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The Kadison-Singer Problem for Strongly Rayleigh Measures and Applications to Asymmetric TSP [article]

Nima Anari, Shayan Oveis Gharan
2015 arXiv   pre-print
Marcus, Spielman, and Srivastava in their seminal work MSS13 resolved the Kadison-Singer conjecture by proving that for any set of finitely supported independently distributed random vectors v_1,..., v_n  ...  As independent distributions are just special cases of strongly Rayleigh measures, it is a natural question to see if the main theorem of MSS13 can be extended to families of vectors assigned to the elements  ...  Acknowledgement We would like to thank Adam Marcus, Dan Spielman, and Nikhil Srivastava for stimulating discussions regarding the main obstacles in generalizing their proof of the Kadison-Singer problem  ... 
arXiv:1412.1143v2 fatcat:bpy7kimwnnagpdlvlkwnt4g64a

Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP [article]

Nima Anari, Shayan Oveis Gharan
2015 arXiv   pre-print
We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is polyloglog(n).  ...  Then, we use a recent extension of the seminal work of Marcus, Spielman, and Srivastava [MSS13] by the authors [AO14] to prove the existence of a polylog(k)/k-spectrally thin tree with respect to D.  ...  We would like to thank James R Lee, Satish Rao, Amin Saberi, Luca Trevisan, and Umesh Vazirani for enlightening and stimulating discussions.  ... 
arXiv:1411.4613v4 fatcat:cafqmktkrvbmfhfti3hadefysa