constraint programming engineers

Abstract: Algorithm Selection is essential for efficient Constraint Programming. Over the years, many algorithm selectors based on machine learning methods have been successfully applied, yet traditional feature extraction methods often rely on manually decided instance-level statistics that fail to capture the underlying problem structure. In this paper we aim to bridge this gap by introducing a novel, automated feature extraction methodology that integrates graph conversion and Weisfeiler-Lehman graph kernels to generate robust structural representations of problem instances. The 1-WL test bounds the graph-distinguishing power of standard message-passing Graph Neural Networks (GNNs), and suitable GNN architectures match this bound \citep{Xuetal2018}. WL-based features offer an alternative that does not require training a GNN. Our primary contribution is a cut-based representation (\texttt{WLc}) designed to model structural partitions and provide a more nuanced predictive signal. We evaluate our approach on instances from the 2023--2025 MiniZinc Challenges across two tasks: maximizing Borda count scores and maximizing predictive accuracy. Experimental results across Support Vector Machines, Random Forests, and Multi-Layer Perceptrons demonstrate that cut-based features outperform \texttt{fzn2feat} with SVMs, while results with RFs and MLPs are closer.

Abstract: Building on the bit-complexity framework of Raghavendra-Weitz and the moment-SOS criteria of Gribling-Polak-Slot, we study the effective use of truncated vanishing identities over Boolean polynomial systems. A complete, constructible identity space gives an augmented moment SDP that can be optimized with exact rational feasibility and arbitrary additive accuracy. We give a self-contained geometric implementation: an explicit affine reduction and a simplex of Boolean evaluations supply the radius bounds required by rational ellipsoids. The identity space can be constructed directly or extracted from a supplied graded Groebner basis, including one supplied at a higher truncation degree. An explicit transfer theorem connects this augmented formulation to the original system. Two-sided SoS derivations eliminate the added equality axioms from certificates, with controlled degree and coefficient growth, and imply containment of a projected higher-level moment relaxation in the augmented body. Together with spectral coefficient bounds, this gives polynomial-time search for rational proofs with an additive perturbation. For Min-closed linear systems, propagation constructs the identity space in polynomial time at fixed degree and gives degree-(4t+4) certificates for both signs of each degree-t basis element. Consequently, a rational degree-(8d+4) proof of f + epsilon >= 0 can be found in polynomial time for fixed d whenever f >= 0 has a degree-2d proof. The augmented degree-2d moment SDP can be optimized in polynomial time with exact rational feasibility and a comparison to the original degree-(8d+4) relaxation. Boolean complementation gives the same results for Max-closed systems, including generalized packing and covering. Min-closed systems thus provide a concrete application of the general criteria. All complexity bounds are in the Turing model.

Abstract: Assuming the Exponential Time Hypothesis (ETH), fixed-parameter tractability and polynomial-time solvability coincide for homomorphism problems specified by classes of pattern hypergraphs of bounded incidence degeneracy or bounded primal independence number. In both cases, tractability is characterised by bounded fractional hypertree width. Grohe (JACM 2007) established the corresponding FPT-PTIME equivalence under bounded arity. Our result allows unbounded arity and covers important cases such as bounded-degree patterns and patterns whose incidence graphs exclude a fixed minor. Building on the recent fractional balanced-separator framework and rounding theorem of Korchemna et al. (FOCS 2024), we prove a near-linear bound on fractional hypertree width ($\mathsf{fhw}$) in terms of adaptive width ($\mathsf{adw}$). For every hypergraph $H$ with $\mathsf{adw}(H)\geq 2$, \[ \mathsf{fhw}(H)=O\bigl(λ(H)\mathsf{adw}(H)\log\mathsf{adw}(H)\bigr), \] where $λ(H)=\min\{μ(H),\max\{1,\logα(H)\}\}$, with $μ(H)$ denoting incidence degeneracy and $α(H)$ the independence number of the primal graph. As a further consequence, we obtain a corresponding FPT-PTIME collapse for exact homomorphism counting on every bounded-$λ$ class. More generally, for every recursively enumerable class of pattern hypergraphs, fixed-parameter tractability of the parameterised homomorphism problem implies quasipolynomial-time solvability of the corresponding unparameterised problem, assuming ETH.