Reflection methods for user-friendly submodular optimization

Recently, it has become evident that submodularity naturally captures widely occurring concepts in machine learning, signal processing and computer vision. Consequently, there is need for efficient optimization procedures for submodular functions, especially for minimization problems. While general submodular minimization is challenging, we propose a new method that exploits existing decomposability of submodular functions. In contrast to previous approaches, our method is neither approximate, nor impractical, nor does it need any cumbersome parameter tuning. Moreover, it is easy to implement and parallelize. A key component of our method is a formulation of the discrete submodular minimization problem as a continuous best approximation problem that is solved through a sequence of reflections, and its solution can be easily thresholded to obtain an optimal discrete solution. This method solves both the continuous and discrete formulations of the problem, and therefore has applications in learning, inference, and reconstruction. In our experiments, we illustrate the benefits of our method on two image segmentation tasks.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013

Vertex Sparsifiers for Hyperedge Connectivity

Recently, Chalermsook et al. [SODA'21(arXiv:2007.07862)] introduces a notion of vertex sparsifiers for $c$-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for $c$-edge st-connectivity [Jin and Sun FOCS'21(arXiv:2004.07650)]. We study a natural extension called vertex sparsifiers for $c$-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph $G=(V,E)$ with $n$ vertices and $m$ hyperedges with $k$ terminal vertices and a parameter $c$, there exists a hypergraph $H$ containing only $O(kc^{3})$ hyperedges that preserves all minimum cuts (up to value $c$) between all subset of terminals. This matches the best bound of $O(kc^{3})$ edges for normal graphs by [Liu'20(arXiv:2011.15101)]. Moreover, $H$ can be constructed in almost-linear $O(p^{1+o(1)} + n(rc\log n)^{O(rc)}\log m)$ time where $r=\max_{e\in E}|e|$ is the rank of $G$ and $p=\sum_{e\in E}|e|$ is the total size of $G$, or in $\text{poly}(m, n)$ time if we slightly relax the size to $O(kc^{3}\log^{1.5}(kc))$ hyperedges.Comment: submitted to ESA 202