A tighter relation between sensitivity complexity and certificate complexity

Kun He, Qian Li, Xiaoming Sun

Abstract: The sensitivity conjecture proposed by Nisan and Szegedy in 1994, which asserts that for any Boolean function, its sensitivity complexity is polynomially related to the block sensitivity complexity, is one of the most important and challenging problems in the study of decision tree complexity. Despite a lot of efforts, the best known upper bound of block sensitivity, as well as the certificate complexity, is still exponential in terms of sensitivity. In this paper, we give a better upper bound for certificate complexity and block sensitivity, bs(f)≤C(f)≤(89+o(1))s(f)2s(f)−1<math><mi is="true">b</mi><mi is="true">s</mi><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo><mo is="true">≤</mo><mi is="true">C</mi><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo><mo is="true">≤</mo><mo stretchy="false" is="true">(</mo><mfrac is="true"><mrow is="true"><mn is="true">8</mn></mrow><mrow is="true"><mn is="true">9</mn></mrow></mfrac><mo is="true">+</mo><mi is="true">o</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">)</mo><mi is="true">s</mi><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">s</mi><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo><mo is="true">−</mo><mn is="true">1</mn></mrow></msup></math>, where bs(f),C(f)<math><mi is="true">b</mi><mi is="true">s</mi><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo><mo is="true">,</mo><mi is="true">C</mi><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo></math> and s(f)<math><mi is="true">s</mi><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo></math> are the block sensitivity, certificate complexity and sensitivity, respectively. The proof is based on a deep investigation on the structure of the sensitivity graph. We also provide a tighter relationship between the 0-certificate complexity C0(f)<math><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo></math> and 0-sensitivity s0(f)<math><msub is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo></math> for functions with small 1-sensitivity s1(f)<math><msub is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo></math>.