Kai Zhe Zheng

Dor Minzer, Kai Zhe Zheng

We show that for all $\varepsilon>0$, for sufficiently large $q\in\mathbb{N}$ power of $2$, for all $δ>0$, it is NP-hard to distinguish whether a given $2$-Prover-$1$-Round projection game with alphabet size $q$ has value at least $1-δ$, or value at most $1/q^{1-\varepsilon}$. This establishes a nearly optimal alphabet-to-soundness tradeoff for $2$-query PCPs with alphabet size $q$, improving upon a result of [Chan, Journal of the ACM 2016]. Our result has the following implications: 1) Near optimal hardness for Quadratic Programming: it is NP-hard to approximate the value of a given Boolean Quadratic Program within factor $(\log n)^{1 - o(1)}$ under quasi-polynomial time reductions. This improves upon a result of [Khot, Safra, ToC 2013] and nearly matches the performance of the best known algorithms due to [Megretski, IWOTA 2000], [Nemirovski, Roos, Terlaky, Mathematical Programming 1999] and [Charikar, Wirth, FOCS 2004] that achieve $O(\log n)$ approximation ratio. 2) Bounded degree $2$-CSPs: under randomized reductions, for sufficiently large $d>0$, it is NP-hard to approximate the value of $2$-CSPs in which each variable appears in at most $d$ constraints within factor $(1-o(1))\frac{d}{2}$, improving upon a result of [Lee, Manurangsi, ITCS 2024]. 3) Improved hardness results for connectivity problems: using results of [Laekhanukit, SODA 2014] and [Manurangsi, Inf. Process. Lett., 2019], we deduce improved hardness results for the Rooted $k$-Connectivity Problem, the Vertex-Connectivity Survivable Network Design Problem and the Vertex-Connectivity $k$-Route Cut Problem.

Esty Kelman, Uri Meir, Kai Zhe Zheng

Motivated by applications to property testing in the online-erasure model of Kalemaj, Raskhodnikova, and Varma (ITCS 2022 and Theory of Computing 2023), we define and analyze {\em semi-sample-based testers} for Reed-Muller codes. The task in Reed-Muller testing is to determine whether an input function $f: \F^n \to \F$ belongs to the Reed-Muller code or is far from it, using as few point queries to $f$ as possible. Reed-Muller testing is a well-studied task with its roots in both the Property Testing and Probabilistically Checkable Proofs literature. The online-erasure model introduces a twist: after each query made, an adversary may erase up to $t$ points of the input function, potentially thwarting any test in which the queries follow a predictable pattern. Semi-sample-based testers are a hybrid between sample-based testers -- which can only make uniformly random queries to the input function -- and standard testers, which can choose their queries freely. They are designed with the online-erasure model in mind and operate by first choosing some subset $S$ of the domain and then making their queries uniformly at random inside of $S$. We describe semi-sample-based testers for the Reed-Muller code and give an optimal analysis of their soundness. Consequently, we show that semi-sample-based testers are indeed effective in the presence of online erasures, and thereby achieve optimal query complexity for testing the Reed-Muller code in the online-erasure model. This result improves upon prior work of Minzer and Zheng (SODA 2024). As an added bonus, we show that semi-sample-based testers also exist for the lifted affine-invariant codes of Guo, Kopparty, and Sudan (ITCS 2013), thereby providing the first known testers for these codes in the online-erasure model.