When training complex neural networks, memory usage can be an important bottleneck.

We study the computational problem of computing a fair means clustering of discrete vectors, which admits an equivalent formulation as editing a colored matrix into one with few distinct color-balanced rows by changing at most $k$ values. While NP-hard in both the fairness-oblivious and the fair settings, the problem is well-known to admit a fixed-parameter algorithm in the former ``vanilla'' setting. As our first contribution, we exclude an analogous algorithm even for highly restricted fair means clustering instances. We then proceed to obtain a full complexity landscape of the problem, and establish tractability results which capture three means of circumventing our obtained lower bound: placing additional constraints on the problem instances, fixed-parameter approximation, or using an alternative parameterization targeting tree-like matrices.

The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. Conditioning WFOMC on evidence -- fixing the truth values of a set of ground literals -- has been shown impossible in time polynomial in the domain size (unless $\mathsf{\#P \subseteq FP}$) even for fragments of logic that are otherwise tractable for WFOMC without evidence. In this work, we address the barrier by restricting the binary evidence to the case where the underlying Gaifman graph has bounded treewidth. We present a polynomial-time algorithm in the domain size for computing WFOMC for the two-variable fragments $\text{FO}^2$ and $\text{C}^2$ conditioned on such binary evidence. Furthermore, we show the applicability of our algorithm in combinatorial problems by solving the stable seating arrangement problem on bounded-treewidth graphs of bounded degree, which was an open problem. We also conducted experiments to show the scalability of our algorithm compared to the existing model counting solvers.

Neural models have shown promise in solving NP-hard graph combinatorial optimization (CO) problems. Once trained, they offer fast inference and reasonably high-quality solutions for in-distribution testing instances, but they generally fall short in terms of absolute solution quality compared to classical search-based algorithms that are admittedly slower but offer optimality guarantee once search finishes. We propose a novel framework that combines the inference efficiency and exploratory power of neural models with the solution quality guarantee of search-based algorithms. In particular, we use parameterized algorithms (PAs) as the search component. PAs are dedicated to identifying easy instances of generally NP-hard problems, and allow for practically efficient search by exploiting structural simplicity (of the identified easy instances). Under our framework, we use parameterized analysis to identify the structurally hard parts of a CO instance. The neural model handles the hard parts by generating advisory signals based on its data-driven understanding. The PA-based search component then integrates the advisory signals to systematically and efficiently searches through the remaining structurally easy parts. Notably, our framework is agnostic to the choice of neural model and produces strictly better solutions than neural solvers alone. We examine our framework on multiple CO tasks. Empirical results show that it achieves superior solution quality, competitive with that of commercial solvers. Furthermore, by using the neural model only for exploratory advisory signals, our framework exhibits improved out-of-distribution generalization, addressing a key limitation of existing neural CO solvers.

Numerical results demonstrate the efficacy of the new penalties.

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