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Representing polynomial of ST-CONNECTIVITY

Jānis Iraids, Juris Smotrovs

TL;DR

This work establishes a deep link between representing polynomials of monotone Boolean functions and Möbius functions on atomistic lattices derived from prime implicants. By tying AQ-Connectivity to flow-polytopes, it derives an explicit polynomial $p_G(x)=\sum_{H\in U(P_G)\setminus\{\varnothing\}}(-1)^{D(H)}\prod_{i\in H} x_i$ with $D(H)$ computable from the subquiver $H$, and shows that only unions of paths contribute nonzero coefficients. The results yield exponential growth in the number of nonzero monomials for grid connectivity, giving both lower and upper bounds on $|U(P_{G_n})|$ and, consequently, implications for quantum-query complexity lower bounds via polynomial degree. The paper also clarifies the dual problem and highlights future directions in identifying other monotone functions with simple Möbius structures and their impact on complexity bounds. Overall, it provides a structural, lattice-theoretic framework for representing and analyzing connectivity-related Boolean functions through polynomials and polytopes.

Abstract

We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the Möbius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional $n \times n$ grid connectivity problem is $2^{Ω(n^2)}$.

Representing polynomial of ST-CONNECTIVITY

TL;DR

This work establishes a deep link between representing polynomials of monotone Boolean functions and Möbius functions on atomistic lattices derived from prime implicants. By tying AQ-Connectivity to flow-polytopes, it derives an explicit polynomial with computable from the subquiver , and shows that only unions of paths contribute nonzero coefficients. The results yield exponential growth in the number of nonzero monomials for grid connectivity, giving both lower and upper bounds on and, consequently, implications for quantum-query complexity lower bounds via polynomial degree. The paper also clarifies the dual problem and highlights future directions in identifying other monotone functions with simple Möbius structures and their impact on complexity bounds. Overall, it provides a structural, lattice-theoretic framework for representing and analyzing connectivity-related Boolean functions through polynomials and polytopes.

Abstract

We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the Möbius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are where is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional grid connectivity problem is .

Paper Structure

This paper contains 6 sections, 12 theorems, 15 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Posets, their Möbius functions, and convex polytopes
  3. Representing monotone functions
  4. Möbius function of AQ-Connectivity
  5. Size of U(P_G) for 2D grids
  6. Conclusion

Key Result

Lemma 1

A non-empty $F\subseteq \reals^n$ is a face of $H_1^+ \cap H_2^+ \cap \cdots \cap H_m^+$ if and only if $F=\bigcap_{i\in M}{H_i}\cap \bigcap_{i\notin M}{H_i^+}$ for some $M\subseteq [m]$.

Figures (6)

  • Figure 1: Atomistic lattice with atoms $a_1$, $a_2$, $a_3$ and equalities $a_1a_2=a_1a_3=a_1a_2a_3$
  • Figure 2: Initial state of the matrix $M$. In addition to row indices the corresponding unions of prime implicants are specified. The rows $b_1b_2$, $b_1b_3$ and $b_1b_2b_3$ must be equal, therefore columns $\{2\}$ and $\{3\}$ making them different must be deleted.
  • Figure 3: State of the matrix $M$ after the deletion of columns $\{2\}$ and $\{3\}$. Now the lattice generated by prime implicants $b_1,b_2,b_3$ is isomorphic to the lattice of Figure \ref{['fig:atomistic']}.
  • Figure 4: The matrix $M$ after removing columns which are unions of other columns: $\{1, 2, 3\}=\{1\}\cup\{2,3\}$ and $\varnothing$ (the empty union). Lattice generated by $\{b_1,b_2,b_3\}$ does not change. A monotone Boolean function with lattice isomorphic to that of Figure \ref{['fig:atomistic']} must have at least $4$ input bits, at least one per each of the remaining columns.
  • Figure 5: Ear decomposition
  • ...and 1 more figures

Theorems & Definitions (22)

  • Lemma 1: Schrijver1986
  • Theorem 1: Euler's relation, Gruenbaum2003Broendsted1983
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 12 more