An optimal lower bound on the number of variables for graph identification

In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-variable language with counting is equivalent to the (k−1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant becausen variables obviously suffice to identify graphs onn vertices.

[1]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[2]  Neil Immerman,et al.  The expressiveness of a family of finite set languages , 1991, PODS '91.

[3]  László Babai,et al.  On the Complexity of Canonical Labeling of Strongly Regular Graphs , 1980, SIAM J. Comput..

[4]  László Babai,et al.  Canonical labelling of graphs in linear average time , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[5]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[6]  D. Djoković,et al.  On regular graphs, VI , 1973 .

[7]  László Babai,et al.  Moderately Exponential Bound for Graph Isomorphism , 1981, FCT.

[8]  Neil Immerman,et al.  On Uniformity within NC¹ , 1990, J. Comput. Syst. Sci..

[9]  Regina Tyshkevich,et al.  Graph isomorphism problem , 1985 .

[10]  A. Ehrenfeucht.  An application of games to the completeness problem for formalized theories , 1961 .

[11]  Neil Immerman,et al.  Relational Queries Computable in Polynomial Time , 1986, Inf. Control..

[12]  B. Weisfeiler.  On construction and identification of graphs , 1976 .

[13]  L. Babai.  On the Order of Uniprimitive Permutation Groups , 1981 .

[14]  Paul Erdös,et al.  Random Graph Isomorphism , 1980, SIAM J. Comput..

[15]  Neil Immerman,et al.  Number of Quantifiers is Better Than Number of Tape Cells , 1981, J. Comput. Syst. Sci..

[16]  Derek G. Corneil,et al.  The graph isomorphism disease , 1977, J. Graph Theory.

[17]  Gary L. Miller,et al.  On the nlog n isomorphism technique (A Preliminary Report) , 1978, STOC.

[18]  D. G. Higman.  Coherent configurations , 1975 .

[19]  Martin Fürer,et al.  Normal forms for trivalent graphs and graphs of bounded valence , 1983, STOC.

[20]  Rudolf Mathon,et al.  A Note on the Graph Isomorphism counting Problem , 1979, Inf. Process. Lett..

[21]  Miklós Ajtai,et al.  Recursive construction for 3-regular expanders , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[22]  László Babai,et al.  Computational complexity and the classification of finite simple groups , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[23]  E. Lander,et al.  Describing Graphs: A First-Order Approach to Graph Canonization , 1990 .

[24]  Neil Immerman.  Upper and lower bounds for first order expressibility , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[25]  László Babai,et al.  Canonical labeling of graphs , 1983, STOC.

[26]  Peter J. Cameron,et al.  6-Transitive graphs , 1980, J. Comb. Theory, Ser. B.

[27]  Ludek Kucera,et al.  Canonical labeling of regular graphs in linear average time , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[28]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[29]  David Harel,et al.  Structure and complexity of relational queries , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).