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arXiv2026

Extending Exact Integrality Gap Computations for the Metric TSP

The subtour relaxation of the traveling salesman problem (TSP) plays a central role in approximation algorithms and polyhedral studies of the TSP. A long-standing conjecture asserts that the integrality gap of the subtour relaxation for the metric TSP is exactly $4/3$. In this paper, we extend the exact verification of this conjecture to a larger number of vertices. Using the framework introduced by Benoit and Boyd in 2008, we confirm their results up to $n=10$. We further show that for $n=11$ and $n=12$, the published lists of extreme points of the subtour polytope are incomplete: one extreme point is missing for $n=11$ and twenty-two extreme points are missing for $n=12$. We extend the enumeration of the extreme points of the subtour polytope to instances with up to $n=16$ vertices in the general case. Restricted to half-integral extreme points, we extend the enumeration to $n=18$. Our results provide additional support for the $4/3$-Conjecture. Our lists of extreme points are available on the public bonndata repository (https://doi.org/10.60507/FK2/JK95PC).

math.CO↗

arXiv2026

Lawler-Moore Speedups via Additive Combinatorics

The Lawler-Moore dynamic programming framework is a classical tool in scheduling on parallel machines. It applies when the objective is regular, i.e. monotone in job completion times, and each machine follows a fixed priority order such as Smith's Rule or Jackson's Rule. For the basic objectives $Pm||\sum w_jC_j$, $Pm||L_{\max}$, and $Pm||\sum w_jU_j$, it gives running times $O(P^{m-1}n)$, $O(P^{m-1}n)$, and $O(P^mn)$, respectively, where $P$ is the total processing time. Recent SETH-based lower bounds indicate that the dependence on $P$ is essentially optimal, but they do not rule out improved dependence on the maximum processing time $p_{\max}$. We give the first major speedup of the Lawler-Moore recurrence. Our main ingredients are a new state-pruning method and a swapping argument based on an additive-combinatorial lemma. We prove that, whenever this swap does not increase the objective value, there exists an optimal schedule in which, for every prefix of jobs, the load difference between any two machines is at most $4p_{\max}^2$. This lets us prune redundant states throughout the dynamic program, replacing the dependence on $P$ by a dependence on $p_{\max}^2$. We show that the swap is non-increasing for all three objectives above. Hence $Pm||\sum w_jC_j$ and $Pm||L_{\max}$ admit algorithms with running time $O(p_{\max}^{2m-2}n)$, while $Pm||\sum w_jU_j$ can be solved in time $O(p_{\max}^{2m-2}Pn)\le O(p_{\max}^{2m-1}n^2)$. These bounds strictly improve the original Lawler-Moore runtimes whenever $p_{\max}=o(\sqrt{P})$. In particular, for $Pm||\sum w_jC_j$ and $Pm||L_{\max}$, we obtain the first near-linear-time algorithms when processing times are polylogarithmic in $n$.

cs.DS↗

arXiv2026

On Solving Problems of Substantially Super-linear Complexity in $N^{o(1)}$ Rounds in the MPC Model

We study the possibility of designing $N^{o(1)}$-round protocols for problems of substantially super-linear polynomial-time (sequential) complexity in the model of Massively Parallel Computation, where $N$ is the input size. We show that if the machines are not equipped with relatively large local memory and their number does not exceed $N$, then the exponent of the average time complexity of the local computation performed by a machine in a round (in terms of local memory size) in such protocols must be larger than the exponent of the time complexity of the given problem.

cs.DC↗

arXiv2026

Sharp Normalized Covariance Bounds and Constant-Stretch Correlated Sampling on the Hypersimplex

We establish the normalized covariance bound conjectured by Anari et al. (2026, Conjecture 3) for fixed-rank external-field measures. Let $d\ge2$ and $m\in[d-1]$. For $w\in(0,+\infty)^d$, let $\mathsf S$ be an $m$-subset of $[d]$ with rank-$m$ external-field law $\mathbb P(\mathsf S=S)=\frac{\prod_{i\in S}w_i}{e_m(w)},S\subseteq[d],|S|=m,$ where $e_m(w):=\sum_{T\subseteq[d], |T|=m}\prod_{\ell\in T}w_\ell$ is the $m$th elementary symmetric polynomial in $w_1,\dots,w_d$. Let $X:=(X_1,\dots,X_d)^\top$ be its indicator vector, i.e., $X_i=\mathbb I\{i\in\mathsf S\},i\in\{1,\dots,d\}.$ Let $Σ:=\mathrm{Cov}(X)$, put $v_i:=Σ_{ii}$ for each $i\in[d]$, and define $v:=(v_1,\dots,v_d)^\top,D:=\mathrm{diag}(v),V:=\sum_{i=1}^dv_i.$ We prove $Σ\succeq D-\frac{vv^\top}{V}.$ This improves the coefficient $1/2$ in the first version of our work (Cesari and Colomboni, 2026, Corollary 1.3) to the optimal universal value $1$. As a first corollary, we improve the coefficient in the pseudoinverse bound of Bacchiocchi et al. (2026, Lemma 2) from $2$ to the optimal value $1$. Specializing this bound to coordinate differences gives an alternative proof of our effective-resistance theorem from the first version of our work (Cesari and Colomboni, 2026, Theorem 1.1). The framework of Anari et al. (2026, Theorem 1 and Corollary 2) also yields unconditional correlated-sampling guarantees with stretch $6$ on the hypersimplex and $12$ on its at-most variant. Unconditional constant-stretch guarantees were first established in the first version of our work (Cesari and Colomboni, 2026, Corollaries 1.4 and 1.5), with constants $16$ and $32$, which we improve here to $6$ and $12$. These improved constants strengthen the positive resolution, established in the first version of our work, of the constant-stretch question posed by Naor et al. (2026, Theorem 2 and Section 5).

math.PR↗

arXiv2026

Adaptive Black-Box Exactness Barriers for Nearest-Source Girth Estimation in CONGEST

Recent multi-scale nearest-source methods give polynomially sublinear approximations for girth in the CONGEST model. We study exactification by adaptive black-box composition while preserving the same fresh exchangeable source-selection primitive. Our scalar-oracle model exposes the sampled source identities and the scalar estimate from every call, allows arbitrary persistent controller state, adaptive source cardinalities and capacities, adaptive stopping, and an arbitrary final decoder; the internal nearest-source tables remain encapsulated. We first construct, for infinitely many $n$, a same-size pair $G_t^0,G_t^1$ of maximum-degree-three, logarithmic-diameter graphs whose girths are distinct and both $Θ(\log n)$. A length-transfer construction makes every bulk source contribute identically on the two graphs. The scalar transcripts can separate the pair only when one of $O(\log n)$ interface sources survives a linear nearest-source rank competition. Coupling the adaptive executions with a conditional permutation-rank bound yields an $Ω(n/\log n)$ expected retained-source workload requirement for constant exactness probability, even with arbitrary final decoding. For the standard sequential packetized realization, the same scale is an expected-round barrier. A complementary bridgeless family $\widehat H_t$ shows the same $Ω(n/\log n)$ direct-retuning barrier on bounded-degree graphs with minimum degree at least two, no bridges, and $2$-core equal to the whole graph. Finally, under a known promise $g\ge h$, one full-source call from $Θ(n/h)$ uniformly sampled sources computes exact girth with constant probability in $O(n/h+D)$ rounds, matching the $n/g$ source scale.

cs.DS↗

arXiv2026

Sparse Polynomial GCD Algorithms Asymptotically Linear in All Fundamental Parameters

Let $A, B \in \mathbb{Z}[x_1, \dots, x_n]$ be multivariate polynomials with integer coefficients and let $G = \gcd(A, B)$. We present an algorithm for computing $G$ whose expected bit complexity is asymptotically linear in all fundamental parameters: the number of variables $n$, the term count $T = \max\{\|A\|_0, \|B\|_0, \|G\|_0\}$, the total degree $D$, and the logarithmic coefficient sizes $\log\Hi$ and $\log\Ho$, where $\Hi$ bounds the coefficients of the inputs and $\Ho$ bounds those of the GCD. The bit complexity is characterized by the clean bound \[ \widetilde{O}\bigl( n \cdot T \cdot D \cdot \log\Hi \cdot \log\Ho \bigr). \] To our knowledge, this is the first sparse GCD algorithm over the integers that achieves linear complexity in all these parameters simultaneously. The integer algorithm is built upon a new field GCD algorithm. For $A, B \in \K[x_1, \dots, x_n]$ over a field $\K$ with $\operatorname{char}(\K) = 0$ or $\operatorname{char}(\K) > °G$, we give the first algorithm that computes $G = \gcd(A,B)$ with expected \[ \widetilde{O}\bigl( n \cdot T \cdot D \bigr) \] field operations, which is both input- and output-sensitive. The key technical contribution behind both algorithms is a derivative-aided separated Hensel lifting technique introduced in this paper. By introducing an auxiliary variable and leveraging derivative information, our scheme extracts all partial exponents via a single $z^2$-lift per variable, achieving constant sequential depth $O(1)$. This stands in sharp contrast to classical Hensel lifting, which requires $O(D)$ sequential lifting steps and suffers from representation densification in the sparse setting. The field algorithm is then extended to the integer case through modular reduction and rational reconstruction.

math.AC↗

arXiv2026

Beyond Cut Balance: Spectral Sparsification of the Nonlinear Directed Laplacian

Digraphs with constant cut balance admit nearly linear directed cut sparsifiers. This condition requires the total arc weights in the two directions of every cut to be within a constant factor of each other. We ask whether this condition also permits nearly linear spectral sparsification with respect to the energy of the nonlinear directed Laplacian. For a weighted digraph $G=(V,E,w)$, let \[ Q_G^+(x)=\sum_{(u,v)\in E}w_{uv}(x_u-x_v)_+^2, \qquad (t)_+:=\max\{t,0\}. \] This energy agrees with the outgoing-cut function on binary vectors. A spectral sparsifier is a nonnegatively reweighted subgraph that preserves $Q_G^+(x)$ within a factor of $1\pm\varepsilon$ simultaneously for all $x\in\mathbb R^V$. We show that cut balance alone does not yield nearly linear spectral sparsifiers: for constant error, the worst-case support size for simple unweighted Eulerian digraphs is $\widetildeΘ(n^{3/2})$, although Eulerian digraphs are perfectly cut-balanced and admit nearly linear directed cut sparsifiers. In contrast, we prove that every $n$-vertex tournament has a spectral sparsifier with $\widetilde O(n/\varepsilon^3)$ arcs, without any assumption on its cut balance. This includes the transitive tournament, whose cut balance is unbounded. Thus perfect balance does not guarantee nearly linear spectral sparsification, while unbounded imbalance does not preclude it. Finally, we use convex duality to show that preserving $Q_G^+$ also preserves, for every feasible demand vector, the optimum quadratic cost of a nonnegative flow. Hence the guarantee contains information beyond directed cut values.

cs.DS↗

arXiv2026

Generalized Graph Search Trees

Graph search algorithms and their corresponding graph search trees are commonly used in algorithmic graph theory. In recent years, the recognition problem of these graph search trees has received significant attention. So far, the research has focused on two types of search trees: first-in trees that behave like BFS-trees and last-in trees that behave like DFS-trees. The search tree paradigms differ from each other by the parent a vertex is connected to. In first-in trees, it is the first visited neighbor, while in last-in trees it is the last neighbor visited before that vertex. Here, we will generalize these concepts of graph search trees by allowing every preceding neighbor of a vertex to be the parent. We study the complexity of the recognition problem of these generalized graph search trees. We present NP-completeness proofs for most searches. We also show that the problem is trivial for Generic Search and polynomial-time solvable for several searches on bipartite graphs and chordal graphs. We also study the question how fixing the start vertex influences the complexity of the problem.

cs.DM↗

arXiv2026

Distributed Quantum Property Testing with Quantum Carrier Pigeons

We introduce a framework for distributed quantum inference under communication constraints. In our model, $m$ distributed nodes each receive one copy of an unknown $d$-dimensional quantum state $ρ$, before communicating via a constrained one-way communication channel with a central node, which aims to infer some property of $ρ$. This framework generalizes the classical distributed inference framework introduced by Acharya, Canonne, and Tyagi [COLT 2019], by allowing quantum resources such as quantum communication and shared entanglement. Within this setting, we focus on the fundamental problem of quantum state certification: Given a complete description of some state $σ$, decide whether $ρ=σ$ or $\|ρ-σ\|_1\geq ε$. Additionally, we focus on the case of limited communication between distributed nodes and the central node: we assume each communication channel is limited to only $n_c$ bits and $n_q$ qubits with $n_c + n_q \leq \log d$. When all nodes can make use of a shared source of randomness, we show that the copy complexity of distributed state certification is $Θ(\frac{d^2}{2^{n_q} 2^{n_c/2}ε^2})$. We further demonstrate that shared randomness is necessary to achieve the above complexity, by proving an $Ω(\frac{d^3}{4^{n_q} 2^{n_c} ε^2})$ lower bound in the $\textit{private-coin}$ setting. Moreover, we develop a private-coin algorithm that matches this bound up to a $\sqrt{\log d}$ factor, showing this complexity is near-optimal. Together, our work establishes a general framework for distributed quantum inference with communication constraints and characterizes the complexity of distributed state certification with limited communication.

quant-ph↗

arXiv2026

High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression

Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ is \emph{highly magnetized}). We use our frameworks to design improved samplers for canonical problems in the study of \emph{Ising models} and \emph{Bayesian sparse linear regression}. Our first main result considers the \emph{Sherrington--Kirkpatrick} (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $β>0$, under arbitrary external fields, provided that $k\le c_βd$ for an appropriate constant $c_β$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$β$ limit, our framework permits sampling at field strengths within constant factors of the \emph{Almeida--Thouless line} delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(β)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical \emph{Gaussian spike-and-slab posterior} with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.

cs.DS↗

arXiv2026

A $(\log n)^{1/4}$ Bound for the Komlós Problem

Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from their bound. The fourth root comes from balancing the logarithmic decrease in the alive dimension against the fourth power of the row thresholds. Historical exponential sums control the covariance budget across size classes with summable thresholds. An exact threshold-sum certificate gives the coefficient $2395$, and rounding at most eight remaining fractional coordinates costs $2\sqrt2$. The finite construction also gives partial colourings from any prescribed starting point and at any prescribed depth, preserving existing signs. We formalize the partial- and full-colouring theorems in Lean, including the finite trajectory, exact threshold sum and final rounding, with Bansal--Jiang Theorem A.4 as the sole external research theorem assumption.

math.CO↗

arXiv2026

Parity and Pattern Detection in Permutation Streams

Consider a permutation of $[n]$ whose values arrive one at a time. We resolve two questions about the space needed to decide natural properties of such input: First, computing the parity of the permutation requires $Θ(n)$ bits, even with randomization and constant error, and a constant number of passes. Second, every permutation pattern of length three can be detected deterministically in one pass using $O(\log n)$ bits. Together with the 2026 lower bounds of Berendsohn, this completes the classification of fixed permutation patterns; The optimal space complexity is $Θ(\log n)$ for monotone patterns and patterns of length at most three, and $Θ(n)$ for every other pattern. As a consequence, we observe that we can verify BST traversals in streaming with logarithmic memory.

cs.DS↗

arXiv2026

Approximate Nearest Neighbor in Ultra-High Dimensional $\ell_\infty$

We study the approximate nearest neighbor problem under $\ell_\infty$ in the ultra-high dimensional setting where the dimension $d$ is significantly larger than the number of points $n$. Thus, we desire data structures with no dependence on $d$ in the query time. [Herold-Nanongkai-Spoerhase-Varma-Wu, SoCG 2025] introduce this problem and give data structures in $\ell_p$: for $p=1,2$, they give $(1+\varepsilon)$-approximation data structures with space $\tilde{O}(n\log d/\text{poly}(\varepsilon))$ and query time $\tilde{O}(n/\text{poly}(\varepsilon))$. Since any data structure must have query time $Ω(\min \{n,d\})$, this query time is nearly tight. However, their results are inefficient for $\ell_\infty$, with query time $Ω(nd)$. In order to handle the challenges of $\ell_\infty$, we introduce a notion of subset embeddings, which embed points by simply selecting a subset of dimensions. In particular, we show one may preserve all pairwise distances of an $n$ point dataset up to a factor of $O(c)$ by computing distances on only $n^{1+1/c}$ coordinates. We also show a matching lower bound: for any $c > 1$, there exists a set of $n$ points in $\mathbb{R}^{d}$ such that any subset embedding for the set with approximation $c$ must have at least $n^{1+Ω(1/c)}$ coordinates. Using our subset embeddings, we give data structures for approximate nearest neighbor in $\ell_\infty$ with space $O(n^2\log d)$, query time $\tilde{O}(n^{1+1/c})$, and approximation $O(c\log\log n)$ for any $c \geq 1$. Finally, we give another data structure for the approximate nearest neighbor under $\ell_\infty$ with the same space and query time as our subset embedding approach, but with approximation $O(c^{\log_2 3}) \approx O(c^{1.58})$. This allows us to achieve $O(1)$-approximation with query time e.g.~$n^{1.01}$

cs.DS↗

arXiv2026

Parameterised algorithms for temporally satisfying reconfiguration problems

Given a static vertex-selection problem (e.g. independent set, dominating set) on a graph, we can define a corresponding temporally satisfying reconfiguration problem on a temporal graph which asks for a sequence of solutions to the vertex-selection problem at each time such that we can reconfigure from one solution to the next. We can think of each solution in the sequence as a set of vertices with tokens placed on them; our reconfiguration model allows us to slide tokens along active edges of a temporal graph at each time-step. We show that it is possible to efficiently check whether one solution can be reconfigured to another, and show that approximation results on the static vertex-selection problem can be adapted with a lifetime factor to the reconfiguration version. Our main contributions are fixed-parameter tractable algorithms with respect to: enumeration time of the related static problem; the combination of temporal neighbourhood diversity and lifetime of the input temporal graph; and the combination of lifetime and treewidth of the footprint graph.

cs.DS↗

arXiv2026

Parameterized Complexity of Scheduling Problems in Robotic Process Automation

This paper studies the growing domain of Robotic Process Automation (RPA) problems. Motivated by scheduling problems arising in RPA, we study the parameterized complexity of the single-machine problem with precedence constraints, release times, and deadlines (i.e., the problem known as $1|\operatorname{prec},r_j,d_j|*$ in the three-field notation). We focus on parameters naturally linked to RPA systems, including chain-like precedences, the number of distinct processing times, and the structure of the time windows. We show that the problem is strongly XNLP-hard parameterized by the number of chains, even with only two prescribed processing times and two distinct time-window lengths. The problem remains XNLP-hard even under prec-consistent time windows. On the positive side, we obtain polynomial-time algorithm when all jobs share a single time-window length and FPT when the processing times, release times and deadlines are chain-uniform. We also show that the problem lies in XNLP when parameterized by the width of the precedence relation either when the instance is encoded in unary or the processing times are bounded.

cs.DS↗

arXiv2026

Separator Theorem for Minor-Free Graphs in Linear Time

The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan's theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC '90, Journal of the AMS '90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.

cs.DS↗

arXiv2026

High-dimensional Linear Bandits with Knapsacks

We investigate the contextual bandits with knapsack (CBwK) problem in a high-dimensional linear setting, where the feature dimension can be very large. Our goal is to harness sparsity to obtain sharper regret guarantees. To this end, we first develop an online variant of the hard thresholding algorithm that performs the sparse estimation in an online manner. We then embed this estimator in a primal-dual scheme: every knapsack constraint is paired with a dual variable, which is updated by an online learning rule to keep the cumulative resource consumption within budget. This integrated approach achieves a two-phase sub-linear regret that scales only logarithmically with the feature dimension, improving on the polynomial dependency reported in prior work. Furthermore, we show that either of the following structural assumptions is sufficient for a sharper regret bound of $\tilde{O}(s_{0} \sqrt{T})$: (i) a diverse-covariate condition; and (ii) a margin condition. When both conditions hold simultaneously, we can further control the regret to $O(s_{0}^{2} \log(dT)\log T)$ by a dual resolving scheme. As a by-product, applying our framework to high-dimensional contextual bandits without knapsack constraints recovers the optimal regret rates in both the data-poor and data-rich regimes. Finally, numerical experiments confirm the empirical efficiency of our algorithms in high-dimensional settings.

cs.LG↗

arXiv2026

An FPTAS for 7/9-Approximation to Maximin Share Allocations

We present a new algorithm that achieves a $\frac{7}{9}$-approximation for the \emph{maximin share (MMS)} allocation of indivisible goods under additive valuations, improving the current best ratio of $\frac{10}{13}$~\cite{conf/soda/HeidariKSS26}. Building on a new analytical framework, we further obtain an FPTAS that achieves a $\frac{7}{9}-\varepsilon$ approximation in $\tfrac{1}{\varepsilon} \cdot \mathrm{poly}(n,m)$ time. The main technical ingredient is a dynamic witness-allocation framework that certifies adaptive reductions throughout the allocation process.

cs.GT↗