Necessary and sufficient conditions for the existence of a combinatorial algorithm for a given problem.
Some people in combinatorial optimization use the word "combinatorial algorithm" to mean "algorithm without using ellipsoid method (or interior-point methods)."  For example, Schrijver's Fulkerson-prize-winning paper has a title "A combinatorial algorithm minimizing submodular functions in strongly polynomial time."  The co-winners (Iwata, Fleischer, and Fujishige) also have "A combinatorial strongly polynomial algorithm for minimizing submodular function."

As far as I see the field of combinatorial optimization, people there actually try to give an answer to the posed question, and still struggle.  Cunningham's survey "Matching, matroids, and extensions" (Mathematical Programming B91 (2002) 515-542) is a bit old by now, but you can see how generalizations of Edmonds' blossom algorithm and concepts on matroid lead to combinatorial polynomial-time algorithms for other problems.  Please note that the field has been advanced since the survey, and good sources are probably some papers from IPCO (Integer Programming and Combinatorial Optimization).

I would like to point out that an big open problem is to find a combinatorial algorithm for the maximum independent set problem for perfect graphs running in strongly polynomial time.