Testing Odd Direct Sums Using High Dimensional Expanders

Authors Roy Gotlib, Tali Kaufman

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Author Details

Roy Gotlib
  • Bar-Ilan University, Ramat Gan, Israel
Tali Kaufman
  • Bar-Ilan University, Ramat Gan, Israel

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Roy Gotlib and Tali Kaufman. Testing Odd Direct Sums Using High Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


In this work, using methods from high dimensional expansion, we show that the property of k-direct-sum is testable for odd values of k . Previous work of [Kaufman and Lubotzky, 2014] could inherently deal only with the case that k is even, using a reduction to linearity testing. Interestingly, our work is the first to combine the topological notion of high dimensional expansion (called co-systolic expansion) with the combinatorial/spectral notion of high dimensional expansion (called colorful expansion) to obtain the result. The classical k-direct-sum problem applies to the complete complex; Namely it considers a function defined over all k-subsets of some n sized universe. Our result here applies to any collection of k-subsets of an n-universe, assuming this collection of subsets forms a high dimensional expander.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • High Dimensional Expanders
  • Property Testing
  • Direct Sum


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