diff --git a/src/sage/combinat/posets/posets.py b/src/sage/combinat/posets/posets.py
index c5747a43c28..ef2233eeb49 100644
--- a/src/sage/combinat/posets/posets.py
+++ b/src/sage/combinat/posets/posets.py
@@ -549,7 +549,8 @@ def Poset(data=None, element_labels=None, cover_relations=False, linear_extensio
 
     # Determine cover relations, if necessary.
     if cover_relations is False:
-        D = D.transitive_reduction()
+        from sage.graphs.generic_graph_pyx import transitive_reduction_acyclic
+        D = transitive_reduction_acyclic(D)
 
     # Check that the digraph does not contain loops, multiple edges
     # and is transitively reduced.
diff --git a/src/sage/graphs/base/static_dense_graph.pyx b/src/sage/graphs/base/static_dense_graph.pyx
index b872d51117b..11e98fe212d 100644
--- a/src/sage/graphs/base/static_dense_graph.pyx
+++ b/src/sage/graphs/base/static_dense_graph.pyx
@@ -41,7 +41,7 @@ Functions
 ---------
 """
 
-cdef dict dense_graph_init(binary_matrix_t m, g, translation = False):
+cdef dict dense_graph_init(binary_matrix_t m, g, translation=False):
     r"""
     Initializes the binary matrix from a Sage (di)graph.
 
diff --git a/src/sage/graphs/generic_graph.py b/src/sage/graphs/generic_graph.py
index 632464aaeae..b0b43e0de10 100644
--- a/src/sage/graphs/generic_graph.py
+++ b/src/sage/graphs/generic_graph.py
@@ -14575,15 +14575,9 @@ def transitive_closure(self):
             sage: digraphs.Path(5).copy(immutable=True).transitive_closure()
             Transitive closure of Path: Digraph on 5 vertices
         """
-        from copy import copy
         G = self.copy(immutable=False)
         G.name('Transitive closure of ' + self.name())
-        for v in G:
-            # todo optimization opportunity: we are adding edges that
-            # are already in the graph and we are adding edges
-            # one at a time.
-            for e in G.breadth_first_search(v):
-                G.add_edge((v,e))
+        G.add_edges((v,e) for v in G for e in G.breadth_first_search(v))
         return G
 
     def transitive_reduction(self):
@@ -14612,14 +14606,17 @@ def transitive_reduction(self):
             sage: g.transitive_reduction().size()
             5
         """
-        from copy import copy
-        from sage.rings.infinity import Infinity
-        G = copy(self)
+        if self.is_directed_acyclic():
+            from sage.graphs.generic_graph_pyx import transitive_reduction_acyclic
+            return transitive_reduction_acyclic(self)
+
+        G = self.copy(immutable=False)
+        n = G.order()
         for e in self.edge_iterator():
             # Try deleting the edge, see if we still have a path
             # between the vertices.
             G.delete_edge(e)
-            if G.distance(e[0],e[1])==Infinity:
+            if G.distance(e[0],e[1]) > n:
                 # oops, we shouldn't have deleted it
                 G.add_edge(e)
         return G
diff --git a/src/sage/graphs/generic_graph_pyx.pyx b/src/sage/graphs/generic_graph_pyx.pyx
index c456ffe29cc..983d600ea9c 100644
--- a/src/sage/graphs/generic_graph_pyx.pyx
+++ b/src/sage/graphs/generic_graph_pyx.pyx
@@ -18,6 +18,7 @@ AUTHORS:
 include "sage/ext/interrupt.pxi"
 include 'sage/ext/cdefs.pxi'
 include 'sage/ext/stdsage.pxi'
+include "sage/misc/binary_matrix.pxi"
 
 # import from Python standard library
 from sage.misc.prandom import random
@@ -1280,3 +1281,66 @@ cpdef tuple find_hamiltonian( G, long max_iter=100000, long reset_bound=30000, l
 
     return (True,output)
 
+def transitive_reduction_acyclic(G):
+    r"""
+    Returns the transitive reduction of an acyclic digraph
+
+    INPUT:
+
+    - ``G`` -- an acyclic digraph.
+
+    EXAMPLE::
+
+        sage: from sage.graphs.generic_graph_pyx import transitive_reduction_acyclic
+        sage: G = posets.BooleanLattice(4).hasse_diagram()
+        sage: G == transitive_reduction_acyclic(G.transitive_closure())
+        True
+    """
+    cdef int  n = G.order()
+    cdef dict v_to_int = {vv:i for i,vv in enumerate(G.vertices())}
+    cdef int  u,v,i
+
+    cdef list linear_extension = G.topological_sort()
+    linear_extension.reverse()
+
+    cdef binary_matrix_t closure
+
+    # Build the transitive closure of G
+    #
+    # A point is reachable from u if it is one of its neighbours, or if it is
+    # reachable from one of its neighbours.
+    binary_matrix_init(closure,n,n)
+    binary_matrix_fill(closure,0)
+    for uu in linear_extension:
+        u = v_to_int[uu]
+        for vv in G.neighbors_out(uu):
+            v = v_to_int[vv]
+            binary_matrix_set1(closure,u,v)
+            for i in range(closure.width):
+                closure.rows[u][i] |= closure.rows[v][i]
+
+    # Build the transitive reduction of G
+    #
+    # An edge uv belongs to the transitive reduction of G if no outneighbor of u
+    # can reach v (except v itself, of course).
+    linear_extension.reverse()
+    cdef list useful_edges = []
+    for uu in linear_extension:
+        u = v_to_int[uu]
+        for vv in G.neighbors_out(uu):
+            v = v_to_int[vv]
+            for i in range(closure.width):
+                closure.rows[u][i] &= ~closure.rows[v][i]
+        for vv in G.neighbors_out(uu):
+            v = v_to_int[vv]
+            if binary_matrix_get(closure,u,v):
+                useful_edges.append((uu,vv))
+
+    from sage.graphs.digraph import DiGraph
+    reduced = DiGraph()
+    reduced.add_edges(useful_edges)
+    reduced.add_vertices(linear_extension)
+
+    binary_matrix_free(closure)
+
+    return reduced