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(√{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.

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).

A multiset of literals, called a clause, is \emph{strongly satisfied} by an assignment if \emph{no} literal evaluates to false. Finding an assignment that maximises the number of strongly satisfied clauses is NP-hard. We present a simple algorithm that finds, given a multiset of clauses that admits an assignment that strongly satisfies $ρ$ of the clauses, an assignment in which at least $ρ$ of the clauses are \emph{weakly satisfied}, in the sense that an \emph{even} number of literals evaluate to false. In particular, this implies an efficient algorithm for finding an undirected cut of value $ρ$ in a graph $G$ given that a directed cut of value $ρ$ in $G$ is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of $G$ with $ρ$ edges under the same promise.

Fix two non-empty loopless graphs $G$ and $H$ such that $G$ maps homomorphically to $H$. The Maximum Promise Constraint Satisfaction Problem parameterised by $G$ and $H$ is the following computational problem, denoted by MaxPCSP($G$, $H$): Given an input (multi)graph $X$ that admits a map to $G$ preserving a $ρ$-fraction of the edges, find a map from $X$ to $H$ that preserves a $ρ$-fraction of the edges. As our main result, we give a complete classification of this problem under Khot's Unique Games Conjecture: The only tractable cases are when $G$ is bipartite and $H$ contains a triangle. Along the way, we establish several results, including an efficient approximation algorithm for the following problem: Given a (multi)graph $X$ which contains a bipartite subgraph with $ρ$ edges, what is the largest triangle-free subgraph of $X$ that can be found efficiently? We present an SDP-based algorithm that finds one with at least $0.8823 ρ$ edges, thus improving on the subgraph with $0.878 ρ$ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson.

The 1 -in- 3 and N ot -A ll -E qual satisfiability problems for Boolean CNF formulas are two well-known NP -hard problems. In contrast, the promise 1 -in- 3 vs . N ot -A ll -E qual problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure and establish a complexity dichotomy for the resulting class of computational problems.

A linearly ordered (LO) $k$-colouring of a hypergraph assigns to each vertex a colour from the set $\{0,1,\ldots,k-1\}$ in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO $k$-colouring of an LO 2-colourable 3-uniform hypergraph for any constant $k\geq 2$ [STACS'21] but even the case $k=3$ is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with $O^*(\sqrt{n})$ colours [ICALP'22] and an LO colouring with $O^*(\sqrt[3]{n})$ colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with $O^*(\sqrt[5]{n})$ colours [FSTTCS'24]. We present two simple polynomial-time algorithms that find an LO colouring with $O(\log_2(n))$ colours, which is an exponential improvement.

The 1-in-3 and Not-All-Eqal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Eqal problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure, and establish a complexity dichotomy for the resulting class of computational problems.

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).

The complexity of the promise constraint satisfaction problem \(\operatorname{(PCSP)}(\mathbf{A},\mathbf{B})\) is largely unknown, even for symmetric \(\mathbf{A}\) and \(\mathbf{B}\) , except for the case when \(\mathbf{A}\) and \(\mathbf{B}\) are Boolean. First, we establish a dichotomy for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) where \(\mathbf{A},\mathbf{B}\) are symmetric, \(\mathbf{B}\) is functional (i.e., any \(r-1\) elements of an \(r\) -ary tuple uniquely determines the last one), and \((\mathbf{A},\mathbf{B})\) satisfies technical conditions we introduce called dependency and additivity . This result implies a dichotomy for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) with \(\mathbf{A},\mathbf{B}\) symmetric and \(\mathbf{B}\) functional if (i) \(\mathbf{A}\) is Boolean, or (ii) \(\mathbf{A}\) is a hypergraph of a small uniformity, or (iii) \(\mathbf{A}\) has a relation \(R^{\mathbf{A}}\) of arity at least three such that the hypergraph diameter of \((A,R^{\mathbf{A}})\) is at most one. Second, we show that for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) , where \(\mathbf{A}\) and \(\mathbf{B}\) contain a single relation, \(\mathbf{A}\) satisfies a technical condition called balancedness , and \(\mathbf{B}\) is arbitrary, the combined basic linear programming relaxation and the affine integer programming (AIP) relaxation is no more powerful than the (in general strictly weaker) \({{\rm AIP}}\) relaxation. Balanced \(\mathbf{A}\) include symmetric \(\mathbf{A}\) or, more generally, \(\mathbf{A}\) preserved by a transitive permutation group.

As our first result, we establish a dichotomy for promise constraint satisfaction problems of the form PCSP(A, B), where A is Boolean and symmetric and B is functional (on a domain of any size); i.e, all but one element of any tuple in a relation in B determine the last element. This includes PCSPs of the form PCSP(q-in-r, B), where B is functional, thus making progress towards a classification of PCSP(1-in-3, B), which were studied by Barto, Battistelli, and Berg [STACS’21] for B on three-element domains.As our second result, we show that for PCSP(A, B), where A contains a single symmetric relation and B is arbitrary (and thus not necessarily functional), the combined basic linear programming relaxation (BLP) and the affine integer programming relaxation (AIP) of Brakensiek et al. [SICOMP’20] is no more powerful than the (in general strictly weaker) AIP relaxation of Brakensiek and Guruswami [SICOMP’21].

A linearly ordered (LO) $k$-colouring of an $r$-uniform hypergraph assigns an integer from $\{1, \ldots, k \}$ to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for $r=3$, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on $3$-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results. First, given a 3-uniform hypergraph that admits an LO $2$-colouring, one can find in polynomial time an LO $k$-colouring with $k=O(\sqrt[3]{n \log \log n / \log n})$. Second, given an $r$-uniform hypergraph that admits an LO $2$-colouring, we establish NP-hardness of finding an LO $k$-colouring for every constant uniformity $r\geq k+2$. In fact, we determine relationships between polymorphism minions for all uniformities $r\geq 3$, which reveals a key difference between $r