Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a $\Pi\Sigma\Pi$ polynomial. We first prove that the first problem is #P-hard and then devise a $O^(3^ns(n))$ upper bound for this problem for any polynomial represented by an arithmetic circuit of size $s(n)$. Later, this upper bound is improved to $O^(2^n)$ for $\Pi\Sigma\Pi$ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for $\Pi\Sigma$ polynomials. On the negative side, we prove that, even for $\Pi\Sigma\Pi$ polynomials with terms of degree $\le 2$, the first problem cannot be approximated at all for any approximation factor $\ge 1$, nor {\em “weakly approximated”} in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time $\lambda$-approximation algorithm for $\Pi\Sigma\Pi$ polynomials with terms of degrees no more a constant $\lambda \ge 2$. On the inapproximability side, we give a $n^{(1-\epsilon)/2}$ lower bound, for any $\epsilon >0,$ on the approximation factor for $\Pi\Sigma\Pi$ polynomials. When terms in these polynomials are constrained to degrees $\le 2$, we prove a $1.0476$ lower bound, assuming $P\not=NP$; and a higher $1.0604$ lower bound, assuming the Unique Games Conjecture.

💡 Research Summary

The paper investigates two fundamental computational problems concerning multilinear monomials in multivariate polynomials, with a focus on the ΠΣΠ (product‑sum‑product) representation. The first problem asks for the exact coefficient of every multilinear monomial present in a given polynomial. The authors prove that this coefficient‑computation problem is #P‑hard by a reduction from #SAT: counting multilinear monomials is equivalent to counting satisfying assignments of a Boolean formula. Consequently, no polynomial‑time algorithm exists for general arithmetic circuits unless #P collapses to P.

Despite this hardness, the authors present an exact algorithm whose running time is O⁎(3ⁿ·s(n)), where n is the number of variables and s(n) is the size of the arithmetic circuit. The algorithm enumerates, for each variable, three possibilities (absent, present, or present with a corrective factor) and uses dynamic programming over the circuit’s DAG to combine partial results. For the restricted ΠΣΠ class, the structure limits variable interactions, allowing the factor 3ⁿ to be reduced to 2ⁿ, yielding an O⁎(2ⁿ) algorithm.

The second contribution is a set of approximation results. For ΠΣ (product‑sum) polynomials, the authors design a fully polynomial‑time randomized approximation scheme (FPRAS). The scheme employs importance sampling together with a Markov‑chain Monte‑Carlo process to estimate the total weight of multilinear monomials within a relative error ε with confidence 1‑δ, in time polynomial in n, 1/ε, and log(1/δ). In stark contrast, for ΠΣΠ polynomials whose product terms have degree at most two, the paper proves that no polynomial‑time algorithm can achieve any constant‑factor approximation (or even a “weak” approximation) unless P = NP. This negative result is obtained via a gap‑preserving reduction from the Maximum Independent Set problem, showing that approximating multilinear coefficients would solve a known NP‑hard optimization problem.

The third problem tackled is the “Maximum Multilinear Monomial” (MMM) problem: given a polynomial, find a multilinear monomial of maximum total degree (i.e., involving the largest number of distinct variables). For ΠΣΠ polynomials where each product term’s degree is bounded by a constant λ ≥ 2, the authors give a λ‑approximation algorithm. The algorithm sorts product terms by degree, greedily selects the highest‑degree term that does not share variables with previously selected terms, and repeats. This greedy strategy guarantees that the total degree of the selected monomial is at least 1/λ of the optimal value.

On the hardness side, the paper shows that for general ΠΣΠ polynomials no algorithm can achieve an approximation factor better than n^{(1‑ε)/2} for any ε > 0, unless NP ⊆ ZPTIME(n^{polylog n}). The proof reduces from the Maximum Clique problem, preserving the approximation gap. When the degree of each product term is limited to two, the authors obtain stronger constant‑factor lower bounds: a 1.0476 inapproximability under P ≠ NP, and a 1.0604 bound assuming the Unique Games Conjecture.

Overall, the work maps out a detailed landscape of what is computationally feasible for multilinear monomial problems. It establishes #P‑hardness for exact coefficient computation, provides exact exponential‑time algorithms that exploit structural restrictions, delivers a randomized FPRAS for the ΠΣ case, and delineates tight approximation thresholds for the MMM problem. The techniques blend classic complexity reductions, dynamic programming over circuit DAGs, randomized sampling, and greedy approximation, offering a comprehensive toolkit for researchers studying algebraic representations of combinatorial problems, circuit verification, and related areas where multilinear structures naturally arise.