Given a graph G and an integer k, the ACHROMATIC COLORING problem asks for a coloring of a graph using at least k colors such that every pair of distinct colors appears on at least one edge. It is a classical and notoriously hard coloring problem with applications in graph labeling and combinatorial optimization. From the parameterized complexity perspective, this problem has recently attracted renewed attention. Jana et al. (ISAAC 2025) showed that ACHROMATIC COLORING is fixed-parameter tractable when parameterized by the vertex cover number cw, via a reduction to SuBGRAPH ISOMORPHISM with a pattern graph of size O(cw2) and constant treewidth, yielding a running time of 2O(cw2) , nO(1). We present a simple dynamic programming based algorithm that runs in time 2vc2/2+O(vc log vc) , nO(1). The problem is NP-hard even when restricted to trees. For forests, Jana et al. in the same paper obtained a kernel of size O(k2), where k is the number of colors, implying an algorithm with running time O(kk2,nO(1)). The problem does not admit a fixed-parameter tractable (FPT) algorithm when parameterized by treewidth unless P=NP. We consider a parameter clique-width that generalizes tree-width and design an FPT algorithm with the combined parameters clique-width cw and the number of colors, running in time O(2 2kcw+k2,(cw + k2) , |E|) where E is expression given as input. As a corollary of this result, we obtain a fixed-parameter algorithm for forests running in time 2k2+O(k) , nO(1), improving upon the previously known kernel-based algorithm.
Given a set of equations over terms, we investigate whether the set of equivalence classes they induce is finite. This problem is undecidable in general but plays an important role in program verification, where finiteness of the set of the equivalence classes ensures termination of the verification. We introduce a semi-algorithm for this problem that uses the Myhill-Nerode Theorem and results from [1] to build a deterministic finite tree automaton recognizing the equivalence classes induced by the equations. This algorithm is correct and semi-complete, i.e., if the set of equivalence classes is finite then it successfully builds the automaton, otherwise it loops.
MAXIMUM MINIMAL SET COVER is the max-min variant of SET COVER, and consists of finding a minimal set cover of maximum cardinality. It generalizes MAX MIN VERTEX COVER and UPPER DOMINATION. Besides that, (MAXIMUM MINIMAL) SET COVER and (MAXIMUM MINIMAL) HITTING SET can be considered similar problems depending on the complexity aspect analyzed. However, when the size of the universe, say n, is considered as a parameter, these families of problems may behave differently. An O*(2n)-time algorithm for (MAXIMUM MINIMAL) HITTING SET is trivial, while exhaustively trying all possible ways to cover the elements takes O*(2m) time, where m is the number of sets. Since either m or n may be significantly larger than the other, it is natural to ask for the best running time that one can achieve in terms of n (the universe size) to MAXIMUM MINIMAL SET COVER. In this regime, we show that such a problem can be solved in O*(2(n)) time, but it cannot be solved in O*(2(sigma(n))), unless ETH fails. We also present an O*(2(k))-time algorithm for MAXIMUM MINIMAL SET COVER, where k is the vertex cover number of the incidence graph of the input. This is an improvement of the O*(2n)-time algorithm, since k <= min{n, m}. Finally, we show that MAXIMUM MINIMAL SET COVER cannot be solved in O*(1.1892k) time, unless SETH fails.
In this article we establish that imposing a minimum length for the runs of TIMSORT's decomposition, as is always done in practice, makes its average-case complexity almost optimal, that is, asymptotically nlog(2) n + Theta(n) comparisons. Theoretically, we need the minimum length to depends on n, specifically it must be Omega(log n/log log n). In practice, however, the fixed threshold adopted in programming languages is sufficient for all realistic input sizes.
Given a budget and an undirected graph with a cost and a profit function on the edges, the Edge Orienteering Problem asks for a closed walk in the graph that collects the most profit while the sum of costs does not exceed the budget. While the profit is collected only once per edge, the cost is incurred each time the edge is traversed by the walk. In this paper, we present a (4 + epsilon)-approximation algorithm for the Edge Orienteering Problem.
