Papers

We present a new generic transformation from weak PRFs computable in depth $d(n) = \Omega(\log n)$ to strong PRFs computable in depth $O(d(n))$. This construction refines the classical tree-based paradigm of GGM by {tapering} the internal state so the per-level depth decreases geometrically. We complement the above with new depth-efficient weak PRF constructions based on various standard assumptions. As a corollary, we obtain new $\mathsf{NC}^1$-computable PRFs from various classical assumptions, resolving several long-standing open problems. Concretely, for the first time, we obtain $\mathsf{NC}^1$-computable PRFs: (1) from the Learning With Errors (LWE) assumption with a polynomial modulus-to-noise ratio, improving upon prior low-depth constructions that required Ring-LWE with super-polynomial ratios [Banerjee-Peikert-Rosen, EUROCRYPT 2012]; (2)from the standard Learning Parity with Noise (LPN) assumption, removing the need for structured LPN variants [Boyle et al., FOCS 2020], [Ding-Jain-Komargodski, STOC 2025]; (3) from the Computational Diffie-Hellman (CDH) assumption; prior works relied on the stronger Decisional Diffie-Hellman (DDH) or generalized Diffie-Hellman (GDH) assumptions [Naor-Reingold, FOCS '97, J. ACM '04].

The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomassé; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.

Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult even for shallow ReLU networks. We study this problem for two-layer input-convex neural networks (ICNNs), a restricted architecture where nonnegative output weights enforce convexity. Computing the $L_p$-Lipschitz constant for these networks is equivalent to maximizing the dual norm over a zonotope. While $L_1$- and $L_\infty$-norm maximization on zonotopes admit fixed-parameter and polynomial-time algorithms, respectively, the parameterized complexity of the remaining $L_p$-norms was open. We prove that, for every fixed $p\in (1,\infty)\cap \mathbb{Q}$, maximizing the $L_p$-norm over a zonotope in $\mathbb{R}^d$ is W[1]-hard with respect to the dimension $d$. Moreover, our hardness results imply that brute-force enumeration algorithms are essentially optimal for this problem under the Exponential Time Hypothesis. By duality, the same hardness results hold for computing the $L_p$-Lipschitz constant of two-layer ReLU ICNNs. Our proof first establishes the result for the $L_2$-norm and then transfers the construction to arbitrary fixed $p\in (1,\infty)\cap\mathbb{Q}$ using a suitable Taylor approximation. These results resolve the corresponding questions regarding the parameterized complexity status for zonotope norm maximization and two-layer ICNN Lipschitz constants. Our paper resolves an open problem posted at COLT'25. There are several independent concurrent papers resolving the same problem. Our paper prioritizes a clear exposition of the underlying mathematics and conceptual intuitions behind the proof. Additionally, we explicitly describe our research process including the use of LLMs.

Many problems in the verification of recursive programs can be reduced to pushdown model checking. In this problem, we are given as input a pushdown automaton (PDA) over a constant-sized stack alphabet, and a description of undesirable behaviors given by an intersection of NFAs, and the problem is to decide if there is a behavior of the PDA that belongs to the set of undesirable behaviors. It is well-known that there is an algorithm for this problem that runs in time $O(n^{2k} |\Sigma| + n^{3k})$, where $n$ is the maximum number of states of the PDA and the NFAs, $\Sigma$ is the common input alphabet, and $k-1$ is the number of NFAs. Despite the importance of this problem, no better algorithm is known for it. In this paper, we provide an explanation for this lack of progress using the lens of fine-grained complexity theory. We prove that if the $3k$-Clique hypothesis (resp. combinatorial $3k$-Clique hypothesis) is true, then for any $\epsilon > 0$, there is no algorithm (resp. combinatorial algorithm) that solves this problem in time $O((n^{(\omega-1)k} |\Sigma| + n^{\omega k})^{1-\epsilon})$ (resp. $O((n^{2k} |\Sigma| + n^{3k})^{1-\epsilon})$) where $\omega$ is the matrix multiplication exponent. Furthermore, using the combinatorial hypothesis, we also show that pushdown model checking over constant-sized input alphabets cannot be solved in time faster than $O(n^{3(k-1)-\epsilon})$ for any $\epsilon > 0$. Finally, we investigate the possibility of an $O(N^{3k-\epsilon})$ time algorithm for this problem where $N$ is the total bit size of the input. We formulate a new hypothesis, the 2NPDA$(k)$ hypothesis, that helps explain the lack of $O(N^{3k-\epsilon})$ time algorithms for this problem. To corroborate this hypothesis, we show a web of linear-time reductions between the 2NPDA$(k)$ hypothesis, pushdown model checking, and other problems in formal language and automata theory.

It is shown in this manuscript that a random graph $G$ drawn from the Erdős–Rényi model $\mathcal{G}(n,p)$ with \[ p=p(n)\leq 1/2, \qquad \lim_{n\to+\infty}(np-\log n-\log\log n)=+\infty, \] is a homomorphic core, i.e., every homomorphism from $G$ to itself is an automorphism. This implies tight ETH-based lower bounds of the subgraph isomorphism problem for almost all $k$-vertex patterns with polynomial average degree.

In the problem of ground-state energy estimation, one aims to estimate the smallest eigenvalue of a Hamiltonian, often given a guiding state, with some promised overlap $\gamma$ with the ground space. The main approach to this problem is to simulate its evolution, and estimate the smallest (or equivalently, largest) eigenphase of the resulting unitary $U$. We give a quantum algorithm that estimates the largest eigenphase of a unitary $U$ in this guided setting using a factor of $\log\frac{1}{\gamma}$ fewer queries to $U$ than the previous best approach. The result matches an existing lower bound, and answers an open question from Mande and de Wolf. The algorithm is based on transducers, which often allow composition of quantum algorithms without overhead from error reduction.

Suppose we can apply the unitary $U=e^{i H}$ for some Hamiltonian $H$, and are given access to a unitary that prepares a guiding state promised to have overlap at least $\gamma >0$ with the ground space of $H$. Our goal is to estimate the ground-state energy of $H$ within additive error $\delta > 0$ and success probability at least $1-\varepsilon$, $\varepsilon>0$. How many applications of $U$ and its inverse $U^{-1}$ are necessary and sufficient? This quantity corresponds to the total Hamiltonian-simulation time needed. An upper bound $O(\log(1/\varepsilon)\log(1/\gamma)/\gamma\delta)$ was known, and was improved to $O(\log(1/\varepsilon)/\gamma\delta)$ very recently [JW26]. A matching lower bound was known whenever one of the three parameters $\delta,\gamma,\varepsilon$ was held constant [MdW26]. In this paper we prove the joint lower bound $\Omega(\log(1/\varepsilon)/\gamma\delta)$ with the tight $\varepsilon$-dependence provided the dimension of $H$ is at least $\log(1/\varepsilon)/\gamma^2$. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and $H$ has a gap of $\delta$ between its first and second eigenvalue; and for ground-state preparation, where $\delta$ denotes the spectral gap and $\varepsilon$ now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of $U$ are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when $H$ is nonnegative and presented as a sum of squares; and our results imply the lower bound $\Omega(\log(1/\varepsilon)/\gamma\sqrt{\delta})$ for this case.

The Post Correspondence Problem is as follows: having a set of dominoes, is there any (maybe repeating) sequence of them such that the words formed by the upper parts and the lower parts by the sequence of dominoes are identical. It is one of the most known problems that is algorithmically undecidable. In this paper, the reverse Post Correspondence Problem is defined, that is, where the two assembled words of the dominos are reversals of each other. Undecidability about this new, modified problem is proven. Further, based on this result, it is also proven that the emptiness problem for 5'->3' String Assembly Systems is undecidable too. 5'->3' String Assembly Systems belong to String Assembly Systems type formal language generating models. The 5'->3' denotes that, in these variants, the derivations of the generated words start from the two extremes. The notation and the new model are bio-motivated as any double stranded DNA has two opposite oriented 5'->3' strands.

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $\rho$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-\rho} \] with success probability $1-\delta$. Provided that $\rho$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/\delta))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

This paper proposes the Coronavirus Optimization Algorithm (COA), a SARS-CoV-2-inspired success-history adaptive evolutionary optimizer for box-constrained continuous global optimization. COA does not model disease transmission; instead, it maps selected coronavirus mechanisms to explicit search operators, including elite-guided attraction, trial-vector generation, adaptive parameter variation, stagnation recovery, and population-size scheduling. The algorithm combines opposition-based initialization, current-to-pbest mutation, binomial crossover, an external archive, success-history adaptation, population reduction, and partial restart. COA is evaluated on 29 CEC 2017 benchmark functions at 10, 30, and 50 dimensions against 15 competitive optimizers. Results show that COA achieves the best overall Friedman rank across all dimensions, with particularly strong performance on composition functions. The findings demonstrate that COA is a compact, transparent, and competitive adaptive evolutionary optimizer, while also highlighting limitations on some hybrid functions and the need for further high-dimensional validation.

We prove that exact Euclidean decision-CVP is $\mathsf{NP}$-complete on the coefficient lattices of nonzero principal ideals in the power-of-two cyclotomic rings $R_d:=\mathbb{Z}[y]/(y^d+1)$. Our deterministic reduction from Exact Cover by 3-Sets (X3C) produces a target and a squared threshold $\Delta$ such that the closest squared distance is exactly $\Delta$ in YES instances and at least $\Delta+4$ in NO instances. This also implies $\mathsf{NP}$-hardness of exact search-CVP under polynomial-time Turing reductions. We also transfer the resulting principal-ideal CVP instances to full-rank principal ideals of the cyclic quotient ring $\mathbb{Z}[X]/(X^D-1)$, where $D:=2d$. Their coefficient lattices are invariant under cyclic coordinate shifts. The lift preserves principality and multiplies corresponding squared distances by eight. Thus, on principal cyclic ideal lattices, exact decision-CVP is $\mathsf{NP}$-complete and exact search-CVP is $\mathsf{NP}$-hard. We also obtain uniformly computable fixed cyclotomic and cyclic families in which only the target and threshold depend on the X3C collection. Consequently, a polynomial-time solution to exact decision-CVPP on either family would imply $\mathsf{NP}\subseteq\mathsf{P}/\mathrm{poly}$ and collapse the polynomial hierarchy to $\Sigma_2^{\mathsf{P}}$. To our knowledge, the cyclic results answer Micciancio's questions of whether exact decision-CVP is $\mathsf{NP}$-hard on cyclic lattices and on a fixed family of cyclic lattices, even when restricted to full-rank principal cyclic ideals. Finally, under the coefficient embedding, we prove that exact decision-module-SIVP is $\mathsf{NP}$-complete on free rank-two modules over the same cyclotomic rings.

We prove that minimum Riesz $s$-energy subset selection in the Euclidean plane is NP-complete already for the fixed exponent $s=2$. To our knowledge, this is the first Euclidean hardness result for exact Riesz-energy subset selection in which both the ambient dimension and the exponent are fixed. The reduction uses Barahona's planar cubic Ising model with uniform field. A spin is encoded by one diagonal of a four-point square. Axis-aligned selector chains implement ferromagnetic consistency, while a $45^\circ$ terminal geometry yields an antiferromagnetic source interaction. Rational diagonal perturbations realize the magnetic field, and all remaining interactions are dominated by polynomial separation. Because $s=2$ and all coordinates are rational, every constructed energy and the decision threshold are rational exactly.

The 17th International Conference on Automata and Formal Languages (AFL 2026) was held in Košice, September 7-10, 2026. It was organized by the Faculty of Science, P. J. Šafárik University in Košice and the Mathematical Institute of the Slovak Academy of Sciences. This volume contains the texts of the three invited lectures and the 18 papers selected by the Program Committee from a total of 27 submissions. The accepted papers, submitted by authors from 14 countries, cover all important areas of automata and formal language theory and their applications such as grammars and acceptors for strings, trees, and graphs, algebraic theories and algorithms for automata and languages, decision problems, relations to complexity theory, picture description and analysis, and other fields.

We prove a common generalization of two structure theorems: the dimension-free decomposition theorem for idempotent Schur multipliers and the idempotent theorem in harmonic analysis. Roughly speaking, our result shows that an invariant integer-valued kernel with Hilbert-space factorization norm $\gamma$ admits a signed decomposition into at most $2^{O(\gamma^4)}$ elementary pieces. In the matrix setting these pieces are blocky matrices, while in the group setting they are indicators of cosets. In the locally compact abelian setting this quantitatively strengthens the theorem of Green and Sanders, while in the non-abelian setting it gives a quantitative strengthening of Host's idempotent theorem and, for finite groups, of Sanders's quantitative result. It also improves the exponent in the dimension-free matrix decomposition from $\gamma^6$ to $\gamma^4$.

We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - \epsilon$ is $(p, C_{p,q}/\epsilon)$-average-radius list-decodable with probability at least $1 - q^{-\Omega(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/\epsilon$ codewords closest to $y$ have average fractional Hamming distance at least $p$ from $y$. This extends a similar result for (standard) list-decoding due to Guruswami, Håstad, and Kopparty (2010) to the stronger average-radius guarantee, with the same $O(1/\epsilon)$ list size. For average-radius list-decoding, such a result was previously known only for binary linear codes (Guruswami, Li, Mosheiff, Resch, Silas, and Wootters, 2021) and for general (non-linear) random codes over arbitrary alphabets (Elias, 1991).

We develop a resource representation for step recursion in which mutable-state width and recursion descent are explicit and independent parameters. A width bound $u$ controls the size of the encoded machine state, while an effective descent $\rho$ determines the available recursion depth $\delta_\rho(u)$. For generalized-inverse descents, we derive the depth directly from generator growth and characterize the increasing sequences that can occur as generator orbits. We then connect this depth–width geometry to standard finite-branching computation. Every deterministic bounded-state dynamics is realizable by a single ordinary bounded step recursion over a fixed finite numerical basis. Using deterministic, existential, universal, or alternating aggregation on the same local dynamics yields the corresponding machine semantics. After closure under the width reparameterizations needed to absorb fixed local cost, the resulting language classes are exactly the machine time–space classes on profiles $(\delta_\rho(u),u)$. Finally, profile domination quotients effective descents by admissible width reparameterization. Some depth curves collapse, yet polynomial widths support an explicit infinite strict hierarchy between the canonical polynomial- and exponential-depth profiles. Thus descent remains a nonredundant resource coordinate after polynomial width reparameterization; standard complexity classes are calibration points.

We prove a strict space separation between static and ordinary dynamic approximate membership at every fixed error rate. For each fixed $\varepsilon\in(0,1)$, a capacity-$n$ ordinary dynamic filter over a universe of size $u$, with zero false negatives, pointwise false-positive probability at most $\varepsilon$, arbitrary history dependence, a free public random tape, and at most $H$ bits of persistent state, satisfies \[ H\ge \bigl(\log_2(1/\varepsilon)+a_\varepsilon^{\rm c}\bigr)n-o(n), \] under only $u/n\to\infty$. The constant $a_\varepsilon^{\rm c}$ is an explicit variational threshold obtained by preserving the dependence between the parent accepted mass and the successor reservoir. The structural step is a common-continuation transport lemma. A joint posterior KL bound gives a branch-specific survivor support; the same legal delete–insert word transports that support to one successor state, forcing an accepted reservoir. We then keep the parent outside mass $1-X$ in the conditional-entropy argument instead of replacing it by $1-\varepsilon$. This yields a two-variable analytic envelope, with no selected thresholds, dyadic witnesses, or numerical assumptions.

We show that approximately counting the vertices of a bounded 0/1 polytope, presented as a system of rational linear inequalities, is, informally speaking, NP-hard. In particular, there is no FPRAS for this problem unless RP=NP. The proof is by a reduction from approximately counting homomorphisms from a given graph to a particular four-vertex graph. The main proof ideas were found using GPT-5.6 Sol Ultra.

Let $k_{1,\varepsilon}(n)$ be the smallest number of real linear measurements needed by a randomized oblivious sketch that estimates the nuclear norm of every fixed real $n\times n$ matrix within a factor $1\pm\varepsilon$, with probability at least $2/3$. For every fixed $0<\varepsilon<1$, we prove \[ \frac{n^2}{(\log n)^{A_\varepsilon}} \le k_{1,\varepsilon}(n) \le C_\varepsilon \frac{n^2\{\log\log(e^e n)\}^2}{\log(e n)}. \] Previously, the best unrestricted bounds for general linear sketches of the Schatten–1 norm were $\Omega(n)$ and the trivial $O(n^2)$ upper bound (Li, Nguyen, Woodruff'19), leaving a polynomial gap. Our bounds close that gap up to polylogarithmic factors and give a nontrivial logarithmic saving below the $n^2$-measurement storage bound. The result extends much further. Write $k_{p,\varepsilon}(n)$ for the analogous sketch dimension for the Schatten–$p$ norm. For every fixed finite $p>0$ that is not a positive even integer, there are positive constants $A_{p,\varepsilon},C_{p,\varepsilon},c_p$ such that \[ \frac{n^2}{(\log n)^{A_{p,\varepsilon}}} \le k_{p,\varepsilon}(n) \le C_{p,\varepsilon}\frac{n^2}{(\log n)^{c_p}}, \] so $k_{p,\varepsilon}(n)=n^{2-o(1)}$ throughout the non-even regime. Together with the known tight bounds $\Theta_{p,\varepsilon}(n^{2-4/p})$ for positive even $p$ and $\Theta_\varepsilon(n^2)$ for $p=\infty$ (Li, Woodruff'16), our results close the remaining polynomial gap across the Schatten family and complete, up to polylogarithmic factors, the polynomial-order classification of general linear sketches for all Schatten-$p$ norms.

The house allocation problem is a classical one-sided matching problem that concerns the assignment of a set of $m$ houses to $n$ agents according to their preferences, where each agent is assigned exactly one house. Among the various objectives studied in this setting, envy-freeness is one of the most widely adopted fairness criteria. As envy-free house allocations do not always exist, we address this challenge by introducing subsidies and aim to compute allocations that achieve envy-freeness with minimum total subsidy. For binary instances, we show that a total subsidy of at most $(n-1)$ suffices to guarantee envy-freeness in house allocation, and this bound is tight. Building on the known NP-hardness for general utilities, we further show that computing an allocation that minimizes the total subsidy is NP-hard, even under binary utilities. However, when there are only a bounded number of types of agents with binary utilities, the problem can be solved in polynomial time. Finally, we present a polynomial time algorithm that computes the minimum subsidy required to achieve envy-freeness for two types of agents with general utilities.

Kuszmaul, Liang, and Zhou (SODA 2026) ask whether succinct constant-time dynamic fusion nodes exist when the number of stored keys is polylogarithmic in the universe size. We give a negative answer for always-correct structures. For n^8 <= U, log_2 U >= 2^70, and redundancy 0 <= R < n, a dynamic dictionary requires at least 2^-26 log_2(1+n/(R+1)) expected-amortized cell probes per operation. The model permits fixed layouts of packed cells of at most one word each, including the short-spill convention used by succinct word-RAM structures, and covers zero-error Las Vegas algorithms with fresh per-invocation randomness and almost-sure termination. The proof repairs a conditioning defect in the inherited communication argument by placing pointwise probe caps inside the consistency event, then extends the lower bound to large universes through a scale-adaptive entropy parameter. Consequently, when U=2^w and n=ceil(w^c) for any fixed c>0, no always-correct predecessor structure can use log_2 binom(U,n)+o(n) persistent mutable bits and support constant-time operations. Constant expected-amortized time requires Omega(n) redundant bits. Lean 4 checks the complete packed-memory and fresh-random indexed models, hard distribution, communication bounds, separator, nested-forest accounting, deterministic and Las Vegas lower bounds, strict-predecessor reduction, and redundancy corollaries.

We study the deterministic norm-query complexity of computational problems in finite-dimensional vector spaces over $\mathbb{Q}_p$ equipped with an arbitrary ultrametric norm. For orthogonalization, we prove that no uniform finite query bound depending only on the dimension exists: for every deterministic algorithm that produces an $N$-orthogonal basis for every ultrametric norm $N$, the number of norm queries is unbounded as $N$ varies. We then study the Longest Vector Problem (LVP) for a rank-$m$ $p$-adic lattice. By adapting a brute-force search to the general norm-query setting and eliminating the scalar redundancy among nonzero coefficient vectors modulo $p$, we obtain an algorithm using exactly $(p^m-1)/(p-1)$ norm queries for $m\ge 2$, and prove that no deterministic norm-query algorithm can use fewer queries in the worst case. Finally, we consider the Closest Vector Problem (CVP). Apart from the trivial cases in which no norm query is needed, we prove that the deterministic worst-case norm-query complexity of the CVP is unbounded, even when the lattice and the target vector are fixed.

We study fixed-layout local verification with target-dependent local tests. Let $M$ be a random variable on $\{0,1\}^K$, and let $S$ record the test selected at each coordinate. For each $s\in\operatorname{supp}(S)$, let $F_s$ be the corresponding target fiber and set $D_{\mathrm{fib}}=\max_s\operatorname{VCdim}(F_s)$. We prove $H(M\mid S)\le \log_2\!\left(\sum_{j=0}^{D_{\mathrm{fib}}}\binom Kj\right)$. A fiber that shatters $d$ coordinates yields a weak relaxed locally decodable code with message length $d$ and block length $d+P$ over the original proof alphabet. For a uniform $K$-bit target and fixed proof alphabet, $Q$, and $\sigma$, the Goldberg–Gur–Saraogi lower bound implies that $I(M;S)\le\gamma K$, for fixed $\gamma<1$, forces $P=\Omega\!\left(K^{1+1/a}/(\log K)^{2+2/a}\right)$, where $a=\lceil Q/\sigma\rceil$. If $P\le K(\log K)^c$, then $I(M;S)\ge K-O\!\left(K^{a/(a+1)}(\log K)^{3+ac/(a+1)}\right)=K-o(K)$. Any discrete verifier state $T$ determining $S$ satisfies the same information lower bound. Bounded-randomness adaptive branches can be simulated nonadaptively by exposing their decision trees. A branch using at most $r$ random bits and $q$ adaptive proof queries yields a decoder with perfect completeness and at most $1+2^{r+1}\sum_{j<q}A^j$ queries. Under $I(M;S)\le\gamma K$, near-linear proof length requires this quantity to be $\Omega(\log K/\log\log K)$; the binary one-query case gives $P=2^{\Omega(K)}$. Applied to a global list-sound dPCP interface of Gur–Minzer–Weissenberg–Zheng, our bound shows that a fixed target-independent menu of $L$ test profiles must satisfy $\log_2L\ge K-o(K)$. Proof-dependent lists and additional target-dependent selection data must be included in the measured state.

We study low-degree testing for group-valued functions over a Boolean slice. Specifically given a degree parameter $d$ and oracle access to a function $f:\{0,1\}^n_{n/2}\to G$ where $\{0,1\}^n_k$ denotes the set of vectors in $\{0,1\}^n$ of Hamming weight $k$ and $G$ is an Abelian group, the low-degree testing problem asks us to distinguish the case where $f$ is a polynomial of degree at most $d$ (with coefficients from $G$) or is $\varepsilon$-far from the set of all such polynomials. Classical works in this area considered functions with domain $\mathbb{F}_q^n$ and range $\mathbb{F}_q$. More recent works have considered the setting where the domain is the Boolean cube [Bafna, Srinivasan, Sudan (Random Struct. Algorithms 2020), Amireddy, Srinivasan, Sudan (RANDOM 2023)], or when the domain is the slice (i.e., $\{0,1\}^n_{k}$) and the range is $\mathbb{F}_2$ [David, Dinur, Goldenberg, Kindler and Shinkar (SIAM J. Comput. 2017), Kalai, Lifshitz, Minzer and Ziegler (FOCS 2024)]. Each of the changes introduces new challenges in designing and analyzing low-degree tests and this happens again in our setting with domain being a slice and range is general. Our main theorem gives a test that makes $O_d(1)$ queries to $f$ and accepts degree-$d$ functions while rejecting functions that are $\varepsilon$-far with probability $\Omega(\varepsilon)$. The central proof idea is to reduce this low-degree testing problem to the problem of low-degree testing on the cube. Specifically we show how to randomly embed the $n/2$-dimensional cube $\{0,1\}^{n/2}$ in the $n$-dimensional slice while nearly preserving the proximity of $f$ to the space of degree-$d$ polynomials on this cube. While the embedding is simple and natural, the analysis involves a careful induction with a novel use of a basis of degree-$d$ polynomials on slices (from a work of Anstee, Rónyai and Sali (Graphs and Combinatorics 2002)).

Anonymous Multi-Agent Path Finding (AMAPF) admits polynomial-time network-flow algorithms for several objectives, including makespan, total distance, and sum-of-costs (SoC) when agents disappear upon reaching goals. We show that standard goal-staying AMAPF is fundamentally different. We first formulate SoC minimization by augmenting the standard time-expanded flow model with goal-settlement constraints and show that the resulting linear programming relaxation is non-integral. We then prove that minimizing SoC in goal-staying AMAPF is NP-hard via a reduction from 3-SAT. Together with the polynomial-time result for the disappearing variant, this establishes a sharp complexity boundary determined by whether completed agents remain at their goals.

We give a catalytic logspace algorithm for the problem of graph spectral sparsification. Given an undirected graph $G$ on $n$ vertices and $\varepsilon>0$, our algorithm outputs an $\varepsilon$-spectral sparsifier of $G$ with $O(n\varepsilon^{-2}\log n)$ edges, matching the effective resistance sampling of Spielman and Srivastava (STOC 2008). This gives a new, natural problem in catalytic logspace that is not known to be in deterministic $\mathbf{NC}$ or $\mathbf{SC}$. Our main contribution is an entirely new technique in the compress–or–random paradigm for catalytic logspace that we believe will have further applications. We first analyze effective-resistance sparsification using a pessimistic estimator that can itself be computed in catalytic logspace. The estimator is motivated by the viewpoint of graph quasirandomness and immediately gives a simple, deterministic greedy algorithm for graph sparsification. Subsequently, we show that such a pessimistic estimator can be transformed into an algorithm that performs an in-place compression of a string with bad potential. Our algorithm is based on using the potential function to define a measure over strings, and implementing arithmetic coding using this measure in-place. This compression technique is substantially distinct from all prior tools in the field of catalytic computation.

Consider the following problem of learning an unknown linear order on $n$ items. In each round, the learner guesses a complete ordering of the items and receives either confirmation that the guess is correct or a counterexample: a pair of items in the wrong order. The goal is to identify the unknown order using as few queries as possible. We study this problem when up to $k$ of the returned counterexamples may be untruthful, where $k$ is not known in advance. We determine the optimal query complexity up to constant factors: \[ \Theta(n\log n + nk). \] Thus, while the noiseless complexity matches the classical complexity of sorting, each untruthful counterexample incurs an additional cost of order $n$. The upper bound is based on a geometric representation of permutations and Grünbaum's theorem, while the lower bound combines sorting arguments with a Condorcet-type construction. We also study the case where the target ranking has a low-dimensional geometric representation: each item is represented by a point in $\mathbb{R}^d$, and the ranking is obtained by projecting the points onto an unknown direction. For these classes we give an upper bound of $O(d^2\log n+dk)$ and a lower bound of $\Omega(d\log n+dk)$, leaving a factor of $d$ gap in the noiseless term.

An $st$-orientation of an undirected graph $G$ is an acyclic digraph with a single source $s$ and a single sink $t$ that can be obtained from $G$ by assigning a direction to each edge. The classical problem of deciding if an undirected graph $G$ has an $st$-orientation can be solved efficiently. On the other hand, deciding if an $st$-orientation of $G$ exists that does not have any transitive arc is NP-complete, even if each vertex of $G$ has degree at most four. Here we show that this last decision problem remains NP-complete if $G$ is cubic, which settles an open question by Binucci et al. (2025). We obtain NP-completeness for two variants of the problem: (i) $s$ and $t$ are fixed and given as part of the input and (ii) $s$ and $t$ can be chosen freely. We then use these results to investigate the computational complexity of a problem that arises in computational evolution. Specifically, we show that the problem of deciding if an unrooted binary phylogenetic network has an orientation as a rooted binary phylogenetic network without any shortcuts (the analog of a transitive arcs in phylogenetics) is NP-complete. Our results connect the two (mostly) distinct research areas of orienting undirected graphs and orienting unrooted phylogenetic networks.

Courcelle's theorem and its optimization variants yield fixed-parameter tractable algorithms for a wide range of graph problems on graphs of bounded treewidth or clique-width. However, the limited counting power of $\mathsf{CMSO}$ poses an obstacle to capturing certain optimization problems and properties within this framework. We introduce a new logic $\mathsf{AmCMSO}$, which extends $\mathsf{CMSO}$ with predicates for membership in the families of minimum- and maximum-cardinality sets satisfying a fixed formula $\phi(X)$. In contrast to most previous extensions of $\mathsf{CMSO}$ with cardinality constraints, we give algorithmic meta-theorems based on fixed-parameter tractable model checking for $\mathsf{AmCMSO}_1$ parameterized by clique-width and the formula, and for $\mathsf{AmCMSO}_2$ parameterized by treewidth and the formula. Our proof is based on the combination of Feferman–Vaught-type decomposition and fundamental techniques for dynamic programming. The meta-theorems yield fixed-parameter tractable algorithms for a wide range of optimization problems involving optimal solutions, including network interdiction, pre-assignment for solution uniquification, and diversity maximization, without parameterizing by the optimum value. Finally, allowing an optimality predicate to depend on even one external set variable makes model checking hard for every level of the polynomial hierarchy, already on trees of depth four.

Kubernetes is the de-facto platform for container orchestration. Its scheduler combines resource capacities with label-based affinity and anti-affinity rules, and the interaction of these features can make the eventual placement of a pod. In this paper, we study the pod-deployability problem: given an initial cluster, a pod type, and a designated node, does some legal sequence of pod deployments and deletions cover the target pair? We give three complexity results. First, when dynamic constraints contain no affinity (anti-affinity is allowed), pod-deployability is decidable in polynomial time. Second, required affinity together with required anti-affinity makes the problem PSPACE-complete. Third, required affinity alone is already enough for PSPACE-completeness on a single node with one scalar capacity. The lower bounds encode, respectively, 1-safe Petri-net coverability and bounded black pebbling. These results isolate two independent sources of state-space complexity in Kubernetes scheduling: logical exclusion and resource-bounded prerequisite management.