1 / 24

On the Hardness of Graph Isomorphism Jacobo Tor á n SIAM J. Comput. Vol 33, p1093-1108, 2004.

On the Hardness of Graph Isomorphism Jacobo Tor á n SIAM J. Comput. Vol 33, p1093-1108, 2004. Presenter: Qingwu Yang April, 2006.

blue
Download Presentation

On the Hardness of Graph Isomorphism Jacobo Tor á n SIAM J. Comput. Vol 33, p1093-1108, 2004.

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. On the Hardness of Graph IsomorphismJacobo ToránSIAM J. Comput. Vol 33, p1093-1108, 2004. Presenter: Qingwu Yang April, 2006

  2. A graph isomorphism is a bijection, i.e., a one-to-one mapping, between the vertices of two graphs G and H: with the property that any two vertices u and v from G are adjacent if and only if f(u) and f(v) are adjacent in H. If an isomorphism can be constructed between two graphs, then we say those graphs are isomorphic. Determining whether two graphs are isomorphic is the graph isomorphism problem. (From Wikipedia, the free encyclopedia) Graph isomorphism (GI)

  3. Example • f(a) = 1 • f(b) = 6 • f(c) = 8 • f(d) = 3 • f(g) = 5 • f(h) = 2 • f(i) = 4 • f(j) = 7 (From Wikipedia, the free encyclopedia)

  4. Why is GI interesting? • Many applications • One of the few problems in NP that has resisted all attempts to be classified as NP-complete, or within P. • Graph isomorphism complete complexity class. • The best existing upper bound for GI is • Polynomial time algorithms for restricted graph classes, such as planar graphs or graphs of bounded degree.

  5. Graph Automorphism An automorphism in an undirected graph G = (V,E) is a permutation ϕ of the nodes that preserves adjacency. That is, for every u, v ∈ V, (u, v) ∈ E ⇔ (ϕ(u), ϕ(v)) ∈ E. http://mathworld.wolfram.com/GraphAutomorphism.html

  6. An example of Graph Automorphism For example, the grid graph has four automorphisms: (1, 2, 3, 4, 5, 6), (2, 1, 4, 3, 6, 5), (5, 6, 3, 4, 1, 2), and (6, 5, 4, 3, 2, 1). http://mathworld.wolfram.com/GraphAutomorphism.html

  7. Logarithmic space modular counting class -- ModkL Class • Functions f: there is a nondeterministic polynomial time Turing machine which, on input x, has exactly f(x) accepting computation paths. • ModkL is the complexity class corresponding to sets recognized by nondeterministic logarithmic space machines such that x  ModkL iff f(x)  0 (mod k) for every natural number k.

  8. AC0 Class and AC0 many-one reductions • AC0 Class: class of problems solvable by a uniform family of Boolean circuits of polynomial size and constant depth with unbounded fan-in AND- and OR- gates, and NOT gates. • AC0 many-one reductions from function f to function g: if there is a uniform AC0 family of circuits  such that f(x) = g((x)) for every input x.

  9. GI is hard for all the logarithmic space modular counting classes ModkL (k≥2) Main idea: • simulate each modular gate with a graph gadget • then combine the gadgets for the different gates into a graph, whose automorphisms simulate the behavior of the modular circuit.

  10. How to generate graph gadgets?

  11. The graph G2simulating a parity gate(k = 2 )

  12. Properties of graph gadgets • The graph gadget for a modular gate has nodes encoding the inputs and outputs of the gate. • Any automorphism in the graph mapping the input nodes in a certain way must map the output nodes according to the value of the modular gate being simulated.

  13. Next we prove ϕis an automorphism, for allv,w, (v,w) ∈ E if and only if (ϕ(v), ϕ(w)) ∈ E. The nodes in graph Gkcan be partitioned in three layers, the x and y nodes (input layer), the u nodes, andthe z nodes (output layer).

  14. Proof • Let k ≥ 2. We reduce the modkcircuit value problem to GI. • We transform an instance C of the circuit value problem for modkcircuits into a graph GCby constructing for every modular gate gj of C a graph gadget. • Moreover, we color the x, y, u, and z nodes of the jth gadget, respectively, with one of the colors (x, j), (y, j), (u, j), and (z, j). The coloring implies that in any automorphism the nodes corresponding to a gate are mapped to nodes from the same gate.

  15. Proof • Connections between gates are translated in the following way: If the output z of a gate in the circuit is connected to one of the inputs x of another gate, the reduction puts k additional edges connecting (for i ∈ {0, . . . , k − 1}) node zifrom the first gate to node xifrom the second gate. • For an input variable vj , k nodes vj0, . . . , vjk−1 are considered in the reduction. • Suppose the input variables of the circuit, v1,…,vn, take values a1,…,an. The output gate z takes value b ∈ {0,…,k−1} if and only if there is an automorphism in GCmapping for all i = 0,…,k−1 and j = 1,…,n, and mapping zi to

  16. Proof • All the steps in the reduction can be done locally by an AC0 circuit. • The question of whether the output of the circuit equals b ∈ {0,…,k−1} can be easily reduced to whether two graphs Gb,Gb’are isomorphic. • This question can be reduced to two graphs pairs ((G,H), (I, J)) ∈ PGI, with G being isomorphic to H if the value of the circuit is b, and I being isomorphic to J otherwise. For this it suffices to define G as Gb, H as Gb, and I and J as the standard OR-function for GI of i=b(Gi,Gi).

  17. Homework • The undirected graph accessibility problem is: given an undirected graph G and two vertices u, vG, determine whether there is a path from u to v. • Show that the undirected graph accessibility problem is reducible to the complement of GI.

More Related