Tamio-Vesa Nakajima

Finn Moltmann, Tamio-Vesa Nakajima, Sebastian Wild

We give a more space-efficient implementation of adaptive mergesort: Virtual-Memory Powersort. Using internal buffering techniques, we significantly reduce the memory consumption of the algorithm; specifically, for sorting $n$ objects the required buffer area is reduced from space for $n/2$ objects to $O(\sqrt{n \log n})$ objects. While this space-efficiency can be achieved (indeed reduced to $O(1)$) conceptually very easily with known inplace merging algorithms, using these as a drop-in replacement for the standard merge algorithm incurs a substantial slow-down. Virtual-Memory Powersort, by contrast, uses the same number of moves and comparisons as previous Powersort implementations up to an additive $O(n)$ term. We report on an empirical running-time study comparing our implementation against other Powersort variants and state-of-the-art stable sorting methods, demonstrating that almost in-place stable sorting can be achieved with negligible overhead in many scenarios.

Benjamin Bedert, Tamio-Vesa Nakajima, Karolina Okrasa et al.

We introduce a new notion of sparsification, called \emph{strong sparsification}, in which constraints are not removed but variables can be merged. As our main result, we present a strong sparsification algorithm for 1-in-3-SAT. The correctness of the algorithm relies on establishing a sub-quadratic bound on the size of certain sets of vectors in $\mathbb{F}_2^d$. This result, obtained using the recent \emph{Polynomial Freiman-Ruzsa Theorem} (Gowers, Green, Manners and Tao, Ann. Math. 2025), could be of independent interest. As an application, we improve the state-of-the-art algorithm for approximating linearly-ordered colourings of 3-uniform hypergraphs (Håstad, Martinsson, Nakajima and{Ž}ivn{ý}, APPROX 2024). We also investigate the existence of strong sparsification algorithms for other constraint satisfaction problems.

Tamio-Vesa Nakajima, Zephyr Verwimp, Marcin Wrochna et al.

Using the algebraic approach to promise constraint satisfaction problems, we establish complexity classifications of three natural variants of hypergraph colourings: standard nonmonochromatic colourings, conflict-free colourings, and linearly-ordered colourings. Firstly, we show that finding an $\ell$-colouring of a $k$-colourable $r$-uniform hypergraph is NP-hard for all constant $2\leq k\leq \ell$ and $r\geq 3$. This provides a shorter proof of a celebrated result by Dinur et al. [FOCS'02/Combinatorica'05]. Secondly, we show that finding an $\ell$-conflict-free colouring of an $r$-uniform hypergraph that admits a $k$-conflict-free colouring is NP-hard for all constant $3\leq k\leq\ell$ and $r\geq 4$, except for $r=4$ and $k=2$ (and any $\ell$); this case is solvable in polynomial time. The case of $r=3$ is the standard nonmonochromatic colouring, and the case of $r=2$ is the notoriously difficult open problem of approximate graph colouring. Thirdly, we show that finding an $\ell$-linearly-ordered colouring of an $r$-uniform hypergraph that admits a $k$-linearly-ordered colouring is NP-hard for all constant $3\leq k\leq\ell$ and $r\geq 4$, thus improving on the results of Nakajima and Živný [ICALP'22/ACM TocT'23].

Demian Banakh, Alexey Barsukov, Tamio-Vesa Nakajima

The logic MMSNP is a well-studied fragment of Existential Second-Order logic that, from a computational perspective, captures finite-domain Constraint Satisfaction Problems (CSPs) modulo polynomial-time reductions. At the same time, MMSNP contains many problems that are expressible as $ω$-categorical CSPs but not as finite-domain ones. We initiate the study of Promise MMSNP (PMMSNP), a promise analogue of MMSNP. We show that every PMMSNP problem is poly-time equivalent to a (finite-domain) Promise CSP (PCSP), thereby extending the classical MMSNP-CSP correspondence to the promise setting. We then investigate the complexity of PMMSNPs arising from forbidding monochromatic cliques, a class encompassing promise graph colouring problems. For this class, we obtain a full complexity classification conditional on the Rich 2-to-1 Conjecture, a recently proposed perfect-completeness surrogate of the Unique Games Conjecture. As a key intermediate step which may be of independent interest, we prove that it is NP-hard, under the Rich 2-to-1 Conjecture, to properly colour a uniform hypergraph even if it is promised to admit a colouring satisfying a certain technical condition called reconfigurability. This proof is an extension of the recent work of Braverman, Khot, Lifshitz and Minzer (Adv. Math. 2025). To illustrate the broad applicability of this theorem, we show that it implies most of the linearly-ordered colouring conjecture of Barto, Battistelli, and Berg (STACS 2021).

Ziad Ismaili Alaoui, Tamio-Vesa Nakajima, Namrata et al.

We combine two methods for the lossless compression of unlabeled graphs - entropy compressing adjacency lists and computing canonical names for vertices - and solve an ensuing novel optimisation problem: Minimum-Entropy Tree-Extraction (MINETREX). MINETREX asks to determine a spanning forest $F$ to remove from a graph $G$ so that the remaining graph $G-F$ has minimal indegree entropy $H(d_1,\ldots,d_n) = \sum_{v\in V} d_v \log_2(m/d_v)$ among all choices for $F$. (Here $d_v$ is the indegree of vertex $v$ in $G-F$; $m$ is the number of edges.) We show that MINETREX is NP-hard to approximate with additive error better than $δn$ (for some constant $δ>0$), and provide a simple greedy algorithm that achieves additive error at most $n / \ln 2$. By storing the extracted spanning forest and the remaining edges separately, we obtain a degree-entropy compressed ("ultrasuccinct") data structure for representing an arbitrary (static) unlabeled graph that supports navigational graph queries in logarithmic time. It serves as a drop-in replacement for adjacency-list representations using substantially less space for most graphs; we precisely quantify these savings in terms of the maximal subgraph density. Our inapproximability result uses an approximate variant of the hitting set problem on biregular instances whose hardness proof is contained implicitly in a reduction by Guruswami and Trevisan (APPROX/RANDOM 2005); we consider the unearthing of this reduction partner of independent interest with further likely uses in hardness of approximation.