Popular conjectures imply strong lower bounds for dynamic problems
  We consider several well-studied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms:   1. Is the 3SUM problem on $n$ numbers in $O(n^{2-\epsilon})$ time for some $\epsilon>0$?   2. Can one determine the satisfiability of a CNF formula on $n$ variables in $O((2-\epsilon)^n poly n)$ time for some $\epsilon>0$?   3. Is the All Pairs Shortest Paths problem for graphs on $n$ vertices in $O(n^{3-\epsilon})$ time for some $\epsilon>0$?   4. Is there a linear time algorithm that detects whether a given graph contains a triangle?   5. Is there an $O(n^{3-\epsilon})$ time combinatorial algorithm for $n\times n$ Boolean matrix multiplication?   The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Pagh's problem defined in a recent paper by Patrascu [STOC 2010]. 
Amir Abboud and Virginia Vassilevska Williams
cs.DS cs.CC