A Distanced Matching Game, Decremental APSP in Expanders, and Faster Deterministic Algorithms for Graph Cut Problems

Expander graphs play a central role in graph theory and algorithms. With a number of powerful algorithmic tools developed around them, such as the Cut-Matching game, expander pruning, expander decomposition, and algorithms for decremental All-Pairs Shortest Paths (APSP) in expanders, to name just a few, the use of expanders in the design of graph algorithms has become ubiquitous. Specific applications of interest to us are fast deterministic algorithms for cut problems in static graphs, and algorithms for dynamic distance-based graph problems, such as APSP. Unfortunately, the use of expanders in these settings incurs a number of drawbacks. For example, the best currently known algorithm for decremental APSP in constant-degree expanders can only achieve a $(\log n)^{O(1/\epsilon^2)}$-approximation with $n^{1+O(\epsilon)}$ total update time for any $\epsilon$. All currently known algorithms for the Cut Player in the Cut-Matching game are either randomized, or provide rather weak guarantees. This, in turn, leads to somewhat weak algorithmic guarantees for several central cut problems: for example, the best current almost linear time deterministic algorithm for Sparsest Cut can only achieve approximation factor $(\log n)^{\omega(1)}$. Lastly, when relying on expanders in distance-based problems, such as dynamic APSP, via current methods, it seems inevitable that one has to settle for approximation factors that are at least $\Omega(\log n)$. In this paper we propose the use of well-connected graphs, and introduce a new algorithmic toolkit for such graphs that, in a sense, mirrors the above mentioned algorithmic tools for expanders. One of these new tools is the Distanced Matching game, an analogue of the Cut-Matching game for well-connected graphs. We demonstrate the power of these new tools by obtaining better results for several of the problems mentioned above.