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arXiv2026

Bridging the Gap Between Stable Marriage and Stable Roommates: A Parameterized Algorithm for Optimal Stable Matchings

In the Stable Roommates Problem (SR), a set of $2n$ agents rank one another in a linear order. The goal is to find a matching that is stable: one that has no pair of agents who mutually prefer each other over their assigned partners. We consider the problem of finding an optimal stable matching. Agents associate weights with each of their potential partners, and the goal is to find a stable matching that minimizes the sum of the associated weights. Efficient algorithms exist for finding an optimal stable matching in the Stable Marriage Problem (SM), but the problem is NP-hard for general SR instances. In this paper, we define a notion of structural distance between SR instances and SM instances, which we call the minimum crossing distance. When an SR instance has minimum crossing distance $0$, the instance is structurally equivalent to an SM instance, and this structure can be exploited to find an optimal stable matching efficiently. More generally, we show that when an SR instance has minimum crossing distance $k$, an optimal stable matching can be computed in time $2^{O(k)} n^{O(1)}$. Thus, the optimal stable matching problem is fixed-parameter tractable (FPT) with respect to minimum crossing distance.

cs.DS↗

arXiv2026

Optimality of Random Regular Graphs in Sparse Network Designs

The problems of designing sparse networks arise frequently in resource allocation and operations research. In production systems, for example, sparse process flexibility designs are used to handle uncertain demand effectively: the goal is to construct the sparsest bipartite graph between supply and demand that still achieves an expected fulfilled demand comparable to that of a fully flexible system. In middle-mile transportation, sparse delivery-route subgraphs that sustain large matchings after random node deletions help reduce delivery costs; here, the goal is to design the sparsest graph whose maximum matching size remains comparable to that of the fully connected graph under node deletions. The design of sparse networks has been studied extensively, with state-of-the-art results providing order-wise optimal designs for both bipartite and unipartite networks (Chen et al., 2015; Feng et al., 2024). However, identifying designs that achieve the sharp theoretical limit -- where the average degree asymptotically matches the lower bound of any graph to achieve a given loss level, has remained open. In this paper, we prove that the random regular graph achieves this sharp optimal condition in both bipartite and unipartite settings. Numerical experiments further validate this optimality. Our results highlight a practical guideline for sparse flexibility networks: designs that combine degree regularity with dispersed edge placement can achieve optimal performance under uncertainty.

cs.DS↗

arXiv2026

Infrequent Resolving Algorithm for Online Linear Programming

Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications such as online auctions, network revenue management, order fulfillment and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former typically guarantees better performance but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithms only require first-order computations but induce a worse performance. In this work, we bridge the gap between these two extremes by proposing a well-performing algorithm that solves LPs at a few selected time points and conducts first-order computations at other time points. Specifically, for the case where the inputs are drawn from an unknown finite-support distribution, the proposed algorithm achieves a constant regret (even for the hard "degenerate" case) while solving LPs only $O(\log\log T)$ times over the time horizon $T$. Moreover, when we are allowed to solve LPs only $M$ times, we design the corresponding schedule such that the proposed algorithm can guarantee a nearly $O\left(T^{(1/2)^{M-1}}\right)$ regret. Our work highlights the value of resolving both at the beginning and the end of the selling horizon, and provides a novel framework to prove the performance guarantee of the proposed policy under different infrequent resolving schedules. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms.

cs.DS↗

arXiv2026

Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration

We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets $A$ and~$B$, where $A$ is an independent set and $B$ induces a graph from some specified graph class ${\cal G}$. We let ${\cal G}$ be the class of $k$-degenerate graphs. This problem is known to be polynomial-time solvable if $k=0$ (bipartite graphs) and NP-complete if $k=1$ (near-bipartite graphs) even for graphs of maximum degree $4$. Yang and Yuan [DM, 2006] showed that the $k=1$ case is polynomial-time solvable for graphs of maximum degree $3$. This also follows from a result of Catlin and Lai [DM, 1995]. We consider graphs of maximum degree $k+2$ on $n$ vertices. We show how to find $A$ and $B$ in $O(n)$ time for $k=1$, and in $O(n^2)$ time for $k\geq 2$. Together, these results provide an algorithmic version of a result of Catlin [JCTB, 1979] and also provide an algorithmic version of a generalization of Brook's Theorem, which was proven in a more general way by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007]. Moreover, the two results enable us to complete the complexity classification of an open problem of Feghali et al. [JGT, 2016]: finding a path in the vertex colouring reconfiguration graph between two given $\ell$-colourings of a graph of maximum degree $k$.

cs.DS↗

arXiv2026

From b-Coloring to $b^*$-Coloring: Large Girth and Parameterized Complexity

A b-coloring is a proper vertex coloring such that every color class contains a vertex, a so-called b-vertex, which sees all colors in its closed neighborhood. This type of coloring has been intensively studied from both structural and algorithmic point of view. Recently, Zaker [DAM 2025] introduced the notion of a b*-coloring, which is a b-coloring in which there is a vertex that sees a b-vertex of every color in its closed neighborhood. The b*-chromatic number is the maximum integer k such that there is a b*-coloring with k colors. We partially answer a question posed by Zaker and prove that graphs of girth at least 7 are b*-monotonic, which means that the b*-chromatic number does not increase by taking an induced subgraph. In addition, we discover a class of d-regular graphs of girth at least 5 with b*-chromatic number d+1, which strengthens a result about b-colorings by Dettlaff, Furmańczyk, Peterin, Roux, and Ziemann [AMC 2024]. We also study the parameterized complexity of finding b*-colorings, and show that for many structural parameters, the complexity coincides with that of finding b-colorings. In particular, the b*-chromatic number can be computed in polynomial time on any class of bounded clique-width. For most parameters, the translation from b-colorings is straightforward but for the feedback edge number, the FPT algorithm for b*-colorings is actually much simpler than that for b-colorings by Balabán [MFCS 2026].

cs.DM↗

arXiv2026

Quantum matrix arithmetics with Hamiltonian evolution

The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a Hamiltonian -- using Hamiltonian evolutions of input operators. We show how to maintain this $\textit{Hamiltonian block encoding}$, so that matrix operations can be composed one after another, and the entire quantum computation takes $\leq 2$ ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadamard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain of interest, while the constructed polynomial is upper bounded by the target function over the entire unit interval. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the doubly factorized tensor hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with $1$ ancilla qubit.

quant-ph↗

arXiv2026

Diversity-Fair Online Selection

Online selection problems arise in applications such as crowdsourcing and recruitment, where decision makers may seek representation across multiple, potentially overlapping demographic or skill dimensions. We study diversity-fair online selection under adversarial arrivals. A recruiter must immediately and irrevocably decide whether to accept each candidate while selecting at most $K$ candidates. Before arrivals begin, the recruiter observes aggregate marginal information: the total number of candidates contributing to each of the $d$ diversity dimensions. When the candidate pool is large, this information may be estimated from demographic statistics of the applicant population. We evaluate the expected utilities across dimensions using the generalized mean $M_p=(d^{-1}\sum_{k=1}^d U_k^p)^{1/p}, -\infty\le p\le 1,$ where $U_k$ denotes the expected utility of dimension $k$. We first study max-min fairness, corresponding to $p=-\infty$. We prove that no online policy can achieve a competitive ratio better than $O(1/\sqrt d)$ and develop a policy with a competitive ratio $1/[4(2+\sqrt2)\sqrt d]$, establishing the optimal dependence on $d$ up to a constant factor. Without exact marginal information, the optimal worst-case rate falls to $Θ(1/d)$, demonstrating the value of this information. We also extend the max-min analysis to nonbinary attributes and characterize the optimal dependence on their value range. Finally, we study generalized-mean objectives. For $0\le p\le1$, we establish an optimal competitive ratio of $Θ(1/\log d)$. For each fixed finite negative mean $p=-q$, where $q>0$, our policy achieves $d^{-q/(2q+1)}$ up to polylogarithmic factors, matching the exponent of the corresponding impossibility bound.

econ.TH↗

arXiv2025

Navigating Uncertainties in Machine Learning for Structural Dynamics: A Comprehensive Survey of Probabilistic and Non-Probabilistic Approaches in Forward and Inverse Problems

In the era of big data, machine learning (ML) has become a powerful tool in various fields, notably impacting structural dynamics. ML algorithms offer advantages by modeling physical phenomena based on data, even in the absence of underlying mechanisms. However, uncertainties such as measurement noise and modeling errors can compromise the reliability of ML predictions, highlighting the need for effective uncertainty awareness to enhance prediction robustness. This paper presents a comprehensive review on navigating uncertainties in ML, categorizing uncertainty-aware approaches into probabilistic methods (including Bayesian and frequentist perspectives) and non-probabilistic methods (such as interval learning and fuzzy learning). Bayesian neural networks, known for their uncertainty quantification and nonlinear mapping capabilities, are emphasized for their superior performance and potential. The review covers various techniques and methodologies for addressing uncertainties in ML, discussing fundamentals and implementation procedures of each method. While providing a concise overview of fundamental concepts, the paper refrains from in-depth critical explanations. Strengths and limitations of each approach are examined, along with their applications in structural dynamic forward problems like response prediction, sensitivity assessment, and reliability analysis, and inverse problems like system identification, model updating, and damage identification. Additionally, the review identifies research gaps and suggests future directions for investigations, aiming to provide comprehensive insights to the research community. By offering an extensive overview of both probabilistic and non-probabilistic approaches, this review aims to assist researchers and practitioners in making informed decisions when utilizing ML techniques to address uncertainties in structural dynamic problems.

cs.LG↗

arXiv2026

Breaking the Exponential Barrier: The First Polynomial-Time Algorithm for the Győri-Lovász Theorem

We give the first polynomial-time algorithm, after half a century, for the celebrated Győri-Lovász theorem, which resolved a conjecture of Frank (1975). The theorem, one of the simplest existential theorems to explain, states that every $k$-connected graph can be partitioned into $k$ disjoint connected subgraphs of arbitrary prescribed positive sizes. This is a fundamental structural result with broad applications, such as flexible allocation of connected subnetworks of prescribed sizes in sufficiently connected cloud infrastructures. While Lovász (1977) gave a highly non-constructive proof for a stronger directed version using algebraic topology, Győri's original constructive proof (1976) requires exponential time. Despite more than 50 years of effort, no polynomial-time algorithm was known even for $k>4$. Determining the computational complexity of the Győri-Lovász theorem---whether it admits even a sub-exponential-time algorithm or is computationally hard (in particular, PLS-complete or PPAD)---has remained one of the central open problems in algorithmic graph theory. In this paper, we finally resolve this long-standing problem by a fundamentally new proof of the existential theorem via introducing the novel concept of \emph{flow-essential assignment}, which genuinely marries matching and cut structures and yields the first polynomial-time constructive algorithm for the Győri-Lovász theorem. In fact, we obtain a polynomial-time algorithm for Lovász's stronger directed version, whose proof was non-constructive even for DAGs; for DAGs, we further obtain a near-linear-time algorithm. We also develop polynomial-time algorithms for weighted generalizations where the seminal work of Chen, Kleinberg, Lovász, Rajaraman, Sundaram, and Vetta (JACM'07) on confluent flows established only existential non-constructive results.

cs.DS↗

arXiv2026

The Cayley Completion of a Graph

A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $γ^{+}$ (additions only) and the Cayley edit distance $γ_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $γ^{+}(G)\ge nΔ^{*}/(2m)-1$, where $Δ^{*}$ is the least $d\geΔ$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $γ^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $γ^{+}$, while $γ_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $γ^{+}(P_n)=γ^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $γ_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.

cs.DM↗

arXiv2026

DS-Lighting: Making Agent Harnesses Explicit for Data-Science Automation

Large Language Model (LLM) agents have shown promise for automating data-science workflows, yet their end-to-end performance depends critically on the agent harness that represents tasks, manages execution state, constrains output artifacts, and provides evaluation feedback. Existing data-science agents often leave this harness implicit, making results difficult to reproduce, compare, and attribute across heterogeneous tasks. We introduce DS-Lighting, a unified harness toolkit that makes harness design explicit for data-science automation. DS-Lighting decomposes the harness into four reusable layers: data, workflow, execution, and evaluation, and represents diverse agents as executable operator programs that support both predefined pipelines and adaptive search. We further integrate multiple open-source data-science benchmarks into an MLE-Bench-style task format, enabling controlled comparison under a shared task interface, sandboxed runtime, and metric protocol. Experiments across agents, harnesses, models, and ablations show that explicit harness design improves reproducibility, comparability, and reliability, while reducing avoidable system-level failures in end-to-end data-science workflows. Our code is available at https://github.com/usail-hkust/dslighting

cs.AI↗

arXiv2026

Disproving the Greedy Superstring Conjecture

The shortest common superstring problem is to find the shortest string that contains every string in a given set as a substring. It is conjectured that the greedy algorithm that repeatedly selects a pair of strings with maximum overlap and merges them is a $2$-approximation algorithm, and this conjecture had remained open for nearly four decades. In this paper, we disprove this conjecture and show that the approximation ratio of this algorithm is at least $9/4$.

cs.DS↗

arXiv2026

The Bron-Kerbosch Algorithm with Vertex Ordering is Output-Sensitive

The Bron-Kerbosch algorithm is a well known maximal clique enumeration algorithm. So far it was unknown whether it was output sensitive or not. In this paper we partially answer this question by proving that the Bron-Kerbosch Algorithm with vertex ordering, first introduced and studied by Eppstein, Löffler and Strash in "Listing all maximal cliques in sparse graphs in near-optimal time. International Symposium on Algorithms and Computation. Springer, Berlin, Heidelberg, 2010" is output sensitive.

cs.DS↗

arXiv2026

Upper and lower bounds on the OBDD-width of a special integer multiplication

We consider the Boolean function ${\rm SMul}_{n-1}^n(\boldsymbol{x},\boldsymbol{y})$, which computes the middle bit of the multiplication of two natural numbers represented as $n$-bit binary strings $\boldsymbol{x}$ and $\boldsymbol{y}$, drawn from a restricted domain. We investigate the width of OBDDs computing ${\rm SMul}_{n-1}^n$. We introduce a combinatorially defined function $s_*(n)$ and show that the width of such OBDDs is $Θ(2^{s_*(n)})$.

cs.CC↗

arXiv2026

Test or Run? Scheduling Jobs of Unknown Length

A machine faces many jobs whose lengths are hidden. Spending one unit of time to inspect a job may reveal a short job that should be finished now, or it may reveal nothing useful while every other job waits. When should the machine keep looking, and when should it start working? We study natural variants of this question and provide optimal algorithms in both the worst-case and instance-optimal frameworks. The resulting algorithms are often quite simple, which may make them useful in practice.

cs.DS↗

arXiv2026

Quadratic Probing Insertions Are $ε^{-(1+o(1))}$

First proposed in 1968, quadratic probing has stood for more than half a century as one of the simplest and most widely used hash-table designs in computer science. It is conjectured that, at load factor $1 - ε$, the hash table achieves $O(ε^{-1})$ expected insertion time. But even proving a bound of the form $f(ε^{-1})$ for any function $f$ has remained open. In this paper, we prove that the expected insertion time is $ε^{-(1 + o(1))}$. This settles the complexity of the data structure up to sub-polynomial factors in $ε^{-1}$.

cs.DS↗

arXiv2026

A Note on Approximating the Rural Postman Problem below 3/2

We give an approximation algorithm for the rural postman problem with approximation ratio strictly smaller than $3/2$. We obtain this result by adapting to the rural postman problem the technique of sampling from maximum entropy distributions for the metric traveling salesman problem of Karlin, Klein, and Oveis Gharan. We also observe that, for every fixed $\varepsilon>0$, any $α$-approximation algorithm for the metric traveling salesman problem yields an $(α+\varepsilon)$-approximation algorithm for the rural postman problem; this implication is already implicit in the treatment of edges that must be traversed in the work of Lampis on the inapproximability of the traveling salesman problem.

cs.DS↗

arXiv2026

On the maximum weight convex problem for some geometric graph-convexities

For a given geometric graph-convexity on a graph $G$ equipped with a weight function on the vertices with value in $\mathbb{Z}$, the Max Weight Convex Set problem consists in determining the convex set $S$ with maximum weight (sum of the weight of the vertices in $S$). Although the problem is NP-complete in general, it remains polynomial for particular cases. After a survey of known results, our main contribution uses a generalisation of the maximum subsequence problem to laminar trees. Then we derive a linear algorithm for proper interval graphs and a quadratic one for interval graphs. Both improve the state of the art.

cs.DS↗