H,\mathcal{H}}(\bar{Y})\mid_{\bar{Z}=\bar{1}}$. Thus upper bounds for $\hat{f}$ give upper bounds for $f$, while lower bounds for $f$ give lower bounds for $\hat{f}$.

We show in Theorem 5.2 that for any $p$-family $(H_{m})$, and any bounded tree-width (path-width, respectively) $p$-family $(G_{m})$, the polynomial family $(f_{m})$ where $f_{m}=\hat{f}_{G_{m},H_{m}}$ is in $\mathsf{VP}$ ($\mathsf{VBP}$, respectively). We then show in Theorem 5.3 that for a specific bounded tree-width family $(G_{m})$, and for $H_{m}=K_{m^{6}}$, the polynomial family $(f_{G_{m},H_{m}})$ is hard, and hence complete, for $\mathsf{VP}$ with respect to projections. An analogous statement is shown in Theorem 5.4 for a specific bounded path-width family $(G_{m})$ and for $H_{m}=K_{m^{2}}$. Over fields of characteristic other than 2, $\mathsf{VBP}$-hardness is obtained for a simpler family of source graphs $G_{m}$, as described in Theorem 5.5.