We present a suite of scalable algorithms for minimizing feedback arcs in large-scale weighted directed graphs, with the goal of revealing biologically meaningful feedforward structure in neural connectomes. Using the FlyWire Connectome Challenge dataset, we demonstrate the effectiveness of our ranking strategies in maximizing the total weight of forward-pointing edges. Our methods integrate greedy heuristics, gain-aware local refinements, and global structural analysis based on strongly connected components. Experiments show that our best solution improves the forward edge weight over previous top-performing methods. All algorithms are implemented efficiently in Python and validated using cloud-based execution on Google Colab Pro+.

Real world tournaments are almost always intransitive. Recent works have noted that parametric models which assume $d$ dimensional node representations can effectively model intransitive tournaments. However, nothing is known about the structure of the class of tournaments that arise out of any fixed $d$ dimensional representations. In this work, we develop a novel theory for understanding parametric tournament representations. Our first contribution is to structurally characterize the class of tournaments that arise out of $d$ dimensional representations. We do this by showing that these tournament classes have forbidden configurations which must necessarily be union of flip classes, a novel way to partition the set of all tournaments. We further characterise rank $2$ tournaments completely by showing that the associated forbidden flip class contains just $2$ tournaments. Specifically, we show that the rank $2$ tournaments are equivalent to locally-transitive tournaments. This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure. For a general rank $d$ tournament class, we show that the flip class associated with a coned-doubly regular tournament of size $\mathcal{O}(\sqrt{d})$ must be a forbidden configuration. To answer a dual question, using a celebrated result of \cite{forster}, we show a lower bound of $\mathcal{O}(\sqrt{n})$ on the minimum dimension needed to represent all tournaments on $n$ nodes. For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the flip class associated with a tournament. We show how our results also shed light on upper bound of sign-rank of matrices.

We study the problem of finding a mapping $f$ from a set of points into the real line, under ordinal triple constraints. An ordinal constraint for a triple of points $(u,v,w)$ asserts that $|f(u)-f(v)|<|f(u)-f(w)|$. We present an approximation algorithm for the dense case of this problem. Given an instance that admits a solution that satisfies $(1-\varepsilon)$-fraction of all constraints, our algorithm computes a solution that satisfies $(1-O(\varepsilon^{1/8}))$-fraction of all constraints, in time $O(n^7) + (1/\varepsilon)^{O(1/\varepsilon^{1/8})} n$.

We present a fast local search algorithm that finds an improved solution (if there is any) in the k-exchange neighborhood of the given solutionto an instance of Weighted Feedback Arc Set in Tournaments. More precisely,given an arc weighted tournament T on n vertices and a feedback arc set F (a set of arcs whose deletion from T turns it into a directed acyclic graph), our algorithm decides in time O(2 o ( k ) n log n) if there is a feedback arc set of smaller weight and that differs from F in at most k arcs. To our knowledge this is the first algorithm searching the k -exchange neighborhood of an NP-complete problem that runs in (parameterized) subexponential time. Using this local search algorithm for Weighted Feedback Arc Set in Tournaments, we obtain subexponential time algorithms for a local search variant of Kemeny Ranking — a problem in social choice theory and of One-Sided Cross Minimization — a problem in graph drawing.