Ultrasound measurement of follicle diameter is essential in IVF monitoring. This study evaluates the analytical performance of follicle counts and size measurements from two-dimensional images using an AI-based platform, compared to assessments by certified sonographers. A total of 5508 TVUS scans from 1689 patients undergoing controlled ovarian stimulation across four IVF centers (Poland, Argentina, Colombia, and the USA) were retrospectively analyzed. All visible follicles were marked using bounding boxes. The dataset included three subsets: training/validation for model development, independent test for evaluating performance across ultrasound systems, and a consensus test set (102 scans from 27 patients) annotated by three expert sonographers. Model performance was assessed using precision, recall, and F1 score. Annotation efficiency was measured by comparing manual and AI-assisted times. Real-world performance was evaluated on a prospective cohort of 904 scans from 269 patients, based on expert adjustments to AI annotations. For follicles ≥ 10 mm, the model achieved 98.2

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 l-colouring of a k-colourable r-uniform hypergraph is NP-hard for all constant 2 <= k <= l and r >= 3. This provides a shorter proof of a celebrated result by Dinur et al. [FOCS'02/Combinatorica'05]. Secondly, we show that finding an l-conflict-free colouring of an r-uniform hypergraph that admits a k-conflict-free colouring is NP-hard for all constant 2 <= k <= l and r >= 4, except for r = 4 and k = 2 (and any l); 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 l-linearly-ordered colouring of an r-uniform hypergraph that admits a k-linearly-ordered colouring is NP-hard for all constant 3 <= k <= l and r >= 4, thus improving on the results of Nakajima and Zivny [ICALP'22/ACM TocT'23].

Abstract Study question How often are fully automated follicle counts and measurements modified in expert review? Summary answer Automatic follicle annotations provided by an artificial intelligence platform in regular operation of a clinic were edited 3.27% of the time. What is known already Ovarian follicle counting is frequently performed, time-consuming, and subject to noticeable inter-observer variability; as such, it is well-suited to automatizing with artificial intelligence. FOLLISCAN (MIM Fertility) is a software platform that automatically annotates follicles on 2D or 3D ultrasound cine videos, with exact outlines and measurements, without any manual pre-processing. It has undergone several retrospective tests of its precision and recall in detecting ovarian follicles, accuracy of measurements, and time-savings compared to manual counting Study design, size, duration The platform was integrated with a clinic’s existing picture archiving and medical record systems. Ultrasound scans of ovaries were performed as part of regular infertility treatment in two IVF centers. Scans were made by 3 experts in a period from November to December 2023, automatically sent to the platform, analyzed, and then immediately visualized for review by the person performing the ultrasound, in place of manual annotation. Participants/materials, setting, methods The study included 294 cine videos from 147 examinations of 101 patients, with 4347 follicles in total (14.7 per ovary on average, from 2mm). The platform allows users to modify proposed measurements and introduce new follicle annotations. After confirming the review, results were automatically sent back to the clinic’s system to be used in medical decisions just as manual annotations would be. Main results and the role of chance Among all videos the average number of editions (follicle additions, modifications, or deletions) was 0.48 (CI: 0.37-0.61). In total, 142 follicles out of 4347 (3.27%, CI: 2.74-3.80) were edited: 66 follicles were added (1.52%, CI: 1.15-1.88), 26 were modified, (0.60%, CI: 0.37-0.83 ), and 50 were deleted (1.15%, CI: 0.83–1.47). Out of the 151 edited follicles, 13 (9.2%) had 2–5 mm, 29 (20.4%) had 5-10 mm, 51 (35.9%) had 10-15 mm, 42 (29.6%) had 15-20 mm, and 7 (4.9%) had 20 mm or more in diameter. Among follicles of 15 mm or more in size, 49 were edited. Out of 294 videos, 87 (29.6%, 95% Confidence Interval: 24.4-34.6) were edited in review in any way. Limitations, reasons for caution The study was limited in scope and did not track patients until success rates in IVF treatments could be observed. A multi-center randomized control trial could compare pregnancy rates for treatments performed with and without the platform. Wider implications of the findings An integrated platform allows for easy review, while significantly reducing the time spent on follicle counting in a real-world setting. Automated annotations result in consistent, reliable, and quick measurements, with the number of expert modifications smaller than the inter-observer reliability reported in previous studies. Trial registration number Project support was provided by the Polish National Center for Research and Development no. POIR.01.01.01-00-1634/20-00 and ERC Consolidator Grant TUgbOAT no. 772346. This study was conducted following the approval of the research protocol by the review board of the Regional Medical Chamber in Gdańsk (approval no. KB – 51/22).

We identify a sufficient condition, treewidth-pliability , that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.

The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a $c$-colouring of a graph that is promised to be $k$-colourable, where $c\geq k$. This problem naturally generalises to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems.

We show a slightly simpler proof the following theorem by I. Dinur, O. Regev, and C. Smyth: for all $c \geq 2$, it is NP-hard to find a $c$-colouring of a 2-coloruable 3-uniform hypergraph. We recast this result in the algebraic framework for Promise CSPs, using only a weaker version of the PCP theorem.

Suppose $\mathcal{F}$ is a finite family of graphs. We consider the following meta-problem, called $\mathcal{F}$-Immersion Deletion: given a graph $G$ and integer $k$, decide whether the deletion of at most $k$ edges of $G$ can result in a graph that does not contain any graph from $\mathcal{F}$ as an immersion. This problem is a close relative of the $\mathcal{F}$-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from $\mathcal{F}$. We prove that whenever all graphs from $\mathcal{F}$ are connected and at least one graph of $\mathcal{F}$ is planar and subcubic, then the $\mathcal{F}$-Immersion Deletion problem admits: a constant-factor approximation algorithm running in time $O(m^3 \cdot n^3 \cdot \log m)$; a linear kernel that can be computed in time $O(m^4 \cdot n^3 \cdot \log m)$; and a $O(2^{O(k)} + m^4 \cdot n^3 \cdot \log m)$-time fixed-parameter algorithm, where $n,m$ count the vertices and edges of the input graph. These results mirror the findings of Fomin et al. [FOCS 2012], who obtained a similar set of algorithmic results for $\mathcal{F}$-Minor Deletion, under the assumption that at least one graph from $\mathcal{F}$ is planar. An important difference is that we are able to obtain a linear kernel for $\mathcal{F}$-Immersion Deletion, while the exponent of the kernel of Fomin et al. for $\mathcal{F}$-Minor Deletion depends heavily on the family $\mathcal{F}$. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of $\mathcal{F}$-Immersion Deletion is quite different than that of $\mathcal{F}$-Minor Deletion.

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin et al. [SICOMP’17] proved a result known as “(2+ɛ)-SAT is NP-hard.” They showed that the problem of distinguishing k -CNF formulas that are g -satisfiable (i.e., some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus, we give a dichotomy for a natural fragment of promise constraint satisfaction problems ( PCSPs ) on arbitrary finite domains. The hardness side is proved using the algebraic approach via a new general NP-hardness criterion on polymorphisms, which is based on a gap version of the Layered Label Cover problem. We show that previously used criteria are insufficient—the problem hence gives an interesting benchmark of algebraic techniques for proving hardness of approximation in problems such as PCSPs.

We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes.Baker's approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and Živný [SODA'21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general "sparsity" condition for which a PTAS is known. We extend these results to general-valued CSPs, which include "crisp" (or "strict") constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element which is at least as feasible as all the others (but possibly less valuable).For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes - a definition by Dvořák [SODA'20] which encompasses all classes where Baker's technique is known to work, except for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability - a property of relational structures connected to logics, statistical physics, and random CSPs.

We identify a sufficient condition, treewidth-pliability , that gives a polynomial-time approximation scheme (PTAS) for a large class of Max-2-CSPs parametrised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing PTASes. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree. The full version [59] containing detailed proofs is available at https://arxiv.org/abs/1911.03204.

We describe a heuristic algorithm for computing treedepth decompositions, submitted for the PACE 2020 challenge. It relies on a variety of greedy algorithms computing elimination orderings, as well as a Divide & Conquer approach on balanced cuts obtained using a from-scratch reimplementation of the 2016 FlowCutter algorithm by Hamann & Strasser [ACM JEA 2018].

In the field of constraint satisfaction problems (CSPs), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem-which has recently seen exciting progress [Bulín, Krokhin, and Oprs̆al, Proceedings of the Symposium on Theory of Computing, 2019, pp. 602--613 and Wrochna and Živný, Proceedings of the Symposium on Discrete Algorithms, 2020, pp. 1426--1435] benefiting from a systematic algebraic approach to promise CSPs based on "polymorphisms," operations that map tuples in the strict form of each constraint to tuples in the corresponding weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, which are invariant under arbitrary coordinate permutations. This generalizes previous work of the first two authors [Brakensiek and Guruswami, Proceedings of the Symposium on Discrete Algorithms, 2019, pp. 436--455]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefer's classic dichotomy theorem and shed further light on how symmetries of polymorphisms enable algorithms. Finally, we show that block symmetric polymorphisms are not only sufficient but also necessary for this algorithm to work, thus establishing its precise power.

We consider the following problem for a fixed graph H: given a graph G and two H-colorings of G, i.e. homomorphisms from G to H, can one be transformed (reconfigured) into the other by changing one color at a time, maintaining an H-coloring throughout. This is the same as finding a path in the Hom(G,H) complex. For H=K_k this is the problem of finding paths between k-colorings, which was shown to be in P for k<=3 and PSPACE-complete otherwise by Cereceda et al. 2011. We generalize the positive side of this dichotomy by providing an algorithm that solves the problem in polynomial time for any H with no C_4 subgraph. This gives a large class of constraints for which finding solutions to the Constraint Satisfaction Problem is NP-complete, but finding paths in the solution space is P. The algorithm uses a characterization of possible reconfiguration sequences (paths in Hom(G,H)), whose main part is a purely topological condition described in algebraic terms of the fundamental groupoid of H seen as a topological space.

In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem--which has recently seen exciting progress [BKO19, WZ20] benefiting from a systematic algebraic approach to promise CSPs based on "polymorphisms," operations that map tuples in the strict form of each constraint to tuples in the corresponding weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, that is the coordinates are permutation invariant. This generalizes previous work of the first two authors [BG19]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefer's classic dichotomy theorem and shed further light on how symmetries of polymorphisms enable algorithms. Finally, we show that block symmetric polymorphisms are not only sufficient but also necessary for this algorithm to work, thus establishing its precise power

We consider the standard ILP F easibility problem: given an integer linear program of the form {A x = b, x ⩾ 0}, where A is an integer matrix with k rows and ℓ columns, x is a vector of ℓ variables, and b is a vector of k integers, we ask whether there exists x ∈ N ℓ that satisfies Ax = b. Each row of A specifies one linear constraint on x; our goal is to study the complexity of ILP F easibility when both k , the number of constraints, and ‖A‖ ∞ , the largest absolute value of an entry in A , are small. Papadimitriou was the first to give a fixed-parameter algorithm for ILP F easibility under parameterization by the number of constraints that runs in time ((‖A‖ ∞ + ‖b‖ ∞ ) ⋅ k ) O ( k 2 ) . This was very recently improved by Eisenbrand and Weismantel, who used the Steinitz lemma to design an algorithm with running time ( k ‖A‖ ∞ ) O ( k ) ⋅ log ‖b‖ ∞ , which was subsequently refined by Jansen and Rohwedder to O (√ k ‖A‖ ∞ ) k ⋅ log (‖ A‖ ∞ + ‖b‖ ∞ ) ⋅ log ‖A‖ ∞ . We prove that for {0, 1}-matrices A , the running time of the algorithm of Eisenbrand and Weismantel is probably optimal: an algorithm with running time 2 o ( k log k ) ⋅ (ℓ + ‖b‖ ∞ ) o ( k ) would contradict the exponential time hypothesis. This improves previous non-tight lower bounds of Fomin et al. We then consider integer linear programs that may have many constraints, but they need to be structured in a “shallow” way. Precisely, we consider the parameter dual treedepth of the matrix A , denoted td D ( A ), which is the treedepth of the graph over the rows of A , where two rows are adjacent if in some column they simultaneously contain a non-zero entry. It was recently shown by Koutecký et al. that ILP F easibility can be solved in time ‖A‖ ∞ 2 O (td D ( A )) ⋅ ( k + ℓ + log ‖b‖ ∞ ) O (1) . We present a streamlined proof of this fact and prove that, again, this running time is probably optimal: even assuming that all entries of A and b are in {−1, 0, 1}, the existence of an algorithm with running time 2 2 o (td D ( A )) ⋅ ( k + ℓ) O (1) would contradict the exponential time hypothesis.

We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring k-colourable graphs with k ((k)(k/2)) - 1 colours for every k >= 4. This improves the result of Bulin, Krokhin, and Oprsal [STOC'19], who gave NP-hardness of colouring k -colourable graphs with 2k - 1 colours for k >= 3, and the result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring k-colourable graphs with 2 Omega(k(1/3)) colours for sufficiently large k. Thus, for k >= 4, we improve from known linear/sub-exponential gaps to exponential gaps. Second, we show that the topology of the box complex of H alone determines whether H -colouring of G-colourable graphs is NP-hard for all (non-bipartite, H-colourable) G. This formalises the topological intuition behind the result of Krokhin and Oprsal [FOCS'19] that 3-colouring of G-colourable graphs is NP-hard for all (3-colourable, nonbipartite) G. We use this technique to establish NP-hardness of H -colouring of G-colourable graphs for H that include but go beyond K-3, including square-free graphs and circular cliques (leaving K-4 and larger cliques open). Underlying all of our proofs is a very general observation that adjoint functors give reductions between promise constraint satisfaction problems. The full version [55] containing detailed proofs is available at https://arxiv.org/abs/1907.00872.

Hedetniemi's conjecture for c-colorings states that the tensor product G × H is c-colorable if and only if G or H is c-colorable. El-Zahar and Sauer proved it for c = 3. In a recent breakthrough, Shitov showed counterexamples, for large c. While Shitov's proof is already remarkably short, Zhu simplified the argument and gave a more explicit counterexample for c=125. Tardif showed that a modification of the arguments allows to use “wide colorings” to obtain counterexamples for c=14, and c=13 with a more involved use of lexicographic products. This note presents two more small modifications, resulting in counterexamples for c=5 (with G and H having 4686 and 30 vertices, respectively).

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.

We consider a natural graph operation Omega k that is a certain inverse (formally: the right adjoint) to taking the k-th power of a graph. We show that it preserves the topology (the Z(2)-homotopy type) of the box complex, a basic tool in applications of topology in combinatorics. Moreover, we prove that the box complex of a graph G admits a Z(2)-map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph Omega(k) (G) admits a homomorphism to H, for high enough k. This allows to show that if Hedetniemi's conjecture on the chromatic number of graph products is true, then the following analogous conjecture in topology is also true: If n is an element of N and X, Y are Z(2)-spaces (finite Z(2)-simplicial complexes) such that X x Y admits a Z(2)-map to the n; dimensional sphere, then X or Y itself admits such a map. We discuss this and other implications, arguing the importance of the topological conjecture. (C) 2019 Elsevier Inc. All rights reserved.