We consider a variety of topology problems on a d-dimensional simplicial complex K given that K ∪ X for X a collapsible simplicial complex embedded in ℝd+1 with known collapsing sequence. Our first result is a solver for the linear system L1x = b, where L1 is the 1-Laplacian of a simplicial complex K with dimH1(K) = 0 and K ∪ X for X a collapsible simplicial complex embedded in ℝ3 with a known collapsing sequence. Our algorithm runs in O(n log2 (nκ/∊)) time, where n is the total number of vertices, edges, and triangles in X, κ is the largest condition number of the two parts of the Laplacian, and ∊ quantifies the approximation quality. This result is a generalization of Cohen et al. [SODA 2014]. The new technical piece of our Laplacian solver, in addition to the machinery described by Cohen et al., is an algorithm to compute a bounding chain of a 1-cycle within k. In addition, we describe faster algorithms for testing null-homology of (d–1)-cycles and null-cohomology of d-cocycles. Our algorithm runs in O(nd) time, where nd is the number of d-simplices in X. Finally, we describe an algorithm to compute a (d–1)-cohomology basis from a given (d–1)-homology basis for a d-simplicial complex K in O(βd–1nd) time; βd–1 is the rank of the (d–1)st homology group of k. In particular, we can obtain a cohomology basis for subcomplexes of a collapsible complex X embedded in ℝ3 in O(nd log nd + βd–1 n) time using a homology basis computed by the algorithm of Dey [SODA 2019]. For all of the problems above, if K ∪ ℝ3 and the collapsible supercomplex X is not provided, we can expand K into a convex ball of possibly quadratic complexity, which is known to be collapsible, resulting in nearly quadratic time algorithms.

The objective of the well-known Tower of Hanoi puzzle is to move a set of discs one at a time from one of a set of pegs to another, while keeping the discs sorted on each peg. We propose an adversarial variation in which the first player forbids a set of states in the puzzle, and the second player must then convert one randomly-selected state to another without passing through forbidden states. Analyzing this version raises the question of the treewidth of Hanoi graphs. We find this number exactly for three-peg puzzles and provide nearly-tight asymptotic bounds for larger numbers of pegs.

We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and show that computing such a cut is NP-hard. However, we provide conditions on the simplicial complex for when the cut found by the linear program is a combinatorial cut. For $d$-dimensional simplicial complexes embedded into $\mathbb{R}^{d+1}$ we provide algorithms operating on the dual graph: computing a maximum flow is dual to computing a shortest path and computing a minimum cut is dual to computing a minimum cost circulation. Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes, prove its correctness, and provide a heuristic which guarantees it to halt.

This paper clarifies a recurrent structural confusion in advanced computation: the tendency to equate Quantum Mechanical (QM) computation and Cognitional Mechanics (CM) solely because both employ non-commutative structures. While the mathematical resemblance is real, the two frameworks operate at fundamentally different ontological layers and therefore govern distinct domains. Quantum computation represents extreme peak performance. Its advantage appears only after a problem has been fully formalized within a closed mathematical system, such as a fixed Hilbert space with well-defined operators and observables. Algorithms like Shor’s and Grover’s demonstrate that, under these conditions, QM can invalidate classical hardness assumptions or achieve dramatic speedups. However, QM neither generates nor reinterprets problems; it presupposes that all semantic uncertainty has already been resolved. Cognitional Mechanics governs the peripheral domain in which problems are formed, redefined, and semantically stabilized. CM models intelligence as a system of non-commutative semantic operations acting on meaning states within an open and evolving semantic manifold. Its non-commutativity is semantic rather than physical: the order of interpretive operations determines which meaning stabilizes, and this process is intrinsically irreversible. Through the example of RSA cryptanalysis, this paper shows that QM and CM are not competing frameworks. QM dominates isolated computational peaks, while CM governs the surrounding periphery that makes such peaks identifiable as problems at all. Recognizing this structural division is necessary to avoid category errors that conflate physical computational power with intelligence itself.

We study two optimization problems on simplicial complexes with homology over ℤ₂, the minimum bounded chain problem: given a d-dimensional complex 𝒦 embedded in ℝ^(d+1) and a null-homologous (d-1)-cycle C in 𝒦, find the minimum d-chain with boundary C, and the minimum homologous chain problem: given a (d+1)-manifold ℳ and a d-chain D in ℳ, find the minimum d-chain homologous to D. We show strong hardness results for both problems even for small values of d; d = 2 for the former problem, and d=1 for the latter problem. We show that both problems are APX-hard, and hard to approximate within any constant factor assuming the unique games conjecture. On the positive side, we show that both problems are fixed-parameter tractable with respect to the size of the optimal solution. Moreover, we provide an O(√{log β_d})-approximation algorithm for the minimum bounded chain problem where β_d is the dth Betti number of 𝒦. Finally, we provide an O(√{log n_{d+1}})-approximation algorithm for the minimum homologous chain problem where n_{d+1} is the number of (d+1)-simplices in ℳ.