2 Search Results for "Gillman, David"


Document
Imperfect Gaps in Gap-ETH and PCPs

Authors: Mitali Bafna and Nikhil Vyas

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
We study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a way to transform a PCP with imperfect completeness to one with perfect completeness, when the initial gap is a constant. We show that PCP_{c,s}[r,q] subseteq PCP_{1,s'}[r+O(1),q+O(r)] for c-s=Omega(1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NP with a O(log n) additive loss in the query complexity q. We show our result by constructing a "robust circuit" using threshold gates. These results are a gap amplification procedure for PCPs, (when completeness is not 1) analogous to questions studied in parallel repetition [Anup Rao, 2011] and pseudorandomness [David Gillman, 1998] and might be of independent interest. We also investigate the time-complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness, i.e. MAX 3SAT(1-epsilon,1-delta), delta > epsilon has 2^{o(n)} algorithms if and only if MAX 3SAT(1,1-delta) has 2^{o(n)} algorithms. We also relate the time complexities of these two problems in a more fine-grained way to show that T_2(n) <= T_1(n(log log n)^{O(1)}), where T_1(n),T_2(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness respectively.

Cite as

Mitali Bafna and Nikhil Vyas. Imperfect Gaps in Gap-ETH and PCPs. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bafna_et_al:LIPIcs.CCC.2019.32,
  author =	{Bafna, Mitali and Vyas, Nikhil},
  title =	{{Imperfect Gaps in Gap-ETH and PCPs}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{32:1--32:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.32},
  URN =		{urn:nbn:de:0030-drops-108545},
  doi =		{10.4230/LIPIcs.CCC.2019.32},
  annote =	{Keywords: PCP, Gap-ETH, Hardness of Approximation}
}
Document
Slow Convergence of Ising and Spin Glass Models with Well-Separated Frustrated Vertices

Authors: David Gillman and Dana Randall

Published in: LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)


Abstract
Many physical models undergo phase transitions as some parameter of the system is varied. This phenomenon has bearing on the convergence times for local Markov chains walking among the configurations of the physical system. One of the most basic examples of this phenomenon is the ferromagnetic Ising model on an n x n square lattice region Lambda with mixed boundary conditions. For this spin system, if we fix the spins on the top and bottom sides of the square to be + and the left and right sides to be -, a standard Peierls argument based on energy shows that below some critical temperature t_c, any local Markov chain M requires time exponential in n to mix. Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered frustrated because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under 90 degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This class includes the ferromagnetic Ising model with mixed boundary conditions described above, where the frustrated faces are on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of Lambda and yields weaker results when they are on the boundary of Lambda but not near the corners. We show that there is a universal temperature T below which M will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the free energy, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing.

Cite as

David Gillman and Dana Randall. Slow Convergence of Ising and Spin Glass Models with Well-Separated Frustrated Vertices. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{gillman_et_al:LIPIcs.AofA.2018.24,
  author =	{Gillman, David and Randall, Dana},
  title =	{{Slow Convergence of Ising and Spin Glass Models with Well-Separated Frustrated Vertices}},
  booktitle =	{29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)},
  pages =	{24:1--24:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{Fill, James Allen and Ward, Mark Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.24},
  URN =		{urn:nbn:de:0030-drops-89170},
  doi =		{10.4230/LIPIcs.AofA.2018.24},
  annote =	{Keywords: Mixing time, spin glass, Ising model, mixed boundary conditions, frustration}
}
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