Additive sparsification of CSPs

Multiplicative cut sparsifiers, introduced by Bencz´ur and Karger [STOC’96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA’17] and non-Boolean domains by Butti and Zivn´y [SIDMA’20]. ˇ Bansal, Svensson and Trevisan [FOCS’19] introduced a weaker notion of sparsification termed “additive sparsification”, which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P : {0, 1} k → {0, 1} of a fixed arity k, we show that CSP(P) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P) admits an additive sparsifier for any predicate P : Dk → {0, 1} of a fixed arity k on an arbitrary finite domain D

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

Maintaining Expander Decompositions via Sparse Cuts

In this article, we show that the algorithm of maintaining expander decompositions in graphs undergoing edge deletions directly by removing sparse cuts repeatedly can be made efficient. Formally, for an $m$-edge undirected graph $G$, we say a cut $(S, \overline{S})$ is $\phi$-sparse if $|E_G(S, \overline{S})| < \phi \cdot \min\{vol_G(S), vol_G(\overline{S})\}$. A $\phi$-expander decomposition of $G$ is a partition of $V$ into sets $X_1, X_2, \ldots, X_k$ such that each cluster $G[X_i]$ contains no $\phi$-sparse cut (meaning it is a $\phi$-expander) with $\tilde{O}(\phi m)$ edges crossing between clusters. A natural way to compute a $\phi$-expander decomposition is to decompose clusters by $\phi$-sparse cuts until no such cut is contained in any cluster. We show that even in graphs undergoing edge deletions, a slight relaxation of this meta-algorithm can be implemented efficiently with amortized update time $m^{o(1)}/\phi^2$. Our approach naturally extends to maintaining directed $\phi$-expander decompositions and $\phi$-expander hierarchies and thus gives a unifying framework while having simpler proofs than previous state-of-the-art work. In all settings, our algorithm matches the run-times of previous algorithms up to subpolynomial factors. Moreover, our algorithm provides stronger guarantees for $\phi$-expander decompositions. For example, for graphs undergoing edge deletions, our approach is the first to maintain a dynamic expander decomposition where each updated decomposition is a refinement of the previous decomposition, and our approach is the first to guarantee a sublinear $\phi m^{1+o(1)}$ bound on the total number of edges that cross between clusters across the entire sequence of dynamic updates

Fully Dynamic Min-Cut of Superconstant Size in Subpolynomial Time

We present a deterministic fully dynamic algorithm with subpolynomial worst-case time per graph update such that after processing each update of the graph, the algorithm outputs a minimum cut of the graph if the graph has a cut of size at most $c$ for some $c = (\log n)^{o(1)}$. Previously, the best update time was $\widetilde O(\sqrt{n})$ for any $c > 2$ and $c = O(\log n)$ [Thorup, Combinatorica'07].Comment: SODA 202

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum