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Xuandi Ren

Publications and source records attributed to Xuandi Ren.

17 recordsLinked to original sources
arXiv2026

Inapproximability of Unique-Machine Precedence Scheduling for Unit-Length Jobs

The Unique-Machine Precedence Scheduling (UMPS) problem, introduced by [DKRSTZ22], seeks a makespan-minimizing schedule of precedence-constrained jobs when each job has a unique eligible machine. On the one hand, UMPS generalizes job shop scheduling by allowing the precedence graph to be an arbitrary DAG rather than a disjoint union of chains. On the other hand, UMPS admits approximation-preserving reductions to scheduling problems with communication delays, including the job-job delay model [DKRSTZ22] and the job-machine delay model [RSY23]. Despite its central role, the approximability of UMPS has remained poorly understood: even for unit-length jobs, known scheduling techniques do not seem to yield a non-trivial approximation, and the existence of a polylogarithmic approximation was left open by [DKRSTZ22]. On the hardness side, the previous best lower bound for unit-length jobs was only the 5/4 inherited from job shop scheduling [WHHHLSS97]. We prove that unit-length UMPS is NP-hard to approximate within any constant factor. We further show that, assuming NP is not in quasi-polynomial time, unit-length UMPS admits no polynomial-time $(\log n)^\gamma$-approximation for some constant $\gamma>0$. Via the known reductions from UMPS, these lower bounds also transfer to the corresponding unit-length communication-delay scheduling models. Our proof proceeds via a reduction from a hypergraph coloring promise problem. In the yes case, the input hypergraph admits a balanced coloring, while in the no case, the hypergraph has no large independent set. Instantiating this reduction with the hardness of [GL18] gives arbitrary constant-factor inapproximability, while combining the $4$-colorable $4$-uniform hypergraph coloring hardness of [GHHSV17] with a certain composition operation for hypergraphs yields the polylogarithmic factor inapproximability.

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arXiv2026

On the Approximability of Parameterized Minimum Monotone Satisfying Assignment

The parameterized Minimum Monotone Satisfying Assignment ($k$-MMSA) problem asks whether a monotone Boolean circuit admits a satisfying assignment of Hamming weight at most $k$. The MMSA hierarchy is defined by allowing a bounded number of alternations between AND and OR gates in the circuit. While the polynomial-time approximability of the MMSA hierarchy has been studied extensively, much less is known in the parameterized setting. In particular, $k$-MMSA$_2$ is the well-known $k$-SetCover problem, whose parameterized inapproximability lies in the $\text{polylog}(n)$ regime. In contrast, $k$-MMSA$_4$ captures $k$-MinLabel, for which known lower bounds give $\text{poly}(n)$ inapproximability. Sandwiched by $k$-MMSA$_2$ and $k$-MMSA$_4$, the inapproximability of $k$-MMSA$_3$ remained comparatively unexplored. In this paper, we give an FPT-time $O(2^k \log n)$-approximation algorithm for $k$-MMSA$_3$, suggesting that in the fixed-parameter regime, the third level of MMSA remains surprisingly close to the second level. Complementing this algorithm, we also give an FPT-time gap-preserving reduction from $k$-MMSA$_3$ to $k$-MMSA$_2$. Thus, stronger inapproximability for $k$-MMSA$_3$ would imply new hardness for $k$-MMSA$_2$, potentially offering a route around the current barriers for the latter problem. Revisiting Marx's reduction from $k$-MMSA$_t$ to gap $k$-MMSA$_{t+2}$, we also show that $k$-MMSA$_4$ admits no $n^{o(1)}$-factor FPT approximation unless W[2]=FPT, and no $n^{O(1/k)}$-factor approximation running in $n^{o(k)}$ time under ETH. These results separate the parameterized approximability behavior of the third and fourth levels and clarify where stronger inapproximability enters the $k$-MMSA hierarchy.

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arXiv2026

Strong Inapproximability for a Promise Rank Problem

Given a linear subspace of $n \times n$ matrices over $\mathbb F_{2^r}$ that is promised to contain a matrix of rank $1$, we prove that it is hard to find a matrix of rank $n^{o(1/\log \log n)}$, assuming NP doesn't have sub-exponential algorithms. In addition to being a basic problem, the hardness of this problem, even for the exact version, drove recent PCP-free inapproximability results for minimum distance and shortest vector problems concerning codes and lattices. The proof combines the concept of superposition soundness introduced by Khot and Saket with moment matrices. To produce a rank-gap of $1$ vs. $k$, the reduction runs in time $n^{O(\log k)}$. We also give another moment-matrix-based construction which runs in time $n^{O(k)}$ but works for any finite field $\mathbb F_q$.

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arXiv2026

Scaling Reasoning Tokens via RL and Parallel Thinking: Evidence From Competitive Programming

We study how to scale reasoning token budgets for competitive programming through two complementary approaches: training-time reinforcement learning (RL) and test-time parallel thinking. During RL training, we observe an approximately log-linear relationship between validation accuracy and the average number of generated reasoning tokens over successive checkpoints, and show two ways to shift this training trajectory: verification RL warmup raises the starting point, while randomized clipping produces a steeper trend in the observed regime. As scaling single-generation reasoning during RL quickly becomes expensive under full attention, we introduce a multi-round parallel thinking pipeline that distributes the token budget across threads and rounds of generation, verification, and refinement. We train the model end-to-end on this pipeline to match the training objective to the test-time structure. Starting from Seed-OSS-36B, the full system with 16 threads and 16 rounds per thread matches the underlying RL model's oracle pass@16 at pass@1 using 7.6 million tokens per problem on average, and surpasses GPT-5-high on 456 hard competitive programming problems from AetherCode.

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arXiv2025

Scheduling Problems with Constrained Rejections

We study bicriteria versions of Makespan Minimization on Unrelated Machines and Santa Claus by allowing a constrained number of rejections. Given an instance of Makespan Minimization on Unrelated Machines where the optimal makespan for scheduling $n$ jobs on $m$ unrelated machines is $T$, (Feige and Vondr\'ak, 2006) gave an algorithm that schedules a $(1-1/e+10^{-180})$ fraction of jobs in time $T$. We show the ratio can be improved to $0.6533>1-1/e+0.02$ if we allow makespan $3T/2$. To the best our knowledge, this is the first result examining the tradeoff between makespan and the fraction of scheduled jobs when the makespan is not $T$ or $2T$. For the Santa Claus problem (the Max-Min version of Makespan Minimization), the analogous bicriteria objective was studied by (Golovin, 2005), who gave an algorithm providing an allocation so a $(1-1/k)$ fraction of agents receive value at least $T/k$, for any $k \in \mathbb{Z}^+$ and $T$ being the optimal minimum value every agent can receive. We provide the first hardness result by showing there are constants $\delta,\varepsilon>0$ such that it is NP-hard to find an allocation where a $(1-\delta)$ fraction of agents receive value at least $(1-\varepsilon) T$. To prove this hardness result, we introduce a bicriteria version of Set Packing, which may be of independent interest, and prove some algorithmic and hardness results for it. Overall, we believe these bicriteria scheduling problems warrant further study as they provide an interesting lens to understand how robust the difficulty of the original optimization goal might be.

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arXiv2025

PCPP-Based Reconfiguration Inapproximability: Query Complexity vs. Soundness Gap Trade-offs

The Reconfiguration Inapproximability Hypothesis (RIH), recently established by Hirahara-Ohsaka (STOC'24) and Karthik-Manurangsi (ECCC'24), studies the hardness of reconfiguring one solution into another in constraint satisfaction problems (CSP) when restricted to approximate intermediate solutions. In this work, we make a tighter connection between RIH's soundness gap and that of probabilistically checkable proofs of proximity (PCPP). Consequently, we achieve an improved trade-off between soundness and query complexity in Gap CSP Reconfiguration. Our approach leverages a parallelization framework, which also appears in some recent parameterized inapproximability results.

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arXiv2025

PCP-free APX-Hardness of Nearest Codeword and Minimum Distance

We give simple deterministic reductions demonstrating the NP-hardness of approximating the nearest codeword problem and minimum distance problem within arbitrary constant factors (and almost-polynomial factors assuming NP cannot be solved in quasipolynomial time). The starting point is a simple NP-hardness result without a gap, and is thus "PCP-free." Our approach is inspired by that of Bhattiprolu and Lee [BL24] who give a PCP-free randomized reduction for similar problems over the integers and the reals. We leverage the existence of $\varepsilon$-balanced codes to derandomize and further simplify their reduction for the case of finite fields.

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arXiv2024

Inapproximability of Finding Sparse Vectors in Codes, Subspaces, and Lattices

Finding sparse vectors is a fundamental problem that arises in several contexts including codes, subspaces, and lattices. In this work, we prove strong inapproximability results for all these variants using a novel approach that even bypasses the PCP theorem. Our main result is that it is NP-hard (under randomized reductions) to approximate the sparsest vector in a real subspace within any constant factor; the gap can be further amplified using tensoring. Our reduction has the property that there is a Boolean solution in the completeness case. As a corollary, this immediately recovers the state-of-the-art inapproximability factors for the shortest vector problem (SVP) on lattices. Our proof extends the range of $\ell_p$ (quasi) norms for which hardness was previously known, from $p\geq 1$ to all $p\geq 0$, answering a question raised by [Khot05]. Previous hardness results for SVP, and the related minimum distance problem (MDP) for error-correcting codes, all use lattice/coding gadgets that have an abundance of codewords in a ball of radius smaller than the minimum distance. In contrast, our reduction only needs many codewords in a ball of radius slightly larger than the minimum distance. This enables an easy derandomization of our reduction for finite fields, giving a new elementary proof of deterministic hardness for MDP. We believe this weaker density requirement might offer a promising approach to showing deterministic hardness of SVP, a long elusive goal. The key technical ingredient underlying our result for real subspaces is a proof that in the kernel of a random Rademacher matrix, the support of any two linearly independent vectors have very little overlap. A broader motivation behind this work is the development of inapproximability techniques for problems over the reals. We hope that the approach we develop could enable progress on analytic variants of sparsest vector.

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arXiv2024

Almost Optimal Time Lower Bound for Approximating Parameterized Clique, CSP, and More, under ETH

The Parameterized Inapproximability Hypothesis (PIH), which is an analog of the PCP theorem in parameterized complexity, asserts that, there is a constant $\varepsilon> 0$ such that for any computable function $f:\mathbb{N}\to\mathbb{N}$, no $f(k)\cdot n^{O(1)}$-time algorithm can, on input a $k$-variable CSP instance with domain size $n$, find an assignment satisfying $1-\varepsilon$ fraction of the constraints. A recent work by Guruswami, Lin, Ren, Sun, and Wu (STOC'24) established PIH under the Exponential Time Hypothesis (ETH). In this work, we improve the quantitative aspects of PIH and prove (under ETH) that approximating sparse parameterized CSPs within a constant factor requires $n^{k^{1-o(1)}}$ time. This immediately implies that, assuming ETH, finding a $(k/2)$-clique in an $n$-vertex graph with a $k$-clique requires $n^{k^{1-o(1)}}$ time. We also prove almost optimal time lower bounds for approximating $k$-ExactCover and Max $k$-Coverage. Our proof follows the blueprint of the previous work to identify a "vector-structured" ETH-hard CSP whose satisfiability can be checked via an appropriate form of "parallel" PCP. Using further ideas in the reduction, we guarantee additional structures for constraints in the CSP. We then leverage this to design a parallel PCP of almost linear size based on Reed-Muller codes and derandomized low degree testing.

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arXiv2024

Baby PIH: Parameterized Inapproximability of Min CSP

The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only $(1-\varepsilon)$-satisfiable (where the parameter is the number of variables) for some constant $0<\varepsilon<1$. We consider a minimization version of CSPs (Min-CSP), where one may assign $r$ values to each variable, and the goal is to ensure that every constraint is satisfied by some choice among the $r \times r$ pairs of values assigned to its variables (call such a CSP instance $r$-list-satisfiable). We prove the following strong parameterized inapproximability for Min CSP: For every $r \ge 1$, it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even $r$-list-satisfiable. We refer to this statement as "Baby PIH", following the recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the combinatorial arguments underlying the Baby PCP theorem, overcoming some basic obstacles that arise in the parameterized setting. Furthermore, our reduction runs in time polynomially bounded in both the number of variables and the alphabet size, and thus implies the Baby PCP theorem as well.

cs.CC↗
arXiv2023

On Inapproximability of Reconfiguration Problems: PSPACE-Hardness and some Tight NP-Hardness Results

Recently, Ohsaka [STACS'23] put forth the Reconfiguration Inapproximability Hypothesis (RIH), which roughly asserts that there is some $\epsilon>0$ such that given as input a $k$-CSP instance (for some constant $k$) over some constant sized alphabet, and two satisfying assignments $\psi_s$ and $\psi_t$, it is PSPACE-hard to find a sequence of assignments starting from $\psi_s$ and ending at $\psi_t$ such that every assignment in the sequence satisfies at least $(1-\epsilon)$ fraction of the constraints and also that every assignment in the sequence is obtained by changing its immediately preceding assignment (in the sequence) on exactly one variable. Assuming RIH, many important reconfiguration problems have been shown to be PSPACE-hard to approximate by Ohsaka [STACS'23; SODA'24]. In this paper, we provide a proof of RIH. Our proof uses known constructions of PCP of Proximity to create the gap, and further leverages a parallelization framework from recent parameterized inapproximability results to analyze the quantitative trade-off between $\epsilon$ and $k$ in RIH. We note that Hirahara and Ohsaka [STOC'24] have also independently proved RIH. We also prove that the aforementioned $k$-CSP Reconfiguration problem is NP-hard to approximate to within a factor of $1/2 + \epsilon$ (for any $\epsilon>0$) when $k=2$. We complement this with a polynomial time $(1/2 - \epsilon)$-approximation algorithm, which improves upon a $(1/4 - \epsilon)$-approximation algorithm of Ohsaka [2023] (again for any $\epsilon>0$). Finally, we show that Set Cover Reconfiguration is NP-hard to approximate to within a factor of $2 - \epsilon$ for any constant $\epsilon > 0$, which matches the simple linear-time 2-approximation algorithm by Ito et al. [TCS'11].

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arXiv2023

Parameterized Inapproximability Hypothesis under ETH

The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an $\varepsilon$ fraction of constraints for some absolute constant $\varepsilon > 0$. PIH plays the role of the PCP theorem in parameterized complexity. However, PIH has only been established under Gap-ETH, a very strong assumption with an inherent gap. In this work, we prove PIH under the Exponential Time Hypothesis (ETH). This is the first proof of PIH from a gap-free assumption. Our proof is self-contained and elementary. We identify an ETH-hard CSP whose variables take vector values, and constraints are either linear or of a special parallel structure. Both kinds of constraints can be checked with constant soundness via a "parallel PCP of proximity" based on the Walsh-Hadamard code.

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arXiv2023

Improved Hardness of Approximating k-Clique under ETH

In this paper, we prove that assuming the exponential time hypothesis (ETH), there is no $f(k)\cdot n^{k^{o(1/\log\log k)}}$-time algorithm that can decide whether an $n$-vertex graph contains a clique of size $k$ or contains no clique of size $k/2$, and no FPT algorithm can decide whether an input graph has a clique of size $k$ or no clique of size $k/f(k)$, where $f(k)$ is some function in $k^{1-o(1)}$. Our results significantly improve the previous works [Lin21, LRSW22]. The crux of our proof is a framework to construct gap-producing reductions for the $k$-Clique problem. More precisely, we show that given an error-correcting code $C:Σ_1^k\toΣ_2^{k'}$ that is locally testable and smooth locally decodable in the parallel setting, one can construct a reduction which on input a graph $G$ outputs a graph $G'$ in $(k')^{O(1)}\cdot n^{O(\log|Σ_2|/\log|Σ_1|)}$ time such that: $\bullet$ If $G$ has a clique of size $k$, then $G'$ has a clique of size $K$, where $K = (k')^{O(1)}$. $\bullet$ If $G$ has no clique of size $k$, then $G'$ has no clique of size $(1-\varepsilon)\cdot K$ for some constant $\varepsilon\in(0,1)$. We then construct such a code with $k'=k^{Θ(\log\log k)}$ and $|Σ_2|=|Σ_1|^{k^{0.54}}$, establishing the hardness results above. Our code generalizes the derivative code [WY07] into the case with a super constant order of derivatives.

cs.CC↗
arXiv2022

Constant Approximating Parameterized $k$-SetCover is W[2]-hard

In this paper, we prove that it is W[2]-hard to approximate k-SetCover within any constant ratio. Our proof is built upon the recently developed threshold graph composition technique. We propose a strong notion of threshold graphs and use a new composition method to prove this result. Our technique could also be applied to rule out polynomial time $o\left(\frac{\log n}{\log \log n}\right)$ ratio approximation algorithms for the non-parameterized k-SetCover problem with $k$ as small as $O\left(\frac{\log n}{\log \log n}\right)^3$, assuming W[1]$\neq$FPT. We highlight that our proof does not depend on the well-known PCP theorem, and only involves simple combinatorial objects.

cs.DS↗
arXiv2022

On Lower Bounds of Approximating Parameterized $k$-Clique

Given a simple graph $G$ and an integer $k$, the goal of $k$-Clique problem is to decide if $G$ contains a complete subgraph of size $k$. We say an algorithm approximates $k$-Clique within a factor $g(k)$ if it can find a clique of size at least $k / g(k)$ when $G$ is guaranteed to have a $k$-clique. Recently, it was shown that approximating $k$-Clique within a constant factor is W[1]-hard [Lin21]. We study the approximation of $k$-Clique under the Exponential Time Hypothesis (ETH). The reduction of [Lin21] already implies an $n^{Ω(\sqrt[6]{\log k})}$-time lower bound under ETH. We improve this lower bound to $n^{Ω(\log k)}$. Using the gap-amplification technique by expander graphs, we also prove that there is no $k^{o(1)}$ factor FPT-approximation algorithm for $k$-Clique under ETH. We also suggest a new way to prove the Parameterized Inapproximability Hypothesis (PIH) under ETH. We show that if there is no $n^{O(\frac{k}{\log k})}$ algorithm to approximate $k$-Clique within a constant factor, then PIH is true.

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arXiv2020

Generalized Sorting with Predictions

Generalized sorting problem, also known as sorting with forbidden comparisons, was first introduced by Huang et al. together with a randomized algorithm which requires $\tilde O(n^{3/2})$ probes. We study this problem with additional predictions for all pairs of allowed comparisons as input. We propose a randomized algorithm which uses $O(n \log n+w)$ probes with high probability and a deterministic algorithm which uses $O(nw)$ probes, where $w$ is the number of mistakes made by prediction.

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