Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis

We study the problem of finding a minimum homology basis, that is, a shortest set of cycles that generates the $1$-dimensional homology classes with $mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in $O(N^omega + N^2 g)$ time, where $N$ denotes the number of simplices in $K$, $g$ denotes the rank of the $1$-homology group of $K$, and $omega$ denotes the exponent of matrix multiplication. In this paper, we present two conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex $K$. The first algorithm runs in $tilde{O}(m^omega)$ time, where $m$ denotes the number of edges in $K$, whereas the second algorithm runs in $O(m^omega + N m^{omega-1})$ time. We also study the problem of finding a minimum cycle basis in an undirected graph $G$ with $n$ vertices and $m$ edges. The best known algorithm for this problem runs in $O(m^omega)$ time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in $tilde{O}(m^omega)$ time.

Parameterized inapproximability of Morse matching

We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of Min-Morse Matching within a factor of $2^{log^{(1-epsilon)}n}$. Our second result shows that Min-Morse Matching is ${bf W{[P]}}$-hard with respect to the standard parameter. Next, we show that Min-Morse Matching with standard parameterization has no FPT approximation algorithm for any approximation factor $rho$. The above hardness results are applicable to complexes of dimension $ge 2$. On the positive side, we provide a factor $O(frac{n}{log n})$ approximation algorithm for Min-Morse Matching on $2$-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa-Farber model of random complexes.

The complexity of high-dimensional cuts

Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. For instance, the minimum cut, the minimum $s$-$t$ cut, the minimum multiway cut, and the minimum $k$-way cut are some of the commonly encountered cut problems. Many of these problems have been extensively studied over several decades. In this paper, we initiate the algorithmic study of some cut problems in high dimensions. The first problem we study, namely, Topological Hitting Set (THS), is defined as follows: Given a nontrivial $r$-cycle $zeta$ in a simplicial complex $mathsf{K}$, find a set $mathcal{S}$ of $r$-dimensional simplices of minimum cardinality so that $mathcal{S}$ meets every cycle homologous to $zeta$. Our main result is that this problem admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the optimal solution is given in terms of the cocycles of the surface. For general complexes, we show that THS is W[1]-hard with respect to the solution size $k$. On the positive side, we show that THS admits an FPT algorithm with respect to $k+d$, where $d$ is the maximum degree of the Hasse graph of the complex $mathsf{K}$. We also define a problem called Boundary Nontrivialization (BNT): Given a bounding $r$-cycle $zeta$ in a simplicial complex $mathsf{K}$, find a set $mathcal{S}$ of $(r+1)$-dimensional simplices of minimum cardinality so that the removal of $mathcal{S}$ from $mathsf{K}$ makes $zeta$ non-bounding. We show that BNT is W[1]-hard with respect to the solution size as the parameter, and has an $O(log n)$-approximation FPT algorithm for $(r+1)$-dimensional complexes with the $(r+1)$-th Betti number $beta_{r+1}$ as the parameter. Finally, we provide randomized (approximation) FPT algorithms for the global variants of THS and BNT.

Approximation Algorithms for Max-Morse Matching

In this paper, we prove that the Max-Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch. We describe two different approximation algorithms for the Max-Morse Matching Problem. For $D$-dimensional simplicial complexes, we obtain a $frac{(D+1)}{(D^2+D+1)}$-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. Our second result is an algorithm that provides a $frac{2}{D}$-factor approximation for simplicial manifolds by processing the simplices in increasing order of dimension. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results.