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arXiv2026

Characterizations and Complexity of Minimum Forward and Integer Cycle Bases

The cycle space of a directed graph is generated by a cycle basis, where, in general, cycles are allowed to have both forward and backward arcs. In a forward cycle, all arcs must follow the given direction. Several open questions remain regarding the complexity of the minimum cycle basis problem, in particular the minimum-weight integral cycle basis problem, and the minimum-weight weakly and strictly fundamental forward cycle basis problems. In this paper, we address these open questions. First, we study the existence, structure, and computational complexity of minimum-weight forward cycle bases. We give a complete structural characterization of digraphs that admit weakly fundamental (and hence integral) forward cycle bases. We further provide a characterization when a strongly connected digraph admits a forward fundamental cycle basis, proving that such a basis exists if and only if the set of directed cycles has cardinality equal to the cycle rank; in this case, the basis is unique. Lastly, we show that while minimum-weight forward fundamental cycle bases can be found in polynomial time whenever they exist, the minimum-weight forward weakly fundamental cycle basis problem is APX-hard via an L-reduction from the minimum-weight weakly fundamental cycle basis problem on digraphs with metric weights. Second, we introduce opt-in graphs, i.e., the family of graphs for which minimum cycle bases are integral for any weight function. We show that this family is minor-closed and hence, by the Robertson-Seymour theorem, is characterized by a finite set of forbidden minors, so that the opt-in recognition problem is solvable in polynomial time. Lastly, we present an algorithm to check whether a graph is opt-in, and if not, to identify which of its minors belong to the set of forbidden minors. Applying this algorithm, we show that the complete graph $K_n$ is opt-in if and only if $n \leq 7$.

math.OC↗
arXiv2026

Joint Alignment and Distillation for Video Generation via Sample-Guided Distribution Matching

Aligning video generative models to human preferences heavily relies on Reinforcement Learning (RL), which suffers from extensive computational overhead. Existing workflows typically treat RL and distillation as disconnected stages: applying RL before distillation incurs prohibitive computational costs, whereas applying RL after distillation frequently leads to model collapse. To overcome these limitations, we propose a unified, single-stage optimization framework grounded in Distribution Matching (DM). In the standard DM framework, distillation updates the model via a gradient direction that minimizes the gap between the real and fake models, guiding generations toward clarity and high fidelity. Building upon this, we introduce DM-Align, which derives a complementary gradient direction to guide the model toward human-preferred samples. Inspired by DPO and GRPO, our method leverages the distributional gap -- formulated from either preference pairs or intra-group exploration -- to directly construct this preference-guided gradient. By synergizing these two gradient directions, our approach eliminates the need for multi-step reward evaluation and complex ODE-SDE conversions inherent in traditional RL. Comprehensive experiments across multiple foundational video models demonstrate that this sample-guided framework robustly enhances both distillation quality and preference alignment, consistently outperforming both standalone variants and sequential two-stage pipelines.

cs.CV↗
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

Layer-Based Width for PAFP

The Path Avoiding Forbidden Pairs problem (PAFP) asks whether, in a directed graph $G$ with terminals $s,t$ and a set $\mathcal{F}$ of forbidden vertex pairs, there is an $s$-$t$ path that contains at most one endpoint from each forbidden pair. We initiate the study of PAFP through a layer-based width measure. Our first focus is the union digraph $G\cup\mathcal{F}$, obtained by adding to $G$ one arc per forbidden pair, oriented according to a fixed reachability-compatible order. Let the BFS layer $L_d$ be all vertices at directed shortest-path distance $d$ from $s$, where the BFS-width from $s$ is $\max_d |L_d|$. We show if $G\cup\mathcal{F}$ has BFS-width $b$ from $s$ and only $β$ arcs going from a later BFS layer to an earlier one, then PAFP is FPT parameterized by $b+β$. The backward-arc hypothesis is essential: we show PAFP remains NP-complete when the union digraph is a DAG with BFS-width 2. We also show if the input DAG has BFS-width at most $2$ and only $k$ backward input arcs, then PAFP can be decided in $2^k |I|^{O(1)}$ time, with unrestricted forbidden pairs. This width-$2$ result is tight: inspection of a classical reduction shows NP-completeness on input DAGs of BFS-width $3$ with no backward input arcs. Moreover, we study exact-length layers in the input graph, where the $d$-th layer consists of the vertices reachable from $s$ by a directed path of length exactly $d$. For DAGs of exact-length width at most $2$, we show PAFP is polynomial-time decidable by a 2-SAT encoding of fixed-length paths. This bound is tight: the same classical reduction yields NP-completeness on DAGs of exact-length width $3$. Unlike previously known polynomial-time regimes for PAFP, which restrict the forbidden-pair set in order to obtain tractability, our two input-graph tractability results allow unrestricted forbidden pairs and input graphs with exponentially many $s$-$t$ paths.

cs.DS↗
arXiv2026

Strong Conflict-Free Vertex-Connection via Twin Cover: Kernelization and Chromatic Bounds

A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a coloring is the strong conflict-free vertex-connection number $\operatorname{svcfc}(G)$. We study this problem under the parameter twin cover. Let $X$ be a twin cover of $G$ of size $t$, and let $k$ be the target number of colors. In our first result, given $(G,k)$ together with a twin cover $X$, we reduce in polynomial time to an equivalent annotated instance on at most $\max\{2,t+(t+1)k2^{t+k-1}\}$ vertices. Hence the annotated version of Strong CFVC Number, in which a twin cover is supplied as part of the input, is fixed-parameter tractable parameterized by $t+k$. Using this bound, we then obtain a kernel parameterized by $\operatorname{tc}(G)+k$; in particular, for every fixed $k$, the problem is fixed-parameter tractable parameterized by the twin-cover number alone. In our second result, we prove every connected graph $G$ with twin cover $X$ of size $t$ satisfies $χ(G)\le \operatorname{svcfc}(G)\le χ(G)+t$. More generally, if $Y\subseteq X$ intersects every shortest path of length at least $3$, then $\operatorname{svcfc}(G)\le χ(G)+|Y|$. We also derive an exact expression for the chromatic number on graphs of bounded twin-cover number: for every proper coloring $φ$ of $G[X]$, the minimum number of colors needed to extend $φ$ to all of $G$ is $K_φ=\max_{S\subseteq X}(|φ(S)|+m(S))$, and hence $χ(G)=\min_{φ\text{ proper on }G[X]} K_φ$. Our results provide the first evidence that twin cover is a useful parameter for strong conflict-free vertex-connection and show that, once a twin cover is fixed, the remaining difficulty is concentrated in a bounded additive gap above the chromatic number.

cs.DM↗
arXiv2026

HyCO: A Hybrid Neural Solver for Combinatorial Optimization

Sequential reinforcement learning (RL) solvers and global diffusion model (DM) solvers for neural combinatorial optimization exhibit complementary failure modes under an optimization-regret view. The former enjoys small marginal regret in the early construction stage, but suffers from horizon-wise compounding errors with super-linear regret growth; the latter avoids horizon compounding but incurs linear or sublinear regret w.r.t. the dimension of the remaining unsolved subspace. We propose Hybrid Neural Solver for Combinatorial Optimization (HyCO), a hybrid inference algorithm that constructs a solution prefix with an RL solver and adaptively switches to a conditional DM to complete the remaining decisions. To characterize why such hybridization helps, when to trigger the handover, and how to realize it in practice, we first develop a unified error-scaling theoretical framework and prove that, under explicit error-scaling assumptions, i) the hybrid structure achieves strictly lower expected regret than either backbone alone, and ii) there exists a unique optimal trigger step that minimizes the hybrid regret. We then design a lightweight adaptive trigger that combines policy entropy and RL-DM disagreement to detect trajectory-level signals of the regime shift as a practical proxy, since the optimal trigger step is defined at the expected-regret level and is not directly computable on individual trajectories. Experimental results on diverse benchmarks demonstrate that HyCO achieves consistent improvements over both backbones and support the empirical effectiveness of adaptive triggering.

cs.LG↗
arXiv2026

Persistent Memory Through Triple-Loop Consolidation Under Stochastic Unit Turnover

Dissipative cognitive architectures maintain computation through continuous energy expenditure, where units that exhaust their energy are stochastically replaced with fresh random state. This creates a fundamental challenge: how can persistent, context-specific memory survive when all learnable state is periodically destroyed? Existing memory mechanisms -- including elastic weight consolidation, synaptic intelligence, and surprise-driven gating -- rely on gradient computation and are inapplicable to systems that do not perform it. We introduce Deep Memory (DM), a backpropagation-free persistent memory mechanism operating through a triple-loop consolidation cycle: (1) recording of expert-specific content centroids, (2) seeding of replaced units with stored representations, and (3) stabilization through continuous re-entry. Discrete expert routing via Mixture-of-Experts (MoE) gating is required, in the regimes tested, to prevent the centroid convergence that would render stored memories identical. We derive a Foster-Lyapunov drift bound for the full triple loop, showing that seeding rescales the turnover noise floor. Across $1{,}007$ simulation runs over thirteen blocks: (i) removing stable context-expert binding removes specialization ($\mathrm{MI}=1.10$ vs. $0.001$; $n=91$); (ii) DM achieves $R=0.984$ vs. $0.385$ without memory ($n=16$); (iii) continuous seeding reconstructs representations after interference ($R_\mathrm{recon}=0.978$; one-shot fails; $n=30$); (iv) the mechanism operates within a characterized $(K,p)$ envelope ($n=350$); (v) recording $\times$ seeding is the minimal critical dyad ($n=40$); (vi) associative and reservoir baselines (Hopfield, ESN) are compared under matched turnover ($n=370$). DM is thus a falsifiable, bounded mechanism for persistent memory in backpropagation-free cognitive systems, with functional parallels to hippocampal consolidation.

cs.NE↗
arXiv2026

Decoupling Disaggregated Memory Optimizations from Indexing: A Compiler-Runtime Approach

Disaggregated memory (DM) decouples compute and memory into independently scalable pools, connected over a slower interconnect rather than a local bus. This decoupling is exactly what makes DM attractive--but it also means that every index must now reason explicitly about remote-memory access and its associated optimizations. State-of-the-art index designs respond to this by embedding remote-memory logic and optimizations directly into their core data structures and concurrency control mechanisms. Consequently, an optimization tuned for one index cannot be lifted and reused in another, and even the same index cannot be ported to a different DM architecture without a fresh round of redesign. This escalating, per-index, per-platform engineering burden is unsustainable as hardware and index requirements evolve. In this paper, we present Nox, a compiler-runtime framework that breaks this coupling by taking an unmodified, concurrent index as input and automatically generating its disaggregated-memory counterpart, without touching the original index logic. A compiler layer rewrites the index's LLVM IR to expose allocation, address, and pointer-dependency information that a centralized runtime uses to drive caching and address translation. Empirically, Nox-generated B+-trees, hash tables, and skip lists scale robustly on real RDMA and CXL hardware across every workload tested. They can also outperform some specialized, hand-crafted indexes and match others, especially on workloads that are closer to real-world ones. These results show that today's fastest disaggregated-memory optimizations need not stay locked inside monolithic, hand-crafted code--a compiler-runtime stack can generalize them while preserving the scalability of proven index implementations, without sacrificing it for portability.

cs.DB↗
arXiv2026

RAPTOR: Role-Aware Private Training for Mixture-of-Experts

Differentially private (DP) fine-tuning methods treat sparse Mixture-of-Experts (MoE) models as a single dense block, ignoring that shared layers see all data while experts only see routed records. We identify and formally characterize three resulting failure modes: global clipping suppresses expert gradients, batch-level normalization dilutes sparse expert updates, and fixed privacy noise degrades signal-to-noise ratio on low-load experts. We introduce RAPTOR - a Role-Aware Private Training framework, which alternates shared and expert optimization and targets each failure directly, using expert-specific clipping and noise together with a public expected-owner denominator and a count-independent update schedule that avoids conditioning on private, realized expert counts. We prove the resulting mechanism satisfies $(\varepsilon,δ)$-DP: because each record is assigned to exactly one owner expert, per-expert mechanisms within a layer compose in parallel, so updating all $E$ experts costs no more, in privacy terms, than updating one, with shared and expert streams composing sequentially across training. We further derive a bias-variance decomposition of the public-denominator estimator showing its bias grows predictably with routing imbalance, yielding a privacy-free rule for selecting which layer to protect from routing entropy measured on a small public corpus. Experiments on Switch Transformer and OLMoE fine-tuning across GLUE tasks, and on DeepSeek-VL2-Tiny, show consistent gains over standard DP baselines across several privacy levels ($\varepsilon$), with the largest margins typically at the tightest budgets. Code and models are publicly available: https://github.com/leduckhai/RAPTOR

cs.LG↗
arXiv2026

Climate Physics Dynamic Matching

Deep generative models such as flow matching and diffusion models have shown potential for learning complex dynamical systems, but typically act as black boxes that neglect underlying physical structure, while physics-based models governed by partial differential equations are often incomplete due to missing source terms, or uncertain parametrisations. We present Climate Physics Dynamic Matching (ClimPhyDM), a variational simulation-free dynamics informed framework for weather forecasting that combines an advection-type physics prior with data-driven components in a variational framework. % to capture the stochasticity and multi-modality of unresolved atmospheric dynamics. On the ERA5 benchmark at hourly (42-hour) and monthly (5-month) resolutions, ClimPhyDM outperforms ClimODE, and GB-DM, keeping the lower error at extended horizon, indicating improved temporal stability and resistance to error accumulation, while its simulation-free paradigm also enables training on a single modest 12 GB consumer GPU.

stat.AP↗
arXiv2026

MGDiff: Multi-Interest Sequence Recommendation with Masking GNN-Guided Diffusion

We propose a novel Multi-Interest Sequence Recommendation Framework with \underline{M}asking \underline{G}NN-Guided \underline{Diff}usion Model (MGDiff), designed to generate accurate, bias-free user interest information during the diffusion process. First, we propose a semantics-enhanced Dual-layer Semantic Guidance (DSG) framework, which decomposes guidance into two synergistic stages: extracting latent item semantics and decoupling multidimensional user intent. We design a Weight-adaptive Masking Graph Neural Network reconstructs missing links to uncover deep item relationships beyond superficial co-occurrence, while a Dynamic Multi-Expert Network projects user preferences into distinct semantic subspaces to suppress irrelevant interference. This hierarchical design yields structured guidance that significantly improves the generation accuracy of diffusion models. Second, We propose a Popularity-Aware Guidance (PAG) mechanism that performs spatial geometric adjustments on the outputs of diffusion models: by using item popularity as a differentiable adjustment signal to recalibrate similarity metrics, we enable DMs to generate diverse recommendations free from popularity bias. Finally, we compare MGDiff with multiple baseline models across four widely used datasets, demonstrating its superior performance and validating its effectiveness.

cs.IR↗
arXiv2026

Exact Recovery Thresholds for Weighted Data Selection in Vector-Valued Linear Regression

We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. In vector-valued linear regression with square loss $\ell_{(x,y)}(W)=|Wx-y|_2^2$, where $x\in\mathbb{R}^d$, $y\in\mathbb{R}^m$ and the learner is the empirical risk minimizer of minimal Frobenius norm, we prove that the minimal budget of weighted examples that recovers the full-data loss on every finite dataset is exactly $n^*(d,m)=(m+1)d$. We further determine two more values of the weighted selection profile $F_w(d,m,n)$: at the near-threshold budget, $F_w(d,m,(m+1)d-1)=1+\frac{1}{dm^2}$, and at the spanning budget, $F_w(d,m,d)=d+1$ for every $m$, while $F_w(d,m,n)=\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_w(2,2,3)\in[13/8,15/8]$ and $F_w(2,2,4)\in[5/4,3/2]$, reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms, and establish strong structural evidence for the conjecture. The upper-bound techniques (a fixed-basis conic compression lemma, a determinant-facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems) are of independent interest. As a byproduct we correct an erroneous claim circulating in a recent unrefereed preprint, exhibiting an explicit dataset with $m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. All results are new only for $m\ge 2$; the scalar case $m=1$ is due to Hanneke et al.

cs.LG↗
arXiv2025

Security Science (SecSci), Basic Concepts and Mathematical Foundations

This textbook compiles the lecture notes from security courses taught at Oxford in the 2000s, at Royal Holloway in the 2010s, and currently in Hawaii. The early chapters are suitable for a first course in security. The middle chapters have been used in advanced courses. Towards the end there are also some research problems.

cs.CR↗
arXiv2026

Fully Distributed GNE Algorithms for Multi-Robot Placement without Consensus on Multipliers

Recent machine learning research has increasingly focused on equilibrium analysis in non-cooperative games rather than solely on optimal solutions. Many such problems involve shared constraints and can be formulated as Generalized Nash Equilibrium Problems (GNEPs). For strongly monotone games, existing methods compute consensus-based variational GNEs (v-GNEs) by exchanging Lagrange multipliers. We propose a fully distributed continuous-time algorithm for shared linear equality constraints that converges without multiplier exchange and reaches any GNE, reducing communication overhead and improving privacy. Discrete-time schemes are also provided, and the method is validated on a multi-robot placement task.

cs.LG↗
arXiv2026

Input-to-State Stability Framework for Fully Distributed Primal-Dual Dynamics for Quadratic GNEPs Without Multiplier Consensus

Generalized Nash Equilibrium Problems (GNEPs) often arise in multi-agent engineering applications that require distributed algorithms. Unlike traditional approaches that enforce consensus on multipliers, our method removes the need to share multipliers, reducing communication and improving privacy. As a result, different initializations can lead to different GNEs, including non-variational ones. We establish convergence under sufficient conditions using an input-to-state stability (ISS) framework.

math.OC↗
arXiv2026

TSExplorer: An interactive data annotation and exploration tool for time-series data

We present TSExplorer, a cross-platform tool for interactive annotation and exploration of time-series data. The tool enables users to inspect high-dimensional datasets through multiple complementary 2D visualizations derived from high-dimensional feature representations. TSExplorer is designed as a general-purpose research tool supporting a wide range of workflows, including exploratory data analysis, annotation of unlabeled or partially-labeled datasets, comparison of feature representations, and post-hoc inspection and refinement of existing labels with interactive visual feedback.

cs.HC↗
arXiv2026

Towards a universal language of concepts: A survey

Humans can learn and generalize novel concepts from sparse data because they express knowledge in rich structural formats. In this paper, we propose that programs are a strong candidate for universal representation of concepts. We review computational models of concept learning that use programs as their concept representation and evaluate their contribution toward a universal representational language.

cs.AI↗
arXiv2026

Hardware Trojan Threats to Multi-Chiplet Photonic Neural Network Accelerators

Multi-chiplet photonic neural network accelerators (MCPNAs) combine photonic computation, photonic communica-tion, and heterogeneous chiplet integration to enable scalable and energy-efficient AI acceleration. However, their distributed archi-tecture and reliance on third-party chiplets introduce significant hardware security risks. This paper examines Hardware Trojan (HT) threats to MCPNAs across three dimensions: confidentiality, integrity, and availability.

cs.AR↗