arXiv subjects

Björn Martinsson

Publications and source records attributed to Björn Martinsson.

3 recordsLinked to original sources

arXiv2025

On the Usefulness of Promises

A Boolean predicate $A$ is defined to be promise-useful if $\operatorname{PCSP}(A,B)$ is tractable for some non-trivial $B$ and otherwise it is promise-useless. We initiate investigations of this notion and derive sufficient conditions for both promise-usefulness and promise-uselessness (assuming $\text{P} \ne \text{NP}$). While we do not obtain a complete characterization, our conditions are sufficient to classify all predicates of arity at most $4$ and almost all predicates of arity $5$. We also derive asymptotic results to show that for large arities a vast majority of all predicates are promise-useless. Our results are primarily obtained by a thorough study of the "Promise-SAT" problem, in which we are given a $k$-SAT instance with the promise that there is a satisfying assignment for which the literal values of each clause satisfy some additional constraint. The algorithmic results are based on the basic LP + affine IP algorithm of Brakensiek et al. (SICOMP, 2020) while we use a number of novel criteria to establish NP-hardness.

cs.CC↗

arXiv2025

A logarithmic approximation of linearly ordered colourings

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.

math.CO↗

arXiv2024

On the NP-Hardness Approximation Curve for Max-2Lin(2)

In the Max-2Lin(2) problem you are given a system of equations on the form $x_i + x_j \equiv b \pmod{2}$, and your objective is to find an assignment that satisfies as many equations as possible. Let $c \in [0.5, 1]$ denote the maximum fraction of satisfiable equations. In this paper we construct a curve $s (c)$ such that it is NP-hard to find a solution satisfying at least a fraction $s$ of equations. This curve either matches or improves all of the previously known inapproximability NP-hardness results for Max-2Lin(2). In particular, we show that if $c \geqslant 0.9232$ then $\frac{1 - s (c)}{1 - c} > 1.48969$, which improves the NP-hardness inapproximability constant for the min deletion version of Max-2Lin(2). Our work complements the work of O'Donnell and Wu that studied the same question assuming the Unique Games Conjecture. Similar to earlier inapproximability results for Max-2Lin(2), we use a gadget reduction from the $(2^k - 1)$-ary Hadamard predicate. Previous works used $k$ ranging from $2$ to $4$. Our main result is a procedure for taking a gadget for some fixed $k$, and use it as a building block to construct better and better gadgets as $k$ tends to infinity. Our method can be used to boost the result of both smaller gadgets created by hand $(k = 3)$ or larger gadgets constructed using a computer $(k = 4)$.

cs.CC↗