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.

Strategy-Stealing Is Non-Constructive

In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing. This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists? We prove that this problem is PSPACE-Complete already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing arguments in the current literature

Testing Core Membership in Public Goods Economies

This paper develops a recent line of economic theory seeking to understand public goods economies using methods of topological analysis. Our first main result is a very clean characterization of the economy\u27s core (the standard solution concept in public goods). Specifically, we prove that a point is in the core iff it is Pareto efficient, individually rational, and the set of points it dominates is path connected. While this structural theorem has a few interesting implications in economic theory, the main focus of the second part of this paper is on a particular algorithmic application that demonstrates its utility. Since the 1960s, economists have looked for an efficient computational process that decides whether or not a given point is in the core. All known algorithms so far run in exponential time (except in some artificially restricted settings). By heavily exploiting our new structure, we propose a new algorithm for testing core membership whose computational bottleneck is the solution of O(n) convex optimization problems on the utility function governing the economy. It is fairly natural to assume that convex optimization should be feasible, as it is needed even for very basic economic computational tasks such as testing Pareto efficiency. Nevertheless, even without this assumption, our work implies for the first time that core membership can be efficiently tested on (e.g.) utility functions that admit ``nice\u27\u27 analytic expressions, or that appropriately defined epsilon-approximate versions of the problem are tractable (by using modern black-box epsilon-approximate convex optimization algorithms)

An Alternate Proof of Near-Optimal Light Spanners

In 2016, a breakthrough result of Chechik and Wulff-Nilsen [SODA '16] established that every $n$-node graph $G$ has a $(1+\varepsilon)(2k-1)$-spanner of lightness $O_{\varepsilon}(n^{1/k})$, and recent followup work by Le and Solomon [STOC '23] generalized the proof strategy and improved the dependence on $\varepsilon$. We give a new proof of this result (with the improved $\varepsilon$-dependence). Our proof is a direct analysis of the often-studied greedy spanner, and can be viewed as an extension of the folklore Moore bounds used to analyze spanner sparsity

Fully Dynamic Spanners with Worst-Case Update Time

An alpha-spanner of a graph G is a subgraph H such that H preserves all distances of G within a factor of alpha. In this paper, we give fully dynamic algorithms for maintaining a spanner H of a graph G undergoing edge insertions and deletions with worst-case guarantees on the running time after each update. In particular, our algorithms maintain: - a 3-spanner with ~O(n^{1+1/2}) edges with worst-case update time ~O(n^{3/4}), or - a 5-spanner with ~O(n^{1+1/3}) edges with worst-case update time ~O (n^{5/9}). These size/stretch tradeoffs are best possible (up to logarithmic factors). They can be extended to the weighted setting at very minor cost. Our algorithms are randomized and correct with high probability against an oblivious adversary. We also further extend our techniques to construct a 5-spanner with suboptimal size/stretch tradeoff, but improved worst-case update time. To the best of our knowledge, these are the first dynamic spanner algorithms with sublinear worst-case update time guarantees. Since it is known how to maintain a spanner using small amortized}but large worst-case update time [Baswana et al. SODA\u2708], obtaining algorithms with strong worst-case bounds, as presented in this paper, seems to be the next natural step for this problem

Folklore Sampling is Optimal for Exact Hopsets: Confirming the n\sqrt{n} Barrier

For a graph $G$, a $D$-diameter-reducing exact hopset is a small set of additional edges $H$ that, when added to $G$, maintains its graph metric but guarantees that all node pairs have a shortest path in $G \cup H$ using at most $D$ edges. A shortcut set is the analogous concept for reachability. These objects have been studied since the early '90s due to applications in parallel, distributed, dynamic, and streaming graph algorithms. For most of their history, the state-of-the-art construction for either object was a simple folklore algorithm, based on randomly sampling nodes to hit long paths in the graph. However, recent breakthroughs of Kogan and Parter [SODA '22] and Bernstein and Wein [SODA '23] have finally improved over the folklore diameter bound of $\widetilde{O}(n^{1/2})$ for shortcut sets and for $(1+\epsilon)$-approximate hopsets. For both objects it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1/3})$. The only setting where folklore sampling remains unimproved is for exact hopsets. Can these improvements be continued? We settle this question negatively by constructing graphs on which any exact hopset of $O(n)$ edges has diameter $\widetilde{\Omega}(n^{1/2})$. This improves on the previous lower bound of $\widetilde{\Omega}(n^{1/3})$ by Kogan and Parter [FOCS '22]. Using similar ideas, we also polynomially improve the current lower bounds for shortcut sets, constructing graphs on which any shortcut set of $O(n)$ edges reduces diameter to $\widetilde{\Omega}(n^{1/4})$. This improves on the previous lower bound of $\Omega(n^{1/6})$ by Huang and Pettie [SIAM J. Disc. Math. '18]. We also extend our constructions to provide lower bounds against $O(p)$-size exact hopsets and shortcut sets for other values of $p$; in particular, we show that folklore sampling is near-optimal for exact hopsets in the entire range of $p \in [1, n^2]$