network infrastructure planners

Abstract: Generalizing from small graphs to larger ones can require computation beyond the depth of a neural model used during training. We study target tasks that admit a common local iterative computation and whose effective dependency radius grows with graph size. The central idea is to learn a local update rule through GNN pretraining, then repeatedly apply the same learned processor beyond the training depth to extend information reach on larger graphs without retraining. Calling such parameter-shared architecture a \emph{looped GNN}, we categorize six schemes for looped GNNs to separate the effects of depth extension, fine-tuning, and freeze-and-steer adaptation. Our analysis establishes fixed-depth information limits and sufficient conditions under which accurate iterative reuse overcomes them. To improve beyond unadapted reuse, we introduce \emph{Loop-Freeze-Steer} (LFS), which freezes the pretrained processor and learns a small controller to steer its repeated application, using much fewer trainable parameters than full fine-tuning. Our finite-sample analysis gives sufficient conditions for adaptation to improve performance on larger graphs. Experiments on synthetic tasks and chip placement evaluate depth extension and adaptation, highlighting LFS's accuracy--trainable-parameter trade-off for size generalization.

Abstract: The Eccentricity Shortest Path problem asks, given a connected graph and a nonnegative integer $R$, whether some shortest path is within distance $R$ of every vertex. We study its structural parameterized complexity. We give fixed-parameter algorithms for vertex-deletion distance to cographs, vertex-deletion distance to interval graphs, and bandwidth. The interval-deletion result answers a question of Bhyravarapu et al. (IWOCA 2023). We obtain XP algorithms for vertex-deletion distance to chordal graphs, clique-width, and rank-width, without requiring expressions or decompositions for the latter two. Consequently, the problem is polynomial-time solvable for every fixed treewidth, answering a question of Chakraborty et al. (2026). We also derive fixed-parameter tractability for treedepth and XP membership for diameter from earlier algorithms. We prove $\mathsf{W}[3]$-hardness for cutwidth, and hence pathwidth and treewidth, even on graphs of maximum degree at most five; $\mathsf{W}[2]$-hardness for vertex clique-cover number; and NP-hardness on graphs of tree-independence number at most two. For maximum leaf number $\ell$, we obtain an $O(\ell^4)$-bit compression and a polynomial kernel. Finally, unless $\mathsf{NP}\subseteq\mathrm{coNP}/\mathrm{poly}$, we exclude polynomial kernels for bandwidth, feedback edge number, vertex-deletion distance to a clique, and vertex cover number. The last bound also excludes polynomial kernels for vertex-deletion distance to proper interval graphs.

Abstract: Discrete diffusion models have emerged as a powerful paradigm for solving combinatorial optimization (CO) problems on graphs by learning to sample high-quality solutions. A common inference-time approach is to generate multiple candidate solutions independently and return the best-performing sample, improving solution quality at the expense of an increase in computational cost. In this work, we introduce PT-Denoise, an inference-time procedure that allows these concurrent denoising trajectories to interact through parallel tempering, without requiring retraining or fine-tuning of the underlying denoiser. Our method assigns a temperature to each diffusion process and allows processes to swap temperatures based on their relative performance. This dynamically reallocates promising, low-energy trajectories to colder, more concentrated sampling regimes while allowing higher-energy states to escape local minima through randomized exploration. Experiments on canonical graph-structured CO problems show that our approach consistently improves the quality of the best solution found, while only adding minimal computational overhead.

Abstract: We study the following class of matroid optimization problems with a linear constraint (P-MOL). Given a matroid M=(E,I), two weight functions $v,w:E\to R_{\ge 0}$, and a threshold $L\in R_{\ge 0}$, find $opt v(S)$ where S is either an independent set or a base of M satisfying a budget-type constraint: $w(S)\le L$ or $w(S)\ge L$, and $opt\in\{min,max\}$. P-MOL provides a unified representation for a broad family of NP-hard optimization problems, including budgeted matroid independent set, constrained minimum-basis, and knapsack-cover variants with a matroid constraint. Also, it naturally extends to multiple matroid constraints. In particular, we consider the matroid intersection cover (MIC) problem, where feasibility is defined by the common independent sets of two matroids and one seeks minimum $v(S)$ subject to $w(S)\ge L$. Our main result is a unified EPTAS for all nontrivial P-MOL variants, obtained by generalizing a technique of Hassin and Levin (SIAM J. Comput., 2004) for solving the constrained minimum spanning tree problem. Specifically, for any fixed $ε>0$, we present an algorithm running in time $|E|^{O(1)} (1/{ε^2})^{O(1/ε)}$ that outputs a feasible solution S whose value is at most $(1+ε)OPT$ for minimization variants and at least $(1-ε)OPT$ for maximization variants. This resolves the complexity status of all members of P-MOL, as none of these problems admits an FPTAS (Doron-Arad, Kulik and Shachnai, ICALP'24). Finally, we separate the P-MOL family from its extension to matroid intersection. We show that an EPTAS is unlikely to exist for the matroid intersection variant of P-MOL under a covering constraint, whereas an EPTAS is known to exist under a budget constraint. This highlights a qualitative difference between these two types of linear constraints that does not arise in the single-matroid setting.