73,248 research outputs found
Graph Isomorphism and the Lasserre Hierarchy
In this paper we show lower bounds for a certain large class of algorithms
solving the Graph Isomorphism problem, even on expander graph instances.
Spielman [25] shows an algorithm for isomorphism of strongly regular expander
graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to
expf O(n^(1/5) [5]). It has since been an open question to remove the
requirement that the graph be strongly regular. Recent algorithmic results show
that for many problems the Lasserre hierarchy works surprisingly well when the
underlying graph has expansion properties. Moreover, recent work of Atserias
and Maneva [3] shows that k rounds of the Lasserre hierarchy is a
generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph
Isomorphism. These two facts combined make the Lasserre hierarchy a good
candidate for solving graph isomorphism on expander graphs. Our main result
rules out this promising direction by showing that even Omega(n) rounds of the
Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism
problem even on expander graphs.Comment: 22 pages, 3 figures, submitted to CC
Isomorphism Checking in GROOVE
In this paper we show how isomorphism checking can be used as an effective technique for symmetry reduction in graph-based state spaces, despite the inherent complexity of the isomorphism problem. In particular, we show how one can use element-based graph certificate mappings to help in recognising nonisomorphic graphs. These are mappings that assign to all elements (edges and nodes) of a given graph a number that is invariant under isomorphism, in the sense that any isomorphism between graphs is sure to preserve this number. The individual element certificates of a graph give rise to a certificate for the entire graph, which can be used as a hash key for the graph; hence, this yields a heuristic to decide whether a graph has an isomorphic representative in a previously computed set of graphs. We report some experiments that show the viability of this method. \u
Two graph isomorphism polytopes
The convex hull of certain tensors was considered
recently in connection with graph isomorphism. We consider the convex hull
of the diagonals among these tensors. We show: 1. The polytope
is a face of . 2. Deciding if a graph has a subgraph
isomorphic to reduces to optimization over . 3. Optimization over
reduces to optimization over . In particular, this implies
that the subgraph isomorphism problem reduces to optimization over
Quantum Algorithms for Tree Isomorphism and State Symmetrization
The graph isomorphism problem is theoretically interesting and also has many
practical applications. The best known classical algorithms for graph
isomorphism all run in time super-polynomial in the size of the graph in the
worst case. An interesting open problem is whether quantum computers can solve
the graph isomorphism problem in polynomial time. In this paper, an algorithm
is shown which can decide if two rooted trees are isomorphic in polynomial
time. Although this problem is easy to solve efficiently on a classical
computer, the techniques developed may be useful as a basis for quantum
algorithms for deciding isomorphism of more interesting types of graphs. The
related problem of quantum state symmetrization is also studied. A polynomial
time algorithm for the problem of symmetrizing a set of orthonormal states over
an arbitrary permutation group is shown
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