New constructions intersect k NFAs in O(m n^{k-1}) transitions for fixed alphabet, enabling faster emptiness algorithms that are optimal unless (k+1)-clique detection admits a combinatorial breakthrough.
If the current clique algorithms are optimal, so is Valiant’s parser
3 Pith papers cite this work. Polarity classification is still indexing.
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Sparsity helps for k-independent set only below certain density thresholds, with new algorithms achieving O(min(n^{ωk/3} + m^{k/3}, n^k)) time and conditional lower bounds showing brute-force necessity above thresholds for many binary constraint families.
Algorithms for LS Vertex Cover achieve ℓ^{f(k)} n^{O(1)} time for ℓ equal to h-index, treewidth, modular-width, or a new modular-decomposition degree parameter, and extend to weighted d-improving swaps.
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Intersecting Dense Automata
New constructions intersect k NFAs in O(m n^{k-1}) transitions for fixed alphabet, enabling faster emptiness algorithms that are optimal unless (k+1)-clique detection admits a combinatorial breakthrough.
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When Does Sparsity Help for k-Independent Set in Hypergraphs and Other Boolean CSPs?
Sparsity helps for k-independent set only below certain density thresholds, with new algorithms achieving O(min(n^{ωk/3} + m^{k/3}, n^k)) time and conditional lower bounds showing brute-force necessity above thresholds for many binary constraint families.