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arXiv2026

Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph $F \in \mathcal{F}$. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems $\textrm{HomInd}(\mathcal{F})$ of deciding homomorphism indistinguishability over $\mathcal{F}$ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of $\textrm{HomInd}(\mathcal{F})$, Seppelt (MFCS 2024) showed that $\textrm{HomInd}(\mathcal{F})$ is in randomised polynomial time for every graph class $\mathcal{F}$ of bounded treewidth that can be defined in counting monadic second-order logic $\mathsf{CMSO}_2$. We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in $\mathsf{PTIME}$. For $\mathsf{CMSO}_2$-definable graph classes $\mathcal{F}$ of bounded pathwidth, we improve the previous complexity upper bound for $\textrm{HomInd}(\mathcal{F})$ from $\mathsf{PTIME}$ to $\mathsf{C}_=\mathsf{L}$ and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as $\mathsf{C}_=\mathsf{L}$-complete.

cs.CC↗
arXiv2026

Ancilla-Efficient QSAMPLE Preparation for Reversible Markov Chains

Preparing quantum samples (QSAMPLES), coherent encodings of stationary distributions of reversible Markov chains, is a fundamental primitive in quantum sampling, particularly for quantum simulated annealing. A central limitation of existing phase-estimation-based frameworks is the ancilla qubit overhead. In this work, we present a new end-to-end framework requiring only one ancilla qubit in the working register. The key technical ingredient is a selective phase compiler circuit using one ancilla qubit, built from a generalized quantum signal processing (GQSP)-based projector onto the 1-eigenspace of the qubitized Szegedy walk. Embedding these selective phase compilers into the fixed-point amplitude amplification (FPAA) procedure and iterating yields a quantum algorithm that, given an initial state, oracle access, lower bounds on the overlaps between adjacent states, and lower bounds on the phase gaps, outputs a QSAMPLE within any desired trace distance and thus total variation distance. The query complexity scales inversely with the square roots of both the minimum overlap and the minimum spectral gap of the Markov chains across the cooling schedule, up to polylogarithmic factors. We also perform simulations to verify how our qubit and query complexity evolve with the trace distance, and how this work compares to the previous framework. These results establish two improvements over the previous framework by Wocjan and Abeyesinghe. First, the working-register ancilla cost is reduced to one. Second, by inserting our GQSP-based selective phase compiler into the FPAA procedure, we improve the QSAMPLE transport overlap dependence from inverse minimum overlap to inverse square-root minimum overlap, relative to their Grover pi-over-three fixed-point method. Finally, as a direct application, we apply the quantum algorithm to prepare a Gibbs QSAMPLE and obtain a rigorous complexity analysis.

quant-ph↗
arXiv2026

Coresets for Continuous $k$-Center in Hyperbolic Space

We study the continuous $k$-center problem in fixed-dimensional hyperbolic space. The input is a finite point set $P \subset \mathbb H^D$, while the centers may be placed anywhere in the ambient space. A nonempty subset $P_\varepsilon\subseteq P$ is an $\varepsilon$-coreset if every exact optimal continuous $k$-center solution for $P_\varepsilon$ is a $(1+\varepsilon)$-approximate solution for $P$. Our construction uses Gonzalez's farthest-first traversal, whose radius $R_G$ gives a constant-factor estimate of the optimum. When $R_G$ is bounded, we map each cluster to the origin of the Poincaré ball model and apply a local Euclidean grid. For larger $R_G$, we partition each cluster into radial shells and project them onto representative hyperbolic spheres, avoiding a discretization of the ambient ball. By the hyperbolic law of cosines, the intersection of a hyperbolic ball with each sphere corresponds to a spherical cap. A recursive $k$-cap coverage certificate then replaces each shell with a radius-independent set of input witnesses that preserves its farthest-point behavior for every set of at most $k$ centers. The resulting coreset $P_\varepsilon$ has size $|P_\varepsilon| = \left({k}/{\varepsilon}\right)^{O(kD)}$ and satisfies $Φ_P(C_{P_\varepsilon}) \le (1+\varepsilon)\operatorname{OPT}_k(P)$ for every optimal continuous $k$-center solution $C_{P_\varepsilon}$ of $P_\varepsilon$. It can be constructed in $O\left( nkD\left({k}/{\varepsilon}\right)^{O(kD)} \right)$ time. Consequently, for fixed $D$, $k$, and $\varepsilon$, the coreset has constant size and can be computed in linear time. We complement this upper bound by showing that, for every fixed $D \ge 1$, every $k \ge 2$, and all sufficiently small $\varepsilon$, there are instances for which every $\varepsilon$-coreset has size $Ω\left(k\varepsilon^{-(D-1)/2}\right)$.

cs.CG↗
arXiv2026

List coloring uncrowded hypergraphs at the shattering threshold

Improving an earlier bound of Frieze and Mubayi, Iliopoulos showed that any $k$-uniform uncrowded hypergraph of maximum degree $Δ$ has list chromatic number at most $(1+o(1))(k-1)\Big(\fracΔ{\logΔ}\Big)^{\frac{1}{k-1}}$. Determining the optimal leading constant in this bound remains a major open problem. A natural target for the constant, suggested by the coupon-collector heuristic, is $(k-1)^{\frac{1}{k-1}}$, which also arises independently as the shattering threshold for coloring and finding large independent sets in sparse random hypergraphs. We show that uncrowded hypergraphs attain this threshold: for $k \ge 2$, every $k$-uniform uncrowded hypergraph $\mathcal{H}$ of maximum degree at most $Δ$ satisfies $$χ_\ell(\mathcal{H})\le(1+o(1))\Big((k-1)\fracΔ{\logΔ}\Big)^{\frac{1}{k-1}}.$$ This matches the coupon-collector heuristic, and establishes a 2019 conjecture of Molloy for uncrowded hypergraphs. As a consequence, we show that every $k$-uniform linear hypergraph of maximum degree at most $Δ$ has chromatic number at most $c_k\Big(\fracΔ{\log Δ}\Big)^{\frac{1}{k-1}}$ with $c_k=1-o_k(1)$, resolving, in a stronger form, a recent conjecture of Verstraëte and Wilson on the independence number of linear hypergraphs. Our techniques yield the first palette sparsification result for hypergraph coloring using $o(Δ^{\frac{1}{k-1}})$ colors per vertex. Specifically, for an uncrowded $k$-graph $\mathcal{H}$ and $q=Θ\Big({\Big(\fracΔ{γ\logΔ}\Big)^{\frac{1}{k-1}}}\Big)$, sampling $Θ(Δ^{\fracγ{k-1}})$ colors per vertex suffices to obtain a proper $q$-coloring of $\mathcal{H}$ w.h.p. As an algorithmic consequence, we obtain a single-pass streaming algorithm using $O\big(n^{1+o(1)}\big)$ space for uncrowded $k$-graphs, and, via hypergraph partitioning, for linear hypergraphs as well.

math.CO↗
arXiv2026

A General Reduction from Near-Additive Emulators to Near-Exact Hopsets

Graph emulators and hopsets are two fundamental concepts for distance approximation. When the multiplicative stretch is $1+ε$ for arbitrarily small $ε>0$, these structures are known as near-additive emulators and near-exact hopsets, respectively. Prior work showed that there is a remarkable similarity between the constructions and guarantees of these two objects. In their survey on this topic, Elkin and Neiman [Bull. EATCS 130, 2020] explicitly asked whether one can obtain a general reduction between near-additive emulators and near-exact hopsets. Following that, Kogan and Parter [FOCS, 2022] provided a general reduction from hopsets to emulators and spanners. In this paper, we address the reverse direction and show that any construction for a near-additive emulator for undirected unweighted graphs can be leveraged as a black box to construct a hopset for an undirected weighted graph with comparable size, stretch, and a hopbound comparable to the emulator's additive stretch. Specifically, we show that any algorithm that constructs a $(1+ε',β)$-emulator, with $0 \le ε' \le 1$ and $β\ge 1$, of size $S_{\mathcal{A}}(n, ε',β)$, can be used to obtain a $(1+ε, O(\frac{β^2}{ε^2} \ln(\frac{n}ε)))$-hopset of size $O((S_{\mathcal{A}}(n+m\fracβ{ε^2}, \fracε{294},β) \frac{1}ε + n)\ln(\frac{n}ε))$, for any $0 < ε\le 1$. Therefore, our reduction answers the question of Elkin and Neiman [Bull. EATCS 130, 2020] for sparse graphs and further advances the understanding of the formal connection between these two structures. Designing a reduction resulting in a hopset size that does not depend on $m$ remains an intriguing open question.

cs.DS↗
arXiv2026

Limited Independence Suffices for Large-k Min-wise Hashing

Min-wise hashing and its $k$-min-wise variant are standard tools in similarity estimation, sampling, sketching, and streaming. A $k$-min-wise family requires every prescribed $r$-subset of a fixed set, for $r\le k$, to appear as the $r$ smallest hash values with approximately the fully random probability, up to multiplicative error $δ$. Previous analyses show that $O(\log(1/δ)+k\log\log(1/δ))$-wise independence suffices. Consequently, for $k=Θ(\log N)$ and $δ=N^{-c}$, the standard polynomial construction uses $O(k\log N\log\log N)$ seed bits. Recent work of Chen, Huang, and Li achieves the optimal $O(k\log N)$ seed length for $k=\log^{O(1)}N$, but only with almost-polynomial error $2^{-O(\log N/\log\log N)}$, leaving open whether polynomially small error is possible with the same seed length. We prove that the standard $s$-wise independent polynomial hash family is $k$-min-wise with multiplicative error $δ$ for $s=O(k+\log(1/δ)).$ Thus, when $k=Ω(\log(1/δ))$, only $O(k)$-wise independence is required. In particular, for $k=Θ(\log N)$ and $δ=N^{-c}$, this gives an explicit family with seed length $O(k\log N)$, matching the support-size lower bound up to constant factors. The proof conditions on the prescribed bottom set and bounds the error only after averaging over the random threshold given by its largest hash value, rather than controlling every threshold separately.

cs.DS↗
arXiv2026

A PTAS for Non-Adaptive Stochastic Top-$k$ Sum under General Combinatorial Constraints

We study non-adaptive selection of a feasible set $S$ that maximizes the expected sum of the $k$ largest realized values among independent nonnegative discrete random variables. The same objective arises in team hiring and as VCG welfare in an $\ell$-unit auction. The main setting is a fixed-dimensional nonnegative packing family, whose natural LP has $d=O(1)$ packing inequalities with binary coefficients. We give a PTAS for every $k\ge 1$ on every such family, including binary one- and two-dimensional knapsack, by approximating the occupancy functional $p\mapsto\mathbb{E}[\min(k,N(p))]$ and realizing the resulting signatures in the packing LP. As a generic guarantee the scheme is essentially optimal: there is no FPTAS that works for every such $\mathcal{F}$ unless $P=NP$, and no EPTAS unless $W[1]=FPT$. An incomparable sufficient condition is a query-weight exact-sum oracle (DAG paths, matchings), which likewise yields a PTAS for every $k$. The same signatures give a PTAS for $\min_{S\in\mathcal{F}}\mathbb{E}[\mathrm{Top}_k(S)]$ on every fixed-$d$ covering family; two-dimensional covering knapsack rules out a generic FPTAS on that class.

cs.DS↗
arXiv2026

Learning-Augmented Algorithms: Guarantees, Construction Mechanisms, and System-Level Implications

Learning-augmented algorithms use fallible predictions while retaining formal performance guarantees. This survey synthesizes prediction interfaces, error measures, consistency--robustness trade-offs, and five representative construction mechanisms across online optimization, caching, learned data structures, graph problems, and mechanism design. An orthogonal theorem-level axis distinguishes achieved upper bounds from matched asymptotic dependence. Formal guarantees are separated from empirical systems evidence, with explicit treatment of prediction cost, feedback, and composition. The resulting synthesis states sufficient conditions for limited end-to-end reasoning and delineates open problems in cost-aware prediction, endogenous error, semantic predictors, and benchmarking.

cs.LG↗
arXiv2026

Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler

We study the problem of sampling from $μ(\mathrm{d}x)\propto e^{-V(x)}\,\mathrm{d}x$ on $\mathbb{R}^d$, where $V$ is $α$-strongly convex and $β$-smooth, and write $κ:=β/α$. We design and analyze the Proximal Bouncy Particle Sampler (Proximal BPS), a new sampler that combines ideas from the proximal sampler and the bouncy particle sampler. From a warm start initialization with $ O(1) $ Rényi divergence w.r.t. $μ$, Proximal BPS returns a sample whose law is $\varepsilon$-close to $μ$ in total variation distance using $\widetilde O(\sqrtκ\,d^{1/4} \,\mathrm{polylog}(1/\varepsilon))$ gradient queries in expectation.

math.ST↗
arXiv2026

Frequency Moments Beyond Equality: Streaming Cosine Density Moments

For a stream of nonzero vectors $x_1,\ldots,x_n\in\mathbb{R}^d$, let $u_i=x_i/\|x_i\|_2$. We define the cosine density of the $i$-th stream element by $D_i:=\sum_{j\in[n]}\langle u_i,u_j\rangle$ and study the density moments $M_p:=\sum_{i\in[n]}D_i^p$ in both the signed- and nonnegative-cosine regimes. These quantities are similarity-aware analogues of classical frequency moments: replacing cosine similarity by equality (that is, $D_i = \sum_{j\in[n]} \mathbf{1}\{u_j = u_i\}$) gives $M_p=F_{p+1}$ and, in particular, $M_{-1}=F_0$, the number of distinct elements. We give one-pass streaming algorithms and lower bounds that are tight or nearly tight in their dependence on the dimension $d$. Our results thus extend several fundamental statistics from the classical data stream literature to cosine similarity, a widely used measure for comparing vector embeddings in modern AI systems. The main challenge in proving a space lower bound for nonnegative cosine is to eliminate unwanted contributions without relying on pairs of opposite vectors. We address this through a construction that we call \emph{equal-sum moment isolation}: two insertion-only prefixes have the same cardinality and vector sum, and a finite-difference comparison cancels their common baseline while isolating the desired higher-order signal. This proof framework may be useful for other insertion-only streaming lower bounds, where direct cancellation is not possible.

cs.DS↗
arXiv2026

Almost Sharp Equivalence between Approximate Message Passing and Low-Degree Polynomials

We prove a sharp lower bound for growing-degree polynomial estimation in the Gaussian planted submatrix model. The observation is $$ \boldsymbol{Y}= \fracλ{\sqrt{n}} \boldsymbolθ \boldsymbolθ^{\top}+\boldsymbol{W}, $$ where the coordinates of $\boldsymbolθ$ are independent $\mathsf{Ber}(ρ)$ variables and $\boldsymbol{W}$ is symmetric with independent standard Gaussian upper-triangular entries. For every fixed $λ>0$ and $ρ\in(0,1)$, we give an explicit finite-dimensional bound implying that every sequence of polynomial estimators of degree $D(n)=o(n^{1/60})$ has normalized mean-square error with limit inferior at least $ρ-q_{\mathsf{amp}}/λ$, the limiting error of Bayes approximate message passing (AMP). This extends the constant-degree result of Montanari and Wein~\cite{montanari2025equivalence} for the Bernoulli prior. Combined with their polynomial approximation of fixed-iteration AMP, the bound identifies the exact limiting low-degree MMSE whenever $D(n)\to\infty$ within this range. It therefore resolves the Bernoulli rank-one case of the growing-degree AMP-equivalence question discussed in~\cite{wein2025computational, maleki2026high}. The proof constructs a low-degree certificate using \emph{conditional} joint cumulants of the signal coordinates and their products. Specifically, we condition on an auxiliary Gaussian channel $\boldsymbol{R}$ calibrated to the AMP fixed point. This retains signal dependence that is lost in unconditional cumulant bounds and produces the cancellations needed for quantitative control as the degree grows. Most of the arguments in this paper were generated using GPT-6 Astra.

math.ST↗
arXiv2026

Maximum Matching-Match: Hardness and Approximation

In this paper, we study \textsc{MaxMMP}, an optimization variant of the Matching-Match Puzzle introduced by Iburi and Uehara (FUN 2024). Given a graph, a partial vertex coloring, and a multiset of colored sticks, the goal is to complete the coloring and assign the sticks to graph edges so as to maximize the number of satisfied edges. We first prove that \textsc{MaxMMP} is APX-hard by an reduction from \textsc{Max-Cut}. The hardness already holds with two colors, no precolored vertices, and only bichromatic sticks. We then give a simple deterministic $\frac{2}{c(c+1)}$-approximation for completely uncolored instances, improving to $\frac{2}{c(c-1)}$ when all sticks are bichromatic. Next, we obtain a randomized $\frac{1-\frac{1}{e}}{2c}$-approximation for arbitrary instances with $c$ colors by reducing the remaining coloring choices to monotone submodular maximization under a partition matroid. On bipartite graphs, the approximation ratio improves to $\frac{1-\frac{1}{e}}{c}$. For every fixed $c$, we further obtain deterministic $\frac{1}{2c}$ and $\frac{1}{c}$-approximations on general and bipartite graphs, respectively, in time $n^{O(c^2)}$. Finally, for every fixed number of colors, we show that \textsc{MaxMMP} can be solved exactly in time $n^{O(c^2)}$ on trees and on cographs.

cs.DS↗
arXiv2026

Dasgupta's Hierarchical Clustering Objective: Geometry and the Price of the Cardinality Constraint

The cost of a hierarchical clustering can be represented by an ultrametric whose lowest-common-ancestor labels are cluster cardinalities. We relate this known representation to the shortest-path geometry of a similarity graph. For a connected support graph $G$, let $d_G$ be its unit-length shortest-path metric and let the edge weights enter only the objective. We prove that the shifted Dasgupta optimum is exactly the minimum edge-weighted cost of a cardinality-realizable ultrametric that dominates $d_G$. Connectedification lemmas put this problem and its freely labeled dominating-ultrametric relaxation on the same class of connected binary hierarchies, labeled respectively by cardinality and graph diameter. As a sharp baseline, we determine the exact worst-case price of cardinality realizability: on every $n$-vertex instance the ratio of the two optima is at most $(2n-1)/3$, with equality on the unweighted complete graph; the sharp factor for the standard unshifted objective is $2(n+1)/3$. Our principal structural result bounds this gap by a hereditary weighted fragmentation profile defined through connected balanced cuts. Uniform local control gives an $O(\log n)$ gap, polynomial decay gives a constant gap, and the logarithmic order is tight even for unweighted trees of maximum degree $3$. On locally regular bounded-degree trees, the hierarchy can be constructed in $O(n\log n)$ time. An energy decomposition and a geometric density bound provide supporting instance-sensitive estimates. Thus the cardinality label has an unavoidable linear worst case but admits substantially smaller bounds on natural sparse graph classes.

cs.DS↗
arXiv2026

The Maximum Mutual Visibility Set on a Cactus Graph and the Self-stabilizing Constructions

Given a graph $G=(V,E)$, let $S$ ($\subseteq V$) be a set of vertices. Two vertices are \emph{mutually visible} if there exists a shortest path in $G$ between them that does not contain any other vertex of $S$. A set $S$ is a \emph{Mutual Visibility Set} (\MVS) if every pair of vertices in $S$ is mutually visible. The concept of \MVS s in graphs has attracted significant attention since its introduction, as it provides an important structural property of graphs. However, determining a maximum \MVS\ in general graphs is computationally intractable; the decision problem of whether a graph admits an \MVS\ of size at least $k$ has been shown to be \emph{NP-complete}. Thus, prior work has focused on finding maximal \MVS s or restricting attention to specific graph classes. Cactus graphs form a fundamental low-treewidth class, yet the maximum \MVS\ problem for this class remains open. In this paper, we first determine the size of maximum \MVS~in cactus graphs, and introduce two self-stabilizing algorithms that construct such sets. The first algorithm uses a single BFS tree and stabilizes in $O(D)$ rounds with $O(\log n)$ bits per process on average; the second one uses parallel BFS trees and stabilizes in $O(|C_{\max}|+|T_{\max}|)$ rounds, which we show to be asymptotically tight as a function of these two parameters, even on graphs where $|C_{\max}|+|T_{\max}| = o(D)$.

cs.DC↗
arXiv2026

Better Late Than Never: Online Flow Time Scheduling with Online Estimates

In the classical online flow-time scheduling problem on a single machine, jobs arrive over time and must be processed to minimize the total time they spend in the system: for over fifty years, we have known that SRPT is an optimal online algorithm. But this algorithm requires exactness in two different ways: (a) job sizes must be known exactly, and (b) they must be revealed as soon as the job arrives. Recent work relaxed each of these assumptions separately: there are algorithms based on knowing approximate sizes (given when the job arrives), or based on knowing (exact) sizes at some point before the remaining size gets too small. Nonetheless, prior to this work, there was no known approach to relax both assumptions simultaneously. In this work, we consider a model that demands much less: When we process a job, at some point in time between when we complete an $\varepsilon$-fraction and a $(1-\varepsilon)$-fraction of its unknown processing requirement, we are informed that the job is ``somewhere in the middle''. Finally, when the job has received its desired amount of processing, we are informed of its completion. No other information is shared about the job. We give an $O(1/\varepsilon^2)$-competitive algorithm for this model. Slightly more generally, we assume that an algorithm receives a $μ$-approximate estimate of each job's processing time at some time before we complete a $(1-\varepsilon)$-fraction of its processing. Our algorithm is $O(μ/\varepsilon)$-competitive, and we show that this is asymptotically optimal. It is a surprisingly natural variant of the multilevel feedback algorithm (MLF) and it is parameter-oblivious: it does not need to know $μ$ or $\varepsilon$ upfront. The core analytical contribution is to robustify the dual-fitting framework for this problem to handle jobs for which we have not yet received estimates.

cs.DS↗
arXiv2026

Algorithms for Finite Group Epimorphism Testing

The Group Epimorphism Problem (GpEpi) asks, given two finite groups $G_1$ and $G_2$, whether there exists a surjective group homomorphism, or epimorphism, from $G_1$ to $G_2$. When the input groups are given by their multiplication (Cayley) tables, the problem admits a quasipolynomial-time algorithm in general, but little is known about its complexity for structured classes of finite groups. In this paper, we study the computational complexity of GpEpi for several well-studied classes of finite groups. Our main results are polynomial-time epimorphism tests for several classes of groups for which polynomial-time isomorphism testing was previously known: Groups with Abelian normal Hall subgroups with cyclic complement; Groups with (product of) elementary Abelian normal Hall subgroup with elementary Abelian complement; and Groups with some constraints on their Abelian chief factors.

cs.DS↗
arXiv2026

Improved Upper Bounds for Dynamic Bin Packing of General, Unit-Fraction, and Power-Fraction Squares

This paper presents significant upper-bound improvements for dynamic 2D square bin packing, where square items arrive and depart over time and the objective is to minimize the peak number of concurrent active unit bins. In our model, repacking is permitted only within a destination bin upon item arrival; migration between active bins is strictly forbidden. By introducing a streamlined two-list algorithm and proving a tight $5/16$ occupied-area bound for Next-Fit Decreasing Height, we reduce the upper bound on the asymptotic competitive ratio for arbitrary squares from 4.2154 down to 3.918, breaking a longstanding theoretical ceiling. For restricted variants, we establish asymptotic competitive ratios of at most 3.356 for unit-fraction side lengths and 2.211 for power-fraction side lengths.

cs.DS↗
arXiv2026

Improved Integrality Gap for Multicommodity Flow on Trees

We improve the best known lower bound on the integrality gap for weighted unit-demand multicommodity flow on trees from $1/4$ to $2/5$, improving on the long-standing bound of Chekuri, Mydlarz, and Shepherd~\cite{CMS}. We give the proof in two stages. First, a surprisingly simple packing lemma and an inductive coloring argument give an intermediate bound of $4/11$. We then refine the argument to obtain $2/5$.

cs.DS↗