Some complexity theoretic aspects of graph isomorphism and related problems[HBNI Th19]

Abstract

The complexity of graph isomorphism problem for restricted classes of graphs are studied and the complexity of group theoretic problems related graph isomorphism are investigated. Several problems closely related to the graph isomorphism problem are classified in Algorithmic graph theory in the classes PZK and SZK. A constant round perfect zero knowledge proof is given for the group isomorphism problem when the groups are given by their multiplication tables. The prover and the verifier in this proof system use only polylogarithmically many random bits. On this motivation, Honest Verifier Statistical Zero Knowledge(HVSZK) proof is studied where the prover, verifier and the simulator use polylogarithmic randomness but also has polylogarithmic message size and only 2 rounds. A polynomial-time oracle algorithm is given for Tournament Canonization that accesses oracles for Tournament Isomorphism and Rigid-Tournament Canonization. Extending the Babai-Luks Tournament Canonization algorithm, an n^O( k^2 + log n) is given for canonization and isomorphism testing of k-hypertournaments, where n is the number of vertices and k is the size of hyper edges. A FPT algorithm is given for the bounded color class hypergraph isomorphism problem which has run-time (b!2^O(b))(N^O(1)), where b is the size of the largest color class and N is the input size. It is proved that the isomorphism and canonization problem for k-tree is in the class StUL which is contained in UL. It is also proved that the isomorphism problem for k-path is complete for L under disjunctive truth-table reductions computable in uniform AC^0.