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It is known that a Cayley digraph of an abelian group A is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup B of A such that the connection set without B is a union of cosets of B in A. We generalize this result to Cayley digraphs of nonabelian groups G by showing that such a digraph is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup H of G such that S without H is a union of double cosets of H in G. This result is proven in the more general situation of a double coset digraph (also known as a Sabidussi coset digraph.) We then give applications of this result which include obtaining a graph theoretic definition of double coset digraphs, and determining the relationship between a double coset digraph and its corresponding Cayley digraph. We further expand the result obtained for double coset digraphs to a collection of bipartite graphs called bi-coset graphs and the bipartite equivalent to Cayley graphs called Haar graphs. Instead of considering when this collection of graphs is a wreath product, we consider the more general graph product known as an X-join by showing that a connected bi-coset graph of a group G with respect to some subgroups L and R of G is isomorphic to an X-join of a collection of empty graphs if and only if the connection set is a union of double cosets of some subgroups N containing L and M containing R in G. The automorphism group of such -joins is also found. We also prove that disconnected bi-coset graphs are always isomorphic to a wreath product of an empty graph with a bi-coset graph.

Every knapsack problem may be relaxed to a cyclic group problem. In 1969, Gomory found the subadditive characterization of facets of the master cyclic group problem. We simplify the subadditive relations by the substitution of complementarities and discover a minimal representation of the subadditive polytope for the master cyclic group problem. By using the minimal representation, we characterize the vertices of cardinality length 3 and implement the shooting experiment from the natural interior point. The shooting from the natural interior point is a shooting from the inside of the plus level set of the subadditive polytope. It induces the shooting for the knapsack problem. From the shooting experiment for the knapsack problem we conclude that the most hit facet is the knapsack mixed integer cut which is the 2-fold lifting of a mixed integer cut. We develop a cutting plane algorithm augmenting cutting planes generated by shooting, and implement it on Wong-Coppersmith digraphs observing that only small number of cutting planes are enough to produce the optimal solution. We discuss a relaxation of shooting as a clue to quick shooting. A max flow model on covering space is shown to be equivalent to the dual of shooting linear programming problem.