Multi-robot task allocation is a practical way to identify synergies between robots. When all the robots within a system fall under the auspices and authority of a single organization, they can simply be compelled to share their information and participate in cooperative protocols. But when, for instance, they are rivals vying in the marketplace, their own private data may be copyrighted or sensitive, so that disclosing information may erode a competitive advantage. Yet, even limited cooperation, by offering some arbitrage of common resources (such as shared infrastructure), often reduces costs for all parties; indeed, competition and cooperation are not mutually exclusive. We examine the question of how to allocate robots to tasks optimally while ensuring that no task valuations, utilities, positions, or related data are released. We do this via an auction-based assignment algorithm implemented using secure multi-party computation operations, without requiring any trusted auctioneer. The approach offers precise and effective privacy guarantees that are stronger than present methods. We demonstrate the feasibility of the approach via tests in a case study inspired by autonomous driving. First, we tested the approach in a single-computer setup, using parties with virtual network interfaces, where we studied the effects of varying the number of parties and the associated parameters of the auction. Next, we tested the approach in a decentralized, physical test-bed using single board computers running over a WiFi LAN network. Finally, we conducted a small proof-of-concept experiment using two autonomous mobile robots performing a decentralized, private auction.

Privacy-Preserving Multi-Robot Task Allocation via Secure Multi-Party Computation

Multi-robot task allocation is a practical way to identify synergies between robots. When all the robots within a system fall under the auspices and authority of a single organization, they can simply be compelled to share their information and participate in cooperative protocols. But when, for instance, they are rivals vying in the marketplace, their own private data may be copyrighted or sensitive, so that disclosing information may erode a competitive advantage. Yet, even limited cooperation, by offering some arbitrage of common resources (such as shared infrastructure), often reduces costs for all parties; indeed, competition and cooperation are not mutually exclusive. We examine the question of how to allocate robots to tasks optimally while ensuring that no task valuations, utilities, positions, or related data are released. We do this via an auction-based assignment algorithm implemented using secure multi-party computation operations, without requiring any trusted auctioneer. The approach offers precise and effective privacy guarantees that are stronger than present methods. We demonstrate the feasibility of the approach via tests in a case study inspired by autonomous driving. First, we tested the approach in a single-computer setup, using parties with virtual network interfaces, where we studied the effects of varying the number of parties and the associated parameters of the auction. Next, we tested the approach in a decentralized, physical test-bed using single board computers running over a WiFi LAN network. Finally, we conducted a small proof-of-concept experiment using two autonomous mobile robots performing a decentralized, private auction. 

We survey a few strengthenings and generalizations of the Combinatorial Nullstellensatz of Alon and the Schwartz-Zippel Lemma. These lemmas guarantee the existence of (a certain number of) nonzeros of a multivariate polynomial when the variables run independently through sufficiently large ranges.

The Generalized Combinatorial Lason-Alon-Zippel-Schwartz Nullstellensatz Lemma

Tools for resolutions in algebra , geometry and their applications

We consider the complexity of computing Chow forms and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants, and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model; our result represents the first Boolean complexity bound. We also extend our algorithm to multiprojective Chow forms and obtain the first computational result in this setting. The motivation for our work comes from incidence geometry where intriguing problems for computational algebraists remain open.

The Boolean complexity of computing Chow forms

Motivated by applications in combinatorial geometry, we consider the following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ be finite sets, and let $S:=S_1 \times S_2 \times \ldots \times S_m \subset \mathbb{C}^n$ be the multi-grid defined by $S_i$. Suppose $p$ is an $n$-variate degree $d$ polynomial. How many zeros does $p$ have on $S$? 
We first develop a multivariate generalization of Combinatorial Nullstellensatz that certifies existence of a point $t \in S$ so that $p(t) \neq 0$. Then we show that a natural multivariate generalization of the DeMillo-Lipton-Schwartz-Zippel lemma holds, except for a special family of polynomials that we call $\lambda$-reducible. This yields a simultaneous generalization of Szemeredi-Trotter theorem and Schwartz-Zippel lemma into higher dimensions, and has applications in incidence geometry. Finally, we develop a symbolic algorithm that identifies certain $\lambda$-reducible polynomials. More precisely, our symbolic algorithm detects polynomials that include a cartesian product of hypersurfaces in their zero set. It is likely that using Chow forms the algorithm can be generalized to handle arbitrary $\lambda$-reducible polynomials, which we leave as an open problem.

pdf/the-multivariate-schwartz-zippel-lemma-3b6jttc5h4.pdf

The Multivariate Schwartz-Zippel Lemma

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The Multivariate Schwartz-Zippel Lemma