Kronecker scaling of tensors with applications to arithmetic circuits and algorithms

TL;DR

The paper connects Strassen-style tensor rank analysis to counting and circuit complexity by establishing a Kronecker-scaling property for balanced tripartitioning tensors $P_n$. This enables uniform arithmetic circuits for the permanent, hafnian, and related DP problems, contingent on low rank bounds for a constant-size tensor $P_d$, and ties these results to Strassen’s asymptotic rank conjecture via the exponent $oldsymbol{\sigma(P_bN)}$. By proving a decomposition of $P_n$ into restricted Kronecker powers through Steinitz balancing, and leveraging Yates-style evaluation, the authors derive exponential-time improvements over Ryser-like methods under the conjectured bounds. The framework further yields parameterized improvements for multilinear-detection tasks and for Hamiltonicity when parameterized by treewidth, through Kronecker-scaling analyses of the matchings connectivity tensor, illustrating a broad impact on arithmetic circuits and combinatorial algorithms.

Abstract

We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser's formula. We show that the same low-rank assumption implies exponential time improvements over the state of the art for a wide variety of other related counting and decision problems. As our main methodological contribution, we show that the tensors $P_n$ have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of $P_d$ for constant $d$. We prove this with a new technique relying on Steinitz's lemma, which we hence call Steinitz balancing. As a consequence of our methods, we show that the mentioned low rank assumption (and hence the improved algorithms) is implied by Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress.

Figures (2)

• Figure 1: Correctness of the hafnian computation: A canonical alternating cycle cover, partitioned in three balanced parts $P_1,P_2,$ and $P_3$, implicitly computed by the three subcircuits $C_1,C_2,$ and $C_3$ in Thm. \ref{['thm:faster-inclusion-exclusion']}, respectively. Every cycle cover represents a unique perfect matching in the underlying complete input graph. The cycles in the cycle cover alternates between actual edges representing entries in the input matrix $A$ (thin edges above the vertices) and auxiliary pairing edges (red edges below the vertices). In the canonical ordering, the cycles are ordered after their anchor (gray), their lowest ranked vertex.
• Figure 2: The graph $Z_{X}$ where $X=\{x_1,\ldots,x_8\}$ with $x_1 < x_2 < \ldots < x_8$.

Theorems & Definitions (71)

• Theorem 1.1: Main; Kronecker scaling for balanced tripartitioning tensors
• Theorem 1.2: Uniform circuits for balanced tripartitioning polynomials
• Theorem 1.3: Main application; Uniform arithmetic circuits for the permanent
• Theorem 1.4: Uniform arithmetic circuits for the hafnian
• Theorem 1.5: Algorithm for counting set partitions
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1: Homogenization (see, e.g., Bürgisser Burgisser2000)
• Lemma 2.2: Steinitz Steinitz1913; Grinberg and Sevast janov GrinbergS1980
• Lemma 3.1: Steinitz concentration