The famous Tower of Hanoi puzzle involves moving n discs of distinct sizes from one of p≥ 3 pegs (traditionally p=3) to another of the pegs, subject to the constraints that only one disc may be moved at a time, and no disc can ever be placed on a disc smaller than itself. Much is known about the Hanoi graph H_p^n, whose p^n vertices represent the configurations of the puzzle, and whose edges represent the pairs of configurations separated by a single legal move. In a previous paper, the present authors presented nearly tight asymptotic bounds of O((p-2)^n) and Ω(n^(1-p)/2(p-2)^n) on the treewidth of this graph for fixed p ≥ 3. In this paper we show that the upper bound is tight, by giving a matching lower bound of Ω((p-2)^n) for the expansion of H_p^n.

We introduce a new quantum algorithm for computing the Betti numbers of a simplicial complex. In contrast to previous quantum algorithms that work by estimating the eigenvalues of the combinatorial Laplacian, our algorithm is an instance of the generic Incremental Algorithm for computing Betti numbers that incrementally adds simplices to the simplicial complex and tests whether or not they create a cycle. In contrast to existing quantum algorithms for computing Betti numbers that work best when the complex has close to the maximal number of simplices, our algorithm works best for sparse complexes. To test whether a simplex creates a cycle, we introduce a quantum span-program algorithm. We show that the query complexity of our span program is parameterized by quantities called the effective resistance and effective capacitance of the boundary of the simplex. Unfortunately, we also prove upper and lower bounds on the effective resistance and capacitance, showing both quantities can be exponentially large with respect to the size of the complex, implying that our algorithm would have to run for exponential time to exactly compute Betti numbers. However, as a corollary to these bounds, we show that the spectral gap of the combinatorial Laplacian can be exponentially small. As the runtime of all previous quantum algorithms for computing Betti numbers are parameterized by the inverse of the spectral gap, our bounds show that all quantum algorithms for computing Betti numbers must run for exponentially long to exactly compute Betti numbers. Finally, we prove some novel formulas for effective resistance and effective capacitance to give intuition for these quantities.

We consider a variety of topology problems on a d-dimensional simplicial complex K given that K subset of 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 L(1)x = b, where L-1 is the 1-Laplacian of a simplicial complex K with dim H-1(K) = 0 and K subset of X for X a collapsible simplicial complex embedded in R-3 with a known collapsing sequence. Our algorithm runs in (O) over tilde (n log(2) (n kappa/epsilon)) time, where n is the total number of vertices, edges, and triangles in X, kappa 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(n(d)) 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(beta(d-1)n(d)) time; beta(d-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 R-3 in O(n(d) log n(d) + beta(d-1)n) time using a homology basis computed by the algorithm of Dey [SODA 2019]. For all of the problems above, if K subset of R-3 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 Towers of Hanoi puzzle is to move a set of disks one at a time from one of a set of pegs to another, while keeping the disks 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.

We investigate generalizations of the graph theoretic notions of effective resistance and capacitance to simplicial complexes and prove analogs of formulas known in the case of graphs. In graphs the effective resistance between two vertices is O ( n ); however, we show that in a simplicial complex the effective resistance of a null-homologous cycle may be exponential. This is caused by relative torsion in the simplicial complex. We provide upper bounds on both effective resistance and capacitance that are polynomial in the number of simplices as well as the maximum cardinality of the torsion subgroup of a relative homology group denoted T max ( K ). We generalize the quantum algorithm deciding st -connectivity in a graph and obtain an algorithm deciding whether or not a ( d − 1)-dimensional cycle γ is null-homologous in a d -dimensional simplicial complex K . The quantum algorithm has query complexity parameterized by the effective resistance and capacitance of γ . Using our upper bounds we find that the query complexity is O (cid:0) n 5 / 2 · d 1 / 2 · T max ( K ) 2 (cid:1) . Under the assumptions that γ is the boundary of a d -simplex (which may or may not be included in the complex) and that K is relative torsion-free, we match the O ( n 3 / 2 ) query complexity obtained for st -connectivity. These assumptions always hold in the case of st -connectivity. We provide an implementation of the algorithm whose running time is polynomial in the size of the complex and the relative torsion. Finally, we prove a duality theorem relating effective resistance and capacitance when K is d -dimensional and admits an embedding into R d +1 .

We study the problem of low-stretch spanning trees in graphs of bounded width: bandwidth, cutwidth, and treewidth. We show that any simple connected graph $G$ with a linear arrangement of bandwidth $b$ can be embedded into a distribution $\mathcal T$ of spanning trees such that the expected stretch of each edge of $G$ is $O(b^2)$. Our proof implies a linear time algorithm for sampling from $\mathcal T$. Therefore, we have a linear time algorithm that finds a spanning tree of $G$ with average stretch $O(b^2)$ with high probability. We also describe a deterministic linear-time algorithm for computing a spanning tree of $G$ with average stretch $O(b^3)$. For graphs of cutwidth $c$, we construct a spanning tree with stretch $O(c^2)$ in linear time. Finally, when $G$ has treewidth $k$ we provide a dynamic programming algorithm computing a minimum stretch spanning tree of $G$ that runs in polynomial time with respect to the number of vertices of $G$.

We study two optimization problems on simplicial complexes with homology over $\mathbb{Z}_2$, the minimum bounded chain problem: given a $d$-dimensional complex $\mathcal{K}$ embedded in $\mathbb{R}^{d+1}$ and a null-homologous $(d-1)$-cycle $C$ in $\mathcal{K}$, find the minimum $d$-chain with boundary $C$, and the minimum homologous chain problem: given a $(d+1)$-manifold $\mathcal{M}$ and a $d$-chain $D$ in $\mathcal{M}$, 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 $O(\sqrt{\log \beta_d})$-approximation algorithm for the minimum bounded chain problem where $\beta_d$ is the $d$th Betti number of $\mathcal{K}$. Finally, we provide an $O(\sqrt{\log n_{d+1}})$-approximation algorithm for the minimum homologous chain problem where $n_{d+1}$ is the number of $d$-simplices in $\mathcal{M}$.

Advanced persistent threats (APT) are stealthy, sophisticated, and unpredictable cyberattacks that can steal intellectual property, damage critical infrastructure, or cause millions of dollars in damage. Detecting APTs by monitoring system-level activity is difficult because manually inspecting the high volume of normal system activity is overwhelming for security analysts. We evaluate the effectiveness of unsupervised batch and streaming anomaly detection algorithms over multiple gigabytes of provenance traces recorded on four different operating systems to determine whether they can detect realistic APT-like attacks reliably and efficiently. This report is the first detailed study of the effectiveness of generic unsupervised anomaly detection techniques in this setting.

Consider a surface $S$ and let $M\subset S$. If $S\setminus M$ is not connected, then we say $M$ \emph{separates} $S$, and we refer to $M$ as a \emph{separating set} of $S$. If $M$ separates $S$, and no proper subset of $M$ separates $S$, then we say $M$ is a \emph{minimal separating set} of $S$. In this paper we use methods of computational combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus $g=2$ and $g=3$. The classification for genus 0 and 1 was done in earlier work, using methods of algebraic topology.