arXiv subjects

Songhua He

Publications and source records attributed to Songhua He.

4 recordsLinked to original sources

arXiv2026

Testing MAX-CUT in the Adjacency Matrix Model: Tight Non-Adaptive Bounds and a Cubic Adaptive Lower Bound

We study the query complexity of approximating the MAX-CUT value to additive error $\varepsilon n^2$ in the adjacency matrix model. This was one of the first problems studied in graph property testing [GGR'98]. Previously, the best known upper bound for both adaptive and non-adaptive algorithms was $O(1/\varepsilon^6)$ [RV'07; MS'08; GMRS'22]. The only known lower bound was a trivial $Ω(1/\varepsilon^2)$ bound. We close this gap from both sides for non-adaptive algorithms, proving that its query complexity is exactly $Θ(1/\varepsilon^4)$. We also prove an adaptive $Ω(1/\varepsilon^3)$ lower bound. Our upper bound is achieved by a canonical tester that samples $O(1/\varepsilon^2)$ vertices in expectation and outputs the rescaled MAX-CUT value of the induced subgraph. We prove its correctness by applying spectral truncation to the adjacency matrix, using a geometric argument to control the sampling error of the retained low-rank part and a cut-norm restriction estimate [RV'07] for the remainder. As an application, an analogous proof also yields a tight $O(1/\varepsilon^4)$ non-adaptive query upper bound for MAX-2CSP. For our lower bounds, we show indistinguishability between a uniform random graph and a random graph with a planted hidden partition. For the non-adaptive lower bound, we observe that one plus the $χ^2$-divergence between the query transcript distributions is exactly a normalized Ising partition function, which can be bounded using Boolean hypercontractivity. For the adaptive lower bound, we relate each query's distinguishing power to the prediction, based on previous answers, of whether its endpoints lie in the same part of the hidden partition. We bound the combined advantage of these predictions using the completed transcript, which lets us analyze the adaptively chosen pairs of vertices together despite their dependence on earlier answers.

cs.DS↗

arXiv2026

Systematic Data Structure Lower Bounds via the Query-with-Sketch Model

We study data structure lower bounds for the Approximate Matrix Powering (AMP) problem. Given a substochastic, symmetric matrix $\mathbf{M}\in\mathbb{R}^{n\times n}$ and parameters $k$ and $α$, the goal is to preprocess $\mathbf{M}$ so as to answer entry queries $(u,v)\mapsto \mathbf{M}^{k}[u,v]$ up to additive error $1/n^α$. We focus on AMP in the succinct and systematic regime, in which the data structure stores $\mathbf{M}$ verbatim, uses an additional $r$ bits of redundancy, and must answer queries by probing only a small number of entries of $\mathbf{M}$. Our main conceptual contribution is a general framework for proving probe--redundancy trade-offs for systematic data structures. We introduce the query-with-sketch model and develop a min-entropy-based approach that lifts conditional min-entropy bounds in the absence of redundancy to probe lower bounds in the presence of redundancy. We then establish these min-entropy bounds using problem-specific analytic and algebraic tools, for the downstream applications to AMP and its variants. As a consequence, our results provide new unconditional evidence toward a conjecture of Patrascu and Roditty (2010) on the space required for constant-time set-disjointness queries.

cs.DS↗

arXiv2026

Should Tables Be Sorted? Revisited with a Large Language Model

We revisit the implicit membership problem in Yao's full-table model [Yao, 1981] and obtain, to our knowledge, the first quantitative improvements to his 45-year-old Ramsey bounds, most notably reducing the two-probe bound from tower-type to polynomial. In this model, an $n$-set $S\subseteq\{1,\ldots,m\}$ is stored as a permutation in an $n$-cell table, and queries decide whether $x\in S$. Let $G_q(n)$ be the largest universe size admitting a $q$-probe membership scheme for all $n$-sets. Yao determined the one-probe case exactly, proving $G_1(n)=2n-2$ for $n>2$, but the behavior for $q\ge2$ remained wide open. Fiat and Naor [1993] constructed schemes for universes of size $\exp(n^c)$ for some constant $c>0$ and sufficiently large constant $q$. For the first adaptive case, $q=2$, we prove $G_2(n)=O(n^2(\log n)^2)$. For every fixed integer $q\ge3$, we show that $G_q(n)$ is at most a tower of height $q-1$ with top $n^{1+o(1)}$; in particular, $G_3(n)\le\exp(n^{1+o(1)})$. The two-probe proof avoids Ramsey theory altogether; for larger fixed $q$, we use Ramsey theory only to make the first $q-1$ probes follow a fixed pattern, and then handle the last probe by the same non-Ramsey argument. Somewhat surprisingly, for each fixed $q$, we also show that implicit membership is as hard as implicit search up to a polynomial loss in universe size. Implicit search must return the cell containing $x$ when present and reject otherwise. For the analogous search threshold $H_q(n)$, we prove $H_q(n)\le G_q(n)\le n^q(H_q(n)+1)^{q+1}$ for every $q,n$. Thus, for every fixed $q$, one threshold is at most $\exp(n^{O(1)})$ if and only if the other is. The proofs were first generated by ChatGPT 5.5 Pro without mathematical hints; the membership-search equivalence emerged while pursuing an improved four-probe bound. The authors have validated and edited the proofs and assume responsibility for all content.

cs.DS↗

arXiv2026

Query Lower Bounds for Correlation Clustering under Memory Constraints

This work initiates the study of memory-query tradeoffs for graph problems, with a focus on correlation clustering. Correlation clustering asks for a partition of the vertices that minimizes disagreements: non-edges inside clusters plus edges across clusters. Our first result is a tight query lower bound: to output a partition whose cost approximates the optimum up to an additive error of $\varepsilon n^2$, any algorithm requires $Ω(n/\varepsilon^2)$ adjacency-matrix queries. Under memory constraints, we show that even for the seemingly easier task of approximating the optimal clustering cost (without producing a partition), any algorithm in the random query model must make $\gg n/\varepsilon^2$ adjacency-matrix queries. Finally, we prove the first general graph model query lower bound for correlation clustering, where algorithms are allowed adjacency-matrix, neighbor, and degree queries. The latter two bounds are not yet tight, leaving room for sharper results.

cs.CC↗