----------------------- REVIEW 1 ---------------------
PAPER: 16
TITLE: Fault-Tolerant Consensus in Directed Graphs
AUTHORS: Lewis Tseng and Nitin Vaidya


----------- REVIEW -----------
This nicely written and presented paper deals with fault-tolerance consensus in directed graphs.
The work identifies necessary and sufficient conditions on the underlying communication graph for achieving consensus in directed networks.
Nice results, certainly fit PODC.
Most Thms and Lemmas seem natural, from graph-theoretic point of view, and I thus believe their proofs are correct (most of them are in Appendixes).


----------------------- REVIEW 2 ---------------------
PAPER: 16
TITLE: Fault-Tolerant Consensus in Directed Graphs
AUTHORS: Lewis Tseng and Nitin Vaidya


----------- REVIEW -----------
This is a nice and well written paper that studies the problem of fault tolerant consensus in graphs that have Directed Links.
The paper provides necessary and sufficient conditions for 3 cases :
1. For exact synchronous consensus
2. For approximate asynchronous consensus
3. For exact synchronous BYzantine consensus.

The paper is very long but well structured. I believe that the proofs are correct.
This paper extends simultaneously many results in the area.
And somehow it unifies approaches.
This can be a typical PODC paper.


----------------------- REVIEW 3 ---------------------
PAPER: 16
TITLE: Fault-Tolerant Consensus in Directed Graphs
AUTHORS: Lewis Tseng and Nitin Vaidya


----------- REVIEW -----------
This paper deals with some flavors of consensus over incomplete directed graphs.
Consensus over incomplete graphs including directed graphs has been already studied from various
perspectives but the present paper finds a niche for which it provides a characterization
at least for the cases of Byzantine synchronous and crash-failure consensus.

For crash-failures the algorithm proceeds in phases where an alternation of min-max functions
are applied in each phase to the values that are received by each node.  This is shown to ensure agreement under appropriate graph conditions. Many of the results are presented in the appendix and only sketched in the main body.  A good feature is that the graph conditions presented are tight.

Overall the paper is well written. The particular flavor of consensus appears to have not been considered in this way before even though quite related variants have appeared in the past. Still the results are reasonably interesting. I lean towards acceptance.

Minor
"we address crash faults and Byzantine faults both" => we address both...