arXiv subjects

Young Kun Ko

Publications and source records attributed to Young Kun Ko.

9 recordsLinked to original sources

arXiv2026

An $Ω((\log n/\log\log n)^2)$ Cell-Probe Lower Bound for Dynamic Boolean Data Structures

We resolve the long-standing open problem of Boolean dynamic data structure hardness, proving an unconditional lower bound of $Ω((\log n/\log\log n)^2)$ for the Multiphase Problem of Patrascu [STOC 2010] (instantiated with Inner Product over $\mathbb{F}_2$). This matches the $\tildeΩ(\log^2 n)$ bound that Larsen [STOC 2012] established for weighted problems, where each query returns $Ω(\log n)$ bits, and closes the gap left by the $\tildeΩ(\log^{1.5} n)$ Boolean bound of Larsen, Weinstein, and Yu [STOC 2018]. It also answers, up to a $(\log\log n)^2$ factor, Open Question 15.24 in Viola's textbook "Mathematics of the Impossible" [Cambridge 2026]. The previous barrier was methodological: all prior works relied on "one-way" communication games, where the inability to verify query simulations necessitated complex machinery (such as the Peak-to-Average Lemma) that hit a hard ceiling at $\log^{1.5} n$. Our key contribution is conceptual: we introduce a 2.5-round Multiphase Communication Game that augments the standard one-way model with a verification round, where Bob confirms the consistency of Alice's simulation against the actual memory. This simple, qualitative change removes the obstruction and yields the full bound directly. As a consequence, our analysis naturally extends to other hard Boolean functions -- those of small discrepancy under product distributions with large min-entropy -- offering a general recipe for translating discrepancy lower bounds into $Ω((\log n/\log\log n)^2)$ dynamic Boolean data structure lower bounds. We also argue that this result likely represents the structural ceiling of the Chronogram framework initiated by Fredman and Saks [STOC 1989]: any $ω(\log^2 n)$ lower bound would require either fundamentally new techniques or major circuit complexity breakthroughs.

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arXiv2026

On the (Classical and Quantum) Fine-Grained Complexity of Approximate CVP and Max-Cut

We show a linear-size reduction from gap Max-2-Lin(2) (a generalization of the gap $\mathrm{Max}$-$\mathrm{Cut}$ problem) to $γ\text{-}\mathrm{CVP}_p$ for $γ= \mathrm{O}(1)$ and finite $p\geq 1$, as well as a no-go theorem against poly-sized non-adaptive quantum reductions from $k$-SAT to $\mathrm{CVP}_2$. This implies three headline results: (i) Faster algorithms for $γ\text{-}\mathrm{CVP}$ are also faster algorithms for Max-2-Lin(2) and Max-Cut. Depending on the approximation regime, even a $2^{0.78n}$-time or $2^{0.3n}$-time algorithm would improve upon the state-of-the-art algorithm such as Williams' 2004 algorithm [Theoretical Computer Science 2005] or Arora et al.'s 2010 algorithm [Journal of the ACM 2015]. This provides evidence that $γ\text{-}\mathrm{CVP}$ for $γ=\mathrm{O}(1)$ requires exponential time, improving upon the previous lower-bound for $γ<3$ by Bennett et al. [arxiv:1704.03928]. (ii) A new almost $2^{(1/2+\varepsilon/4ς+o(1))n}$-time classical algorithm and a new almost $2^{(1/3+\varepsilon/6ς+o(1))n}$-time quantum algorithm for $(1-\varepsilon,1-ς)$-gap Max-2-Lin(2). This algorithm is faster than the algorithm of Arora et al., as well as the algorithm of Williams, and the algorithm of Manurangsi and Trevisan [arxiv:1807.09898] when $c_0 \varepsilon<ς 0$, it must be via an adaptive quantum reduction unless $\mathrm{NP} \subseteq \mathrm{pr}\text{-}\mathrm{QSZK}$. This illuminates some difficulties in characterizing the hardness of approximate CSPs and shows that the post-quantum security of lattice-based cryptography likely cannot be supported by QSETH.

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arXiv2025

Unifying the Landscape of Super-Logarithmic Dynamic Cell-Probe Lower Bounds

We prove a general translation theorem for converting one-way communication lower bounds over a product distribution to dynamic cell-probe lower bounds. Specifically, we consider a class of problems considered in [Pat10] where: 1. $S_1, \ldots, S_m \in \{0, 1\}^n$ are given and publicly known. 2. $T \in \{0, 1\}^n$ is a sequence of updates, each taking $t_u$ time. 3. For a given $Q \in [m]$, we must output $f(S_Q, T)$ in $t_q$ time. Our main result shows that for a "hard" function $f$, for which it is difficult to obtain a non-trivial advantage over random guessing with one-way communication under some product distribution over $S_Q$ and $T$ (for example, a uniform distribution), then the above explicit dynamic cell-probe problem must have $\max \{ t_u, t_q \} \geq \tildeΩ(\log^{3/2}(n))$ if $m = Ω(n^{0.99})$. This result extends and unifies the super-logarithmic dynamic data structure lower bounds from [LWY20] and [LY25] into a more general framework. From a technical perspective, our approach merges the cell-sampling and chronogram techniques developed in [LWY20] and [LY25] with the new static data structure lower bound methods from [KW20] and [Ko25], thereby merging all known state-of-the-art cell-probe lower-bound techniques into one. As a direct consequence of our method, we establish a super-logarithmic lower bound against the Multiphase Problem [Pat10] for the case where the data structure outputs the Inner Product (mod 2) of $S_Q$ and $T$. We suspect further applications of this general method towards showing super-logarithmic dynamic cell-probe lower bounds. We list some example applications of our general method, including a novel technique for a one-way communication lower bound against small-advantage protocols for a product distribution using average min-entropy, which could be of independent interest.

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arXiv2025

Lower Bounds for Linear Operators

We consider a static data structure problem of computing a linear operator under cell-probe model. Given a linear operator $M \in \mathbb{F}_2^{m \times n}$, the goal is to pre-process a vector $X \in \mathbb{F}_2^n$ into a data structure of size $s$ to answer any query $\langle M_i , X \rangle$ in time $t$. We prove that for a random operator $M$, any such data structure requires: $$ t \geq Ω( \min \{ \log (m/s) , n / \log s \} ).$$ This result overcomes the well-known logarithmic barrier in static data structures [MNSW98, Sie04, PD06, PTW08, Pat11, DGW19] by using a random linear operator. Furthermore, it provides the first significant progress toward confirming a decades-old folklore conjecture: that non-linear pre-processing does not substantially help in computing most linear operators. A straightforward modification of our proof also yields a wire lower bound of $Ω(n \cdot \log^{1/d}(n))$ for depth-$d$ circuits with arbitrary gates that compute a specific linear operator $M \in \mathbb{F}_2^{O(n) \times n}$, even against some small constant advantage over random guessing. This bound holds even for circuits with only a small constant advantage over random guessing, improving upon longstanding results [RS03, Che08a, Che08b, GHK+13] for a random operator. Finally, our work partially resolves the communication form of the Multiphase Conjecture [Pat10] and makes progress on Jukna-Schnitger's Conjecture [JS11, Juk12]. We address the former by considering the Inner Product (mod 2) problem (instead of Set Disjointness) when the number of queries $m$ is super-polynomial (e.g., $2^{n^{1/3}}$), and the total update time is $m^{0.99}$. Our result for the latter also applies to cases with super-polynomial $m$.

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arXiv2020

Hardness of Approximate Nearest Neighbor Search under L-infinity

We show conditional hardness of Approximate Nearest Neighbor Search (ANN) under the $\ell_\infty$ norm with two simple reductions. Our first reduction shows that hardness of a special case of the Shortest Vector Problem (SVP), which captures many provably hard instances of SVP, implies a lower bound for ANN with polynomial preprocessing time under the same norm. Combined with a recent quantitative hardness result on SVP under $\ell_\infty$ (Bennett et al., FOCS 2017), our reduction implies that finding a $(1+\varepsilon)$-approximate nearest neighbor under $\ell_\infty$ with polynomial preprocessing requires near-linear query time, unless the Strong Exponential Time Hypothesis (SETH) is false. This complements the results of Rubinstein (STOC 2018), who showed hardness of ANN under $\ell_1$, $\ell_2$, and edit distance. Further improving the approximation factor for hardness, we show that, assuming SETH, near-linear query time is required for any approximation factor less than $3$ under $\ell_\infty$. This shows a conditional separation between ANN under the $\ell_1/ \ell_2$ norm and the $\ell_\infty$ norm since there are sublinear time algorithms achieving better than $3$-approximation for the $\ell_1$ and $\ell_2$ norm. Lastly, we show that the approximation factor of $3$ is a barrier for any naive gadget reduction from the Orthogonal Vectors problem.

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arXiv2019

An Adaptive Step Toward the Multiphase Conjecture

In 2010, Pǎtraşcu proposed the following three-phase dynamic problem, as a candidate for proving polynomial lower bounds on the operational time of dynamic data structures: I: Preprocess a collection of sets $\vec{S} = S_1, \ldots , S_k \subseteq [n]$, where $k=\operatorname{poly}(n)$. II: A set $T\subseteq [n]$ is revealed, and the data structure updates its memory. III: An index $i \in [k]$ is revealed, and the data structure must determine if $S_i\cap T=^? \emptyset$. Pǎtraşcu conjectured that any data structure for the Multiphase problem must make $n^ε$ cell-probes in either Phase II or III, and showed that this would imply similar unconditional lower bounds on many important dynamic data structure problems. Alas, there has been almost no progress on this conjecture in the past decade since its introduction. We show an $\tildeΩ(\sqrt{n})$ cell-probe lower bound on the Multiphase problem for data structures with general (adaptive) updates, and queries with unbounded but "layered" adaptivity. This result captures all known set-intersection data structures and significantly strengthens previous Multiphase lower bounds, which only captured non-adaptive data structures. Our main technical result is a communication lower bound on a 4-party variant of Pǎtraşcu's Number-On-Forehead Multiphase game, using information complexity techniques. We also show that a lower bound on Pǎtraşcu's original NOF game would imply a polynomial ($n^{1+ε}$) lower bound on the number of wires of any constant-depth circuit with arbitrary gates computing a random $\tilde{O}(n)\times n$ linear operator $x \mapsto Ax$, a long-standing open problem in circuit complexity. This suggests that the NOF conjecture is much stronger than its data structure counterpart.

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arXiv2016

Hardness Results for Signaling in Bayesian Zero-Sum and Network Routing Games

We study the optimization problem faced by a perfectly informed principal in a Bayesian game, who reveals information to the players about the state of nature to obtain a desirable equilibrium. This signaling problem is the natural design question motivated by uncertainty in games and has attracted much recent attention. We present new hardness results for signaling problems in (a) Bayesian two-player zero-sum games, and (b) Bayesian network routing games. For Bayesian zero-sum games, when the principal seeks to maximize the equilibrium utility of a player, we show that it is NP-hard to obtain an additive FPTAS. Our hardness proof exploits duality and the equivalence of separation and optimization in a novel way. Further, we rule out an additive PTAS assuming planted clique hardness, which states that no polynomial time algorithm can recover a planted clique from an Erdős-Rényi random graph. Complementing these, we obtain a PTAS for a structured class of zero-sum games (where obtaining an FPTAS is still NP-hard) when the payoff matrices obey a Lipschitz condition. Previous results ruled out an FPTAS assuming planted-clique hardness, and a PTAS only for implicit games with quasi-polynomial-size strategy sets. For Bayesian network routing games, wherein the principal seeks to minimize the average latency of the Nash flow, we show that it is NP-hard to obtain a (multiplicative) $(4/3 - ε)$-approximation, even for linear latency functions. This is the optimal inapproximability result for linear latencies, since we show that full revelation achieves a $(4/3)$-approximation for linear latencies.

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arXiv2015

Near-optimal bounds on bounded-round quantum communication complexity of disjointness

We prove a near optimal round-communication tradeoff for the two-party quantum communication complexity of disjointness. For protocols with $r$ rounds, we prove a lower bound of $\tildeΩ(n/r + r)$ on the communication required for computing disjointness of input size $n$, which is optimal up to logarithmic factors. The previous best lower bound was $Ω(n/r^2 + r)$ due to Jain, Radhakrishnan and Sen [JRS03]. Along the way, we develop several tools for quantum information complexity, one of which is a lower bound for quantum information complexity in terms of the generalized discrepancy method. As a corollary, we get that the quantum communication complexity of any boolean function $f$ is at most $2^{O(QIC(f))}$, where $QIC(f)$ is the prior-free quantum information complexity of $f$ (with error $1/3$).

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arXiv2015

ETH Hardness for Densest-$k$-Subgraph with Perfect Completeness

We show that, assuming the (deterministic) Exponential Time Hypothesis, distinguishing between a graph with an induced $k$-clique and a graph in which all k-subgraphs have density at most $1-ε$, requires $n^{\tilde Ω(log n)}$ time. Our result essentially matches the quasi-polynomial algorithms of Feige and Seltser [FS97] and Barman [Bar15] for this problem, and is the first one to rule out an additive PTAS for Densest $k$-Subgraph. We further strengthen this result by showing that our lower bound continues to hold when, in the soundness case, even subgraphs smaller by a near-polynomial factor ($k' = k 2^{-\tilde Ω(log n)}$) are assumed to be at most ($1-ε$)-dense. Our reduction is inspired by recent applications of the "birthday repetition" technique [AIM14,BKW15]. Our analysis relies on information theoretical machinery and is similar in spirit to analyzing a parallel repetition of two-prover games in which the provers may choose to answer some challenges multiple times, while completely ignoring other challenges.

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