4 research outputs found
On topological relaxations of chromatic conjectures
There are several famous unsolved conjectures about the chromatic number that
were relaxed and already proven to hold for the fractional chromatic number. We
discuss similar relaxations for the topological lower bound(s) of the chromatic
number. In particular, we prove that such a relaxed version is true for the
Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of
Hadwiger from this point of view. For the latter, a similar statement was
already proven in an earlier paper of the first author with G. Tardos, our main
concern here is that the so-called odd Hadwiger conjecture looks much more
difficult in this respect. We prove that the statement of the odd Hadwiger
conjecture holds for large enough Kneser graphs and Schrijver graphs of any
fixed chromatic number
Random-order bin packing
AbstractThe average-case analysis of algorithms usually assumes independent, identical distributions for the inputs. In [C. Kenyon, Best-fit bin-packing with random order, in: Proc. of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 1996, pp. 359–364] Kenyon introduced the random-order ratio, a new average-case performance metric for bin packing heuristics, and gave upper and lower bounds for it for the Best Fit heuristics. We introduce an alternative definition of the random-order ratio and show that the two definitions give the same result for Next Fit. We also show that the random-order ratio of Next Fit equals to its asymptotic worst-case, i.e., it is 2