TY - GEN
T1 - A generalized turán problem and its applications
AU - Gishboliner, Lior
AU - Shapira, Asaf
N1 - Publisher Copyright:
© 2018 Copyright held by the owner/author(s).
PY - 2018/6/20
Y1 - 2018/6/20
N2 - Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f , there is a graph property whose 1-sided-error query complexity is precisely f (Θ(1/)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/ε). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/ε). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem States that for any function f there is a graph property whose 1-sided-error query complexity is f (Θ(1/)) while its 2-sided-error query complexity is only poly(1/). This is the first indication of the surprising power that 2-sided-error testing algorithms have over 1-sided-error ones, even when restricted to properties that are testable with 1-sided error. Again, no result of this type was previously known for any f that is super polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turán problem: for fixed graphs H andT, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T, H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on ex(n,C3,C2ℓ+1). Our third theorem in this paper gives tight bounds for ex(n,Ck,Cℓ) for all the remaining values of k and ℓ.
AB - Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f , there is a graph property whose 1-sided-error query complexity is precisely f (Θ(1/)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/ε). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/ε). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem States that for any function f there is a graph property whose 1-sided-error query complexity is f (Θ(1/)) while its 2-sided-error query complexity is only poly(1/). This is the first indication of the surprising power that 2-sided-error testing algorithms have over 1-sided-error ones, even when restricted to properties that are testable with 1-sided error. Again, no result of this type was previously known for any f that is super polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turán problem: for fixed graphs H andT, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T, H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on ex(n,C3,C2ℓ+1). Our third theorem in this paper gives tight bounds for ex(n,Ck,Cℓ) for all the remaining values of k and ℓ.
KW - Generalized turán problem
KW - Graph property testing
UR - http://www.scopus.com/inward/record.url?scp=85049898512&partnerID=8YFLogxK
U2 - 10.1145/3188745.3188778
DO - 10.1145/3188745.3188778
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AN - SCOPUS:85049898512
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 376
EP - 389
BT - STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
A2 - Henzinger, Monika
A2 - Kempe, David
A2 - Diakonikolas, Ilias
PB - Association for Computing Machinery
T2 - 50th Annual ACM Symposium on Theory of Computing, STOC 2018
Y2 - 25 June 2018 through 29 June 2018
ER -