The Sparsest Cut is a fundamental optimization problem that have been extensively studied. For planar inputs the problem is in P and can be solved in Õ(n 3 ) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90’s and can only achieve O(log n)approximation in Õ(n) time or 3.5approximation in Õ(n 2 ) time [Rao, STOC92]. Our main result is an Ω(n 2−ε ) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)Convolution conjecture, showing that approxima tions are inevitable in the nearlinear time regime. To complement the lower bound, we provide a 3.3approximation in nearlinear time, improving upon the 25year old result of Rao in both time and accuracy. We also show that our lower bound is not far from optimal by observing an exact algorithm with running time Õ(n 5/2 ) improving upon the Õ(n 3 ) algorithm of Park and Phillips [STOC93]. Our lower bound accomplishes a repeatedly raised challenge by being the first finegrained lower bound for a natural planar graph problem in P. Building on our construction we prove nearquadratic lower bounds under SETHmore »
Universallyoptimal distributed algorithms for known topologies
Many distributed optimization algorithms achieve existentiallyoptimal running times, meaning that there exists some pathological worstcase topology on which no algorithm can do better. Still, most networks of interest allow for exponentially faster algorithms. This motivates two questions:
(i) What network topology parameters determine the complexity of distributed optimization?
(ii) Are there universallyoptimal algorithms that are as fast as possible on every topology?
We resolve these 25yearold open problems in the knowntopology setting (i.e., supported CONGEST) for a wide class of global network optimization problems including MST, (1+є)min cut, various approximate shortest paths problems, subgraph connectivity, etc.
In particular, we provide several (equivalent) graph parameters and show they are tight universal lower bounds for the above problems, fully characterizing their inherent complexity. Our results also imply that algorithms based on the lowcongestion shortcut framework match the above lower bound, making them universally optimal if shortcuts are efficiently approximable.
 Publication Date:
 NSFPAR ID:
 10271616
 Journal Name:
 Symposium on Theory of Computing (STOC)
 Page Range or eLocationID:
 1166 to 1179
 Sponsoring Org:
 National Science Foundation
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