arXiv subjects

Yang P. Liu

Publications and source records attributed to Yang P. Liu.

At least 19 recordsLinked to original sources

arXiv2026

A Counting Lemma for Somewhat Restricted 3-APs

For a prime $p\geq 3$, a somewhat restricted $3$-AP in $\mathbb{F}_p^n$ is a triplet $(x,x+a,x+2a)$, where $x\in\mathbb{F}_p^n$ and $a\in \{0,1,2\}^n$. We prove a counting lemma for somewhat restricted $3$-APs in dense sets in $\mathbb{F}_p^n$. More precisely, we prove that for all $\alpha>0$, there exists $\beta>0$, such that for sufficiently large $n$, if a set $A\subseteq \mathbb{F}_p^n$ has density at least $\alpha$, then it contains at least $\beta$ fraction of all somewhat restricted $3$-APs. Our proof builds on recently developed machinery from [Bhangale, Khot, Minzer, 2026]. Our main new ingredient is an arithmetic regularity lemma for patterns such as somewhat restricted 3-APs. This result is in the spirit of arithmetic regularity lemmas from the theory of Gowers uniformity norms [Green, Tao, 2010] and may be of independent interest.

math.CO↗

arXiv2026

Streaming Complexity Separations for Dense and Sparse Graphs

We identify a sharp separation in the streaming space complexity of Maximum Cut when the algorithm must output an approximate cut (rather than only the approximate value). For dense graphs, we show that $O(n/\varepsilon^2)$ space is sufficient and that $Ω(n)$ space is necessary. In contrast, for graphs with $Θ(n/\varepsilon^2)$ edges, the situation is markedly different: we show that the problem requires $Ω(n \log(\varepsilon^2 n)/\varepsilon^2)$ space for any $\varepsilon=ω(1/\sqrt{n})$, which is tight for the full range of $\varepsilon$. We also give an $Ω(n \log n/\varepsilon^2)$-space lower bound against deterministic algorithms for outputting a $(1-\varepsilon)$ approximation to the value of the maximum cut. Using similar techniques we prove an analogous sharp separation in the streaming space complexity of Densest Subgraph and show that for every constant-arity CSP over a constant-size alphabet and the Similarity problem the space complexity in dense streams can be improved by shaving a logarithmic factor.

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arXiv2026

Improved Parallel Repetition for GHZ-Supported Games via Spreadness

We prove that for any 3-player game $\mathcal G$, whose query distribution has the same support as the GHZ game (i.e., all $x,y,z\in \{0,1\}$ satisfying $x+y+z=0\pmod{2}$), the value of the $n$-fold parallel repetition of $\mathcal G$ decays exponentially fast: \[ \text{val}(\mathcal G^{\otimes n}) \leq \exp(-n^c)\] for all sufficiently large $n$, where $c>0$ is an absolute constant. We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant $ε>0$, the probability that the players win at least a $\left(\frac{3}{4}+ε\right)$ fraction of the $n$ coordinates is at most $\exp(-n^c)$, where $c=c(ε)>0$ is a constant. In both settings, our work exponentially improves upon the previous best known bounds which were only polynomially small, i.e., of the order $n^{-Ω(1)}$. Our key technical tool is the notion of \emph{algebraic spreadness} adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.

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arXiv2026

Sparsifying Sums of Positive Semidefinite Matrices

In this paper, we revisit spectral sparsification for sums of arbitrary positive semidefinite (PSD) matrices. Concretely, for any collection of PSD matrices $\mathcal{A} = \{A_1, A_2, \ldots, A_r\} \subset \mathbb{R}^{n \times n}$, given any subset $T \subseteq [r]$, our goal is to find sparse weights $μ\in \mathbb{R}_{\geq 0}^r$ such that $(1 - ε) \sum_{i \in T} A_i \preceq \sum_{i \in T} μ_i A_i \preceq (1 + ε) \sum_{i \in T} A_i.$ This generalizes spectral sparsification of graphs which corresponds to $\mathcal{A}$ being the set of Laplacians of edges. It also captures sparsifying Cayley graphs by choosing a subset of generators. The former has been extensively studied with optimal sparsifiers known. The latter has received attention recently and was solved for a few special groups (e.g., $\mathbb{F}_2^n$). Prior work shows any sum of PSD matrices can be sparsified down to $O(n)$ elements. This bound however turns out to be too coarse and in particular yields no non-trivial bound for building Cayley sparsifiers for Cayley graphs. In this work, we develop a new, instance-specific (i.e., specific to a given collection $\mathcal{A}$) theory of PSD matrix sparsification based on a new parameter $N^*(\mathcal{A})$ which we call connectivity threshold that generalizes the threshold of the number of edges required to make a graph connected. Our main result gives a sparsifier that uses at most $O(ε^{-2}N^*(\mathcal{A}) (\log n)(\log r))$ matrices and is constructible in randomized polynomial time. We also show that we need $N^*(\mathcal{A})$ elements to sparsify for any $ε< 0.99$. As the main application of our framework, we prove that any Cayley graph can be sparsified to $O(ε^{-2}\log^4 N)$ generators. Previously, a non-trivial bound on Cayley sparsifiers was known only in the case when the group is $\mathbb{F}_2^n$.

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arXiv2025

Adaptive Matrix Sparsification and Applications to Empirical Risk Minimization

Consider the empirical risk minimization (ERM) problem, which is stated as follows. Let $K_1, \dots, K_m$ be compact convex sets with $K_i \subseteq \mathbb{R}^{n_i}$ for $i \in [m]$, $n = \sum_{i=1}^m n_i$, and $n_i\le C_K$ for some absolute constant $C_K$. Also, consider a matrix $A \in \mathbb{R}^{n \times d}$ and vectors $b \in \mathbb{R}^d$ and $c \in \mathbb{R}^n$. Then the ERM problem asks to find \[ \min_{\substack{x \in K_1 \times \dots \times K_m\\ A^\top x = b}} c^\top x. \] We give an algorithm to solve this to high accuracy in time $\widetilde{O}(nd + d^6\sqrt{n}) \le \widetilde{O} (nd + d^{11})$, which is nearly-linear time in the input size when $A$ is dense and $n \ge d^{10}$. Our result is achieved by implementing an $\widetilde{O}(\sqrt{n})$-iteration interior point method (IPM) efficiently using dynamic data structures. In this direction, our key technical advance is a new algorithm for maintaining leverage score overestimates of matrices undergoing row updates. Formally, given a matrix $A \in \mathbb{R}^{n \times d}$ undergoing $T$ batches of row updates of total size $n$ we give an algorithm which can maintain leverage score overestimates of the rows of $A$ summing to $\widetilde{O}(d)$ in total time $\widetilde{O}(nd + Td^6)$. This data structure is used to sample a spectral sparsifier within a robust IPM framework to establish the main result.

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arXiv2025

An Analytical Approach to Parallel Repetition via CSP Inverse Theorems

Let $\mathcal{G}$ be a $k$-player game with value $<1$, whose query distribution is such that no marginal on $k-1$ players admits a non-trivial Abelian embedding. We show that for every $n\geq N$, the value of the $n$-fold parallel repetition of $\mathcal{G}$ is $$ \text{val}(\mathcal{G}^{\otimes n}) \leq \frac{1}{\underbrace{\log\log\cdots\log}_{C\text{ times}} n}, $$ where $N=N(\mathcal{G})$ and $1\leq C\leq k^{O(k)}$ are constants. As a consequence, we obtain a parallel repetition theorem for all $3$-player games whose query distribution is pairwise-connected. Prior to our work, only inverse Ackermann decay bounds were known for such games [Ver96]. As additional special cases, we obtain a unified proof for all known parallel repetition theorems, albeit with weaker bounds: (1) A new analytic proof of parallel repetition for all 2-player games [Raz98, Hol09, DS14]. (2) A new proof of parallel repetition for all $k$-player playerwise connected games [DHVY17, GHMRZ22]. (3) Parallel repetition for all $3$-player games (in particular $3$-XOR games) whose query distribution has no non-trivial Abelian embedding into $(\mathbb{Z}, +)$ [BKM23c, BBKLM25]. (4) Parallel repetition for all 3-player games with binary inputs [HR20, GHMRZ21, GHMRZ22, GMRZ22].

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arXiv2025

Quasipolynomial bounds for the corners theorem

Let $G$ be a finite abelian group and $A$ be a subset of $G \times G$ which is corner--free, meaning that there are no $x, y \in G$ and $d \in G \setminus \{0\}$ such that $(x, y)$, $(x+d, y)$, $(x, y+d) \in A$. We prove that \[|A| \le |G|^2 \cdot \exp(-(\log |G|)^{Ω(1)}).\] As a consequence, we obtain polynomial (in the input length) lower bounds on the nondeterministic communication complexity of Exactly-N in the 3-player Number-on-Forehead model. We also obtain the first "reasonable'' lower bounds on the coloring version of the $3$-dimensional corners problem, as well as on the nondeterministic communication complexity of Exactly-N in the 4-player Number-on-Forehead model.

math.CO↗

arXiv2025

Incremental Shortest Paths in Almost Linear Time via a Modified Interior Point Method

We give an algorithm that takes a directed graph $G$ undergoing $m$ edge insertions with lengths in $[1, W]$, and maintains $(1+ε)$-approximate shortest path distances from a fixed source $s$ to all other vertices. The algorithm is deterministic and runs in total time $m^{1+o(1)}\log W$, for any $ε> \exp(-(\log m)^{0.99})$. This is achieved by designing a nonstandard interior point method to crudely detect when the distances from $s$ other vertices $v$ have decreased by a $(1+ε)$ factor, and implementing it using the deterministic min-ratio cycle data structure of [Chen-Kyng-Liu-Meierhans-Probst, STOC 2024].

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arXiv2025

Approximate Spanning Tree Counting from Uncorrelated Edge Sets

We show an $\widetilde{O}(m^{1.5} ε^{-1})$ time algorithm that on a graph with $m$ edges and $n$ vertices outputs its spanning tree count up to a multiplicative $(1+ε)$ factor with high probability, improving on the previous best runtime of $\widetilde{O}(m + n^{1.875}ε^{-7/4})$ in sparse graphs. While previous algorithms were based on computing Schur complements and determinantal sparsifiers, our algorithm instead repeatedly removes sets of uncorrelated edges found using the electrical flow localization theorem of Schild-Rao-Srivastava [SODA 2018].

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arXiv2024

On Approximability of Satisfiable $k$-CSPs: VI

We prove local and global inverse theorems for general $3$-wise correlations over pairwise-connected distributions. Let $μ$ be a distribution over $Σ\times Γ\times Φ$ such that the supports of $μ_{xy}$, $μ_{xz}$, and $μ_{yz}$ are all connected, and let $f: Σ^n \to \mathbb{C}$, $g: Γ^n \to \mathbb{C}$, $h: Φ^n \to \mathbb{C}$ be $1$-bounded functions satisfying \[ \left|\mathbb{E}_{(x,y,z) \sim μ^{\otimes n}}[f(x)g(y)h(z)]\right| \geq \varepsilon. \] In this setting, our local inverse theorem asserts that there is $δ:=\textsf{exp}(-\varepsilon^{-O_μ(1)})$ such that with probability at least $δ$, a random restriction of $f$ down to $δn$ coordinates $δ$-correlates to a product function. To get a global inverse theorem, we prove a restriction inverse theorem for general product functions, stating that if a random restriction of $f$ down to $δn$ coordinates is $δ$-correlated with a product function with probability at least $δ$, then $f$ is $2^{-\textsf{poly}(\log(1/δ))}$-correlated with a function of the form $L\cdot P$, where $L$ is a function of degree $\textsf{poly}(1/δ)$, $\|L\|_2\leq 1$, and $P$ is a product function. We show applications to property testing and to additive combinatorics. In particular, we show the following result via a density increment argument. Let $Σ$ be a finite set and $S \subseteq Σ\times Σ\times Σ$ such that: (1) $(x, x, x) \in S$ for all $x \in S$, and (2) the supports of $S_{xy}$, $S_{xz}$, and $S_{yz}$ are all connected. Then, any set $A \subseteq Σ^n$ with $|Σ|^{-n}|A| \geq Ω((\log \log \log n)^{-c})$ contains $x, y, z \in A$, not all equal, such that $(x_i,y_i,z_i) \in S$ for all $i$. This gives the first reasonable bounds for the restricted 3-AP problem over finite fields.

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arXiv2024

On Approximability of Satisfiable $k$-CSPs: VII

Let $Σ_1,\ldots,Σ_k$ be finite alphabets, and let $μ$ be a distribution over $Σ_1 \times \dots \times Σ_k$ in which the probability of each atom is at least $α$. We prove that if $μ$ does not admit Abelian embeddings, and $f_i: Σ_i \to \mathbb{C}$ are $1$-bounded functions (for $i=1,\ldots,k$) such that \[ \left|\mathbb{E}_{(x_1,\dots,x_k) \sim μ^{\otimes n}}\Big[f_1(x_1) \dots f_k(x_k)\Big]\right| \geq \varepsilon, \] then there exists $L\colon Σ_1^n\to\mathbb{C}$ of degree at most $d$ and $\|L\|_2\leq 1$ such that $|\langle f_1, L\rangle|\geq δ$, where $d$ and $δ>0$ depend only on $k, α$ and $\varepsilon$. This answers the analytic question posed by Bhangale, Khot, and Minzer (STOC 2022). We also prove several extensions of this result that are useful in subsequent applications.

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arXiv2024

Reasonable Bounds for Combinatorial Lines of Length Three

We prove that any subset $A \subseteq [3]^n$ with $3^{-n}|A| \ge (\log\log\log\log n)^{-c}$ contains a combinatorial line of length $3$, i.e., $x, y, z \in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i = 1, 2, \dots, n$. This improves on the previous best bound of $3^{-n}|A| \ge Ω((\log^* n)^{-1/2})$ of [D.H.J. Polymath, Ann. of Math. 2012].

math.CO↗

arXiv2024

Parallel Repetition for $3$-Player XOR Games

In a $3$-$\mathsf{XOR}$ game $\mathcal{G}$, the verifier samples a challenge $(x,y,z)\sim μ$ where $μ$ is a probability distribution over $Σ\timesΓ\timesΦ$, and a map $t\colon Σ\timesΓ\timesΦ\to\mathcal{A}$ for a finite Abelian group $\mathcal{A}$ defining a constraint. The verifier sends the questions $x$, $y$ and $z$ to the players Alice, Bob and Charlie respectively, receives answers $f(x)$, $g(y)$ and $h(z)$ that are elements in $\mathcal{A}$ and accepts if $f(x)+g(y)+h(z) = t(x,y,z)$. The value, $\mathsf{val}(\mathcal{G})$, of the game is defined to be the maximum probability the verifier accepts over all players' strategies. We show that if $\mathcal{G}$ is a $3$-$\mathsf{XOR}$ game with value strictly less than $1$, whose underlying distribution over questions $μ$ does not admit Abelian embeddings into $(\mathbb{Z},+)$, then the value of the $n$-fold repetition of $\mathcal{G}$ is exponentially decaying. That is, there exists $c=c(\mathcal{G})>0$ such that $\mathsf{val}(\mathcal{G}^{\otimes n})\leq 2^{-cn}$. This extends a previous result of [Braverman-Khot-Minzer, FOCS 2023] showing exponential decay for the GHZ game. Our proof combines tools from additive combinatorics and tools from discrete Fourier analysis.

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arXiv2024

Almost-Linear Time Algorithms for Decremental Graphs: Min-Cost Flow and More via Duality

We give the first almost-linear total time algorithm for deciding if a flow of cost at most $F$ still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, and cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and $s$-$t$ distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of $m^{1+o(1)}$ dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized $m^{o(1)}$ time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [Goranci-Räcke-Saranurak-Tan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.

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arXiv2024

On further questions regarding unit fractions

We prove that a subset $A\subseteq [1, N]$ with \[\sum_{n\in A}\frac{1}{n} \ge (\log N)^{4/5 + o(1)}\] contains a subset $B$ such that \[\sum_{n\in B} \frac{1}{n} = 1.\] Our techniques refine those of Croot and of Bloom. Using our refinements, we additionally consider a number of questions regarding unit fractions due to Erdős and Graham.

math.NT↗

arXiv2024

On Approximate Fully-Dynamic Matching and Online Matrix-Vector Multiplication

We study connections between the problem of fully dynamic $(1-ε)$-approximate maximum bipartite matching, and the dual $(1+ε)$-approximate vertex cover problem, with the online matrix-vector ($\mathsf{OMv}$) conjecture which has recently been used in several fine-grained hardness reductions. We prove that there is an online algorithm that maintains a $(1+ε)$-approximate vertex cover in amortized $n^{1-c}ε^{-C}$ time for constants $c, C > 0$ for fully dynamic updates if and only if the $\mathsf{OMv}$ conjecture is false. Similarly, we prove that there is an online algorithm that maintains a $(1-ε)$-approximate maximum matching in amortized $n^{1-c}ε^{-C}$ time if and only if there is a nontrivial algorithm for another dynamic problem, which we call dynamic approximate $\mathsf{OMv}$, that has seemingly no matching structure. This provides some evidence against achieving amortized sublinear update times for approximate fully dynamic matching and vertex cover. Leveraging these connections, we obtain faster algorithms for approximate fully dynamic matching in both the online and offline settings. 1. We give a randomized algorithm that with high probability maintains a $(1-ε)$-approximate bipartite matching and $(1+ε)$-approximate vertex cover in fully dynamic graphs, in amortized $O(ε^{-O(1)} \frac{n}{2^{Ω(\sqrt{\log n})}})$ update time. Our algorithm leverages fast algorithms for $\mathsf{OMv}$ due to Larsen-Williams [SODA 2017]. 2. We give a randomized offline algorithm for $(1-ε)$-approximate maximum matching with amortized runtime $O(n^{.58}ε^{-O(1)})$ by using fast matrix multiplication, significantly improving over the runtimes achieved via online algorithms. We also give an offline algorithm that maintains a $(1+ε)$-approximate vertex cover in amortized $O(n^{.723}ε^{-O(1)})$ time.

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arXiv2023

Parallel Repetition of k-Player Projection Games

We study parallel repetition of k-player games where the constraints satisfy the projection property. We prove exponential decay in the value of a parallel repetition of projection games with value less than 1.

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arXiv2023

Sparsifying sums of norms

For any norms $N_1,\ldots,N_m$ on $\mathbb{R}^n$ and $N(x) := N_1(x)+\cdots+N_m(x)$, we show there is a sparsified norm $\tilde{N}(x) = w_1 N_1(x) + \cdots + w_m N_m(x)$ such that $|N(x) - \tilde{N}(x)| \leq εN(x)$ for all $x \in \mathbb{R}^n$, where $w_1,\ldots,w_m$ are non-negative weights, of which only $O(ε^{-2} n \log(n/ε) (\log n)^{2.5} )$ are non-zero. Additionally, if $N$ is $\mathrm{poly}(n)$-equivalent to the Euclidean norm on $\mathbb{R}^n$, then such weights can be found with high probability in time $O(m (\log n)^{O(1)} + \mathrm{poly}(n)) T$, where $T$ is the time required to evaluate a norm $N_i$. This immediately yields analogous statements for sparsifying sums of symmetric submodular functions. More generally, we show how to sparsify sums of $p$th powers of norms when the sum is $p$-uniformly smooth.

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