A Unified View of Graph Regularity via Matrix Decompositions

We prove algorithmic weak and \Szemeredi{} regularity lemmas for several classes of sparse graphs in the literature, for which only weak regularity lemmas were previously known. These include core-dense graphs, low threshold rank graphs, and (a version of) $L^p$ upper regular graphs. More precisely, we define \emph{cut pseudorandom graphs}, we prove our regularity lemmas for these graphs, and then we show that cut pseudorandomness captures all of the above graph classes as special cases. The core of our approach is an abstracted matrix decomposition, roughly following Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy [Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm by Charikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs, and using work of Oveis Gharan and Trevisan [TOC '15], it also implies new PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expanded class of input graphs. (It is NP Hard to get PTASes for these graphs in general.

A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs

Kolla and Tulsiani [KT07,Kolla11} and Arora, Barak and Steurer [ABS10] introduced the technique of subspace enumeration, which gives approximation algorithms for graph problems such as unique games and small set expansion; the running time of such algorithms is exponential in the threshold-rank of the graph. Guruswami and Sinop [GS11,GS12], and Barak, Raghavendra, and Steurer [BRS11] developed an alternative approach to the design of approximation algorithms for graphs of bounded threshold-rank, based on semidefinite programming relaxations in the Lassere hierarchy and on novel rounding techniques. These algorithms are faster than the ones based on subspace enumeration and work on a broad class of problems. In this paper we develop a third approach to the design of such algorithms. We show, constructively, that graphs of bounded threshold-rank satisfy a weak Szemeredi regularity lemma analogous to the one proved by Frieze and Kannan [FK99] for dense graphs. The existence of efficient approximation algorithms is then a consequence of the regularity lemma, as shown by Frieze and Kannan. Applying our method to the Max Cut problem, we devise an algorithm that is faster than all previous algorithms, and is easier to describe and analyze

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Inapproximability of Maximum Biclique Problems, Minimum kk-Cut and Densest At-Least-kk-Subgraph from the Small Set Expansion Hypothesis

The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose edge expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph $G$, find a complete bipartite subgraph of $G$ with maximum number of edges. - Maximum Balanced Biclique (MBB): given a bipartite graph $G$, find a balanced complete bipartite subgraph of $G$ with maximum number of vertices. - Minimum $k$-Cut: given a weighted graph $G$, find a set of edges with minimum total weight whose removal partitions $G$ into $k$ connected components. - Densest At-Least-$k$-Subgraph (DAL$k$S): given a weighted graph $G$, find a set $S$ of at least $k$ vertices such that the induced subgraph on $S$ has maximum density (the ratio between the total weight of edges and the number of vertices). We show that, assuming SSEH and NP $\nsubseteq$ BPP, no polynomial time algorithm gives $n^{1 - \varepsilon}$-approximation for MEB or MBB for every constant $\varepsilon > 0$. Moreover, assuming SSEH, we show that it is NP-hard to approximate Minimum $k$-Cut and DAL$k$S to within $(2 - \varepsilon)$ factor of the optimum for every constant $\varepsilon > 0$. The ratios in our results are essentially tight since trivial algorithms give $n$-approximation to both MEB and MBB and efficient $2$-approximation algorithms are known for Minimum $k$-Cut [SV95] and DAL$k$S [And07, KS09]. Our first result is proved by combining a technique developed by Raghavendra et al. [RST12] to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test [BK09] whereas our second result is shown via elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

Towards a better approximation for sparsest cut?

We give a new $(1+\epsilon)$-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size $n/r$ expand by a factor $\sqrt{\log n\log r}$ bigger, for some small $r$; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-$r$ Lasserre relaxation. The other is combinatorial and involves a new notion called {\em Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which we show exists in the input graph. Both algorithms run in time $2^{O(r)} \mathrm{poly}(n)$. We also show similar approximation algorithms in graphs with genus $g$ with an analogous local expansion condition. This is the first algorithm we know of that achieves $(1+\epsilon)$-approximation on such general family of graphs