On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter

Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this paper we consider partially dynamic versions of three classic problems in graph theory. Based on popular conjectures we show that: - No algorithm with amortized update time O(n^{1-epsilon}) exists for incremental or decremental maximum cardinality bipartite matching. This significantly improves on the O(m^{1/2-epsilon}) bound for sparse graphs of Henzinger et al. [STOC\u2715] and O(n^{1/3-epsilon}) bound of Kopelowitz, Pettie and Porat. Our linear bound also appears more natural. In addition, the result we present separates the node-addition model from the edge insertion model, as an algorithm with total update time O(m*sqrt(n)) exists for the former by Bosek et al. [FOCS\u2714]. - No algorithm with amortized update time O(m^{1-epsilon}) exists for incremental or decremental maximum flow in directed and weighted sparse graphs. No such lower bound was known for partially dynamic maximum flow previously. Furthermore no algorithm with amortized update time O(n^{1-epsilon}) exists for directed and unweighted graphs or undirected and weighted graphs. - No algorithm with amortized update time O(n^{1/2-epsilon}) exists for incremental or decremental (4/3 - epsilon\u27)-approximating the diameter of an unweighted graph. We also show a slightly stronger bound if node additions are allowed. The result is then extended to the static case, where we show that no O((n*sqrt(m))^{1-epsilon}) algorithm exists. We also extend the result to the case when an additive error is allowed in the approximation. While our bounds are weaker than the already known bounds of Roditty and Vassilevska Williams [STOC\u2713], it is based on a weaker conjecture of Abboud et al. [STOC\u2715] and is the first known reduction from the 3SUM and APSP problems to diameter. Showing an equivalence between APSP and diameter is a major open problem in this area (Abboud et al. [SODA\u2715]), and thus showing even a weak connection in this direction is of interest

New Unconditional Hardness Results for Dynamic and Online Problems

There has been a resurgence of interest in lower bounds whose truth rests on the conjectured hardness of well known computational problems. These conditional lower bounds have become important and popular due to the painfully slow progress on proving strong unconditional lower bounds. Nevertheless, the long term goal is to replace these conditional bounds with unconditional ones. In this paper we make progress in this direction by studying the cell probe complexity of two conjectured to be hard problems of particular importance: matrix-vector multiplication and a version of dynamic set disjointness known as Patrascu's Multiphase Problem. We give improved unconditional lower bounds for these problems as well as introducing new proof techniques of independent interest. These include a technique capable of proving strong threshold lower bounds of the following form: If we insist on having a very fast query time, then the update time has to be slow enough to compute a lookup table with the answer to every possible query. This is the first time a lower bound of this type has been proven

Algorithms and Conditional Lower Bounds for Planning Problems

We consider planning problems for graphs, Markov decision processes (MDPs), and games on graphs. While graphs represent the most basic planning model, MDPs represent interaction with nature and games on graphs represent interaction with an adversarial environment. We consider two planning problems where there are k different target sets, and the problems are as follows: (a) the coverage problem asks whether there is a plan for each individual target set, and (b) the sequential target reachability problem asks whether the targets can be reached in sequence. For the coverage problem, we present a linear-time algorithm for graphs and quadratic conditional lower bound for MDPs and games on graphs. For the sequential target problem, we present a linear-time algorithm for graphs, a sub-quadratic algorithm for MDPs, and a quadratic conditional lower bound for games on graphs. Our results with conditional lower bounds establish (i) model-separation results showing that for the coverage problem MDPs and games on graphs are harder than graphs and for the sequential reachability problem games on graphs are harder than MDPs and graphs; (ii) objective-separation results showing that for MDPs the coverage problem is harder than the sequential target problem.Comment: Accepted at ICAPS'1