Minor changes.

diff --git a/PLAN b/PLAN
index b57a77edd16239179169b1f38345f40a8e287634..b4699d68bf66adb38cfcda039bcbd4f2a6a95de2 100644 (file)
--- a/PLAN
+++ b/PLAN
@@ -66,6 +66,7 @@ Ranking:
 - the general perspective: is it only a technical trick?
 - move description of the ranking set to the chapter on models?
 - ranking of permutations on general sets, relationship with integer sorting
+- mention approximation of permanent
 
 Notation:
 

diff --git a/rank.tex b/rank.tex
index 73c3479f44b8eb3633d173c7f2a973d761f8e0ec..2606aba195b5730161fc3435b1cec5a67a2136e4 100644 (file)
--- a/rank.tex
+++ b/rank.tex
@@ -426,11 +426,16 @@ in decreasing order and stopping as soon as we find~$\pi[1]$ which can be
 extended to a~complete permutation. We can do this for example by using the
 Dinic's algorithm as described above on~the graph of remaining restrictions
 (i.e., $G$ with the vertices 1 and~$\pi[1]$ and removed together with the corresponding
-edges). Once we have~$\pi[1]$, we can fix it and proceed with finding $\pi[2]$
-using the reduced graph. This way we construct the whole maximal permutation~$\pi$
-in~$\O(n^2)$ calls to the Dinic's algorithm.
+edges). Once we have~$\pi[1]$, we can fix it and proceed by finding $\pi[2]$
+in the same way, using the reduced graph. This way we construct the whole
+maximal permutation~$\pi$ in~$\O(n^2)$ calls to the Dinic's algorithm.
 \qed
 
+\para
+However, the hardness of computing the permanent is the worst obstacle.
+We will show that whenever we are given a~set of restrictions for which
+the counting problem is easy (and it is also easy for subgraphs obtained
+by deleting vertices), ranking is easy as well.
 
 
 \endpart