A Space-Efficient Algebraic Approach to Robotic Motion Planning

TL;DR

This paper addresses exponential memory usage in exact robotic motion planning by recasting Graph Inspection as Multilinear Detection on compact arithmetic circuits. The authors introduce tree certificates to repair monomial-detection flaws and enable efficient solution recovery, yielding the ALG-IPA pipeline that operates in randomized time ${\\tilde{O}}(2^t (\\ell t^3 n^2 + t^3 |\\mathcal{C}| n))$ with space ${\\tilde{O}}(\\ell t n^2 + t|\\mathcal{C}| n)$. They present four circuit constructions and three search strategies, plus a two-phase recovery method to extract walks from recovered certificates. Empirical results show substantial memory savings over DP-based approaches, confirming the practicality of algebraic, memory-efficient methods while outlining areas for engineering optimizations and potential GPU acceleration.

Abstract

We consider efficient route planning for robots in applications such as infrastructure inspection and automated surgical imaging. These tasks can be modeled via the combinatorial problem Graph Inspection. The best known algorithms for this problem are limited in practice by exponential space complexity. In this paper, we develop a memory-efficient approach using algebraic tools related to monomial testing on the polynomials associated with certain arithmetic circuits. Our contributions are two-fold. We first repair a minor flaw in existing work on monomial detection using a new approach we call tree certificates. We further show that, in addition to detection, these tools allow us to efficiently recover monomials of interest from circuits, opening the door for significantly broadened application of related algebraic tools. For Graph Inspection, we design and evaluate a complete algebraic pipeline. Our engineered implementation demonstrates that circuit-based algorithms are indeed memory-efficient in practice, thus encouraging further engineering efforts.

Figures (19)

• Figure 1: An arithmetic circuit (left) with 2 variables, 3 addition gates, and 2 multiplication gates, and its associated polynomial $P_\mathcal{F}(X)$ and fingerprint polynomial $P_\mathcal{F}(X,A)$ (right). This is a counterexample of a claim in koutis2016limitsapplications.
• Figure 2: Tree (red) and non-tree (blue) certificates for the same circuit, drawn with polynomials.
• Figure 3: An example of an arithmetic circuit (CompactCircuit with UnifiedSearch), encoding the graph instance $G$ with colors $\mathcal{C}=\{c_1,c_2,c_3\}$, illustrated on the left, with $t=3$. The circuit consists of variable nodes (gray) for each color in $\mathcal{C}$, auxiliary nodes (green), receivers (purple), transmitters (blue), and output nodes (red). Notice that each receiver/transmitter pair is identifiable by a layer, index ($V \setminus \{s\}$ for CompactCircuit), and walk weight from the starting vertex $s$. A tree certificate, corresponding to walk $(s,u,w,s)$ with weight $7$, is highlighted in bold.
• Figure 4: UnifiedSearch
• Figure 5: Edge count (left) and search time (right) for each circuit type, tested on CRISP and DRONE instances with different $k$ values and scaling factors $\lambda$. Figures follow the standard box plot convention. SemiCompactCircuit performs well on both instances (discussions in \ref{['sec:circuit-type-experiment']}).

Theorems & Definitions (45)

• Lemma 3.1: $\bigstar$
• Lemma 3.2: koutis2016limitsapplicationswilliams2009findingpaths
• Lemma 3.3: $\bigstar$
• Lemma 3.4: $\bigstar$
• Lemma 3.5
• Lemma 4.1: $\bigstar$
• Lemma 4.2: $\bigstar$
• Lemma 4.3: $\bigstar$
• Lemma 6.1: $\bigstar$
• Lemma 6.2: $\bigstar$