The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials

• URL: http://arxiv.org/abs/2106.02129v1
• Date: Thu, 3 Jun 2021 21:01:02 GMT
• Title: The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials
• Authors: Guy Bresler, Brice Huang
• Abstract summary: It is known that a satisfying assignment exists with high probability at clause density $m/n 2k log 2 - frac12. We show that low degree algorithms can find a satisfying assignment at clause density $(1 - o_k(1)) 2k log k / k$.
• Score: 18.00315760946089
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $\Phi$ be a uniformly random $k$-SAT formula with $n$ variables and $m$ clauses. We study the algorithmic task of finding a satisfying assignment of $\Phi$. It is known that a satisfying assignment exists with high probability at clause density $m/n < 2^k \log 2 - \frac12 (\log 2 + 1) + o_k(1)$, while the best polynomial-time algorithm known, the Fix algorithm of Coja-Oghlan, finds a satisfying assignment at the much lower clause density $(1 - o_k(1)) 2^k \log k / k$. This prompts the question: is it possible to efficiently find a satisfying assignment at higher clause densities? To understand the algorithmic threshold of random $k$-SAT, we study low degree polynomial algorithms, which are a powerful class of algorithms including Fix, Survey Propagation guided decimation, and paradigms such as message passing and local graph algorithms. We show that low degree polynomial algorithms can find a satisfying assignment at clause density $(1 - o_k(1)) 2^k \log k / k$, matching Fix, and not at clause density $(1 + o_k(1)) \kappa^* 2^k \log k / k$, where $\kappa^* \approx 4.911$. This shows the first sharp (up to constant factor) computational phase transition of random $k$-SAT for a class of algorithms. Our proof establishes and leverages a new many-way overlap gap property tailored to random $k$-SAT.

Related papers

• Local Quantum Search Algorithm for Random $k$-SAT with $Ω(n^{1+ε})$ Clauses [0.0]
  We show that max-k-SSAT is the computational algorithm on average when $m=Omega(n2+delta+epsilon)$ based on the average-case complexity theory. We also prove that max-k-SSAT is on average when $m=Omega(n2+delta+epsilon)$ based on the average-case complexity theory.
  arXiv  Detail & Related papers  (2024-03-05T15:03:47Z)
• A Scalable Algorithm for Individually Fair K-means Clustering [77.93955971520549]
  We present a scalable algorithm for the individually fair ($p$, $k$)-clustering problem introduced by Jung et al. and Mahabadi et al. A clustering is then called individually fair if it has centers within distance $delta(x)$ of $x$ for each $xin P$. We show empirically that not only is our algorithm much faster than prior work, but it also produces lower-cost solutions.
  arXiv  Detail & Related papers  (2024-02-09T19:01:48Z)
• Algoritmo de Contagem Qu\^antico Aplicado ao Grafo Bipartido Completo [0.0]
  Grover's algorithm is capable of finding $k$ elements in an unordered database with $N$ elements using $O(sqrtN/k)$ steps. This work tackles the problem of using the quantum counting algorithm for estimating the value $k$ of marked elements in other graphs.
  arXiv  Detail & Related papers  (2023-12-05T21:15:09Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise [50.64137465792738]
  We show that any efficient SQ algorithm for the problem requires sample complexity at least $Omega(d1/2/(maxp, epsilon)2)$. Our lower bound suggests that this quadratic dependence on $1/epsilon$ is inherent for efficient algorithms.
  arXiv  Detail & Related papers  (2023-07-13T18:59:28Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• Near-Optimal Lower Bounds For Convex Optimization For All Orders of Smoothness [26.71898403195793]
  We study the complexity of optimizing highly smooth convex functions. For a positive integer $p$, we want to find an $epsilon$-approximate minimum of a convex function $f$. We prove a new lower bound that matches this bound (up to log factors), and holds not only for randomized algorithms, but also for quantum algorithms.
  arXiv  Detail & Related papers  (2021-12-02T10:51:43Z)
• Logarithmic Regret from Sublinear Hints [76.87432703516942]
  We show that an algorithm can obtain $O(log T)$ regret with just $O(sqrtT)$ hints under a natural query model. We also show that $o(sqrtT)$ hints cannot guarantee better than $Omega(sqrtT)$ regret.
  arXiv  Detail & Related papers  (2021-11-09T16:50:18Z)
• Classical algorithms and quantum limitations for maximum cut on high-girth graphs [6.262125516926276]
  We show that every (quantum or classical) one-local algorithm achieves on $D$-regular graphs of $> 5$ a maximum cut of at most $1/2 + C/sqrtD$ for $C=1/sqrt2 approx 0.7071$. We show that there is a classical $k$-local algorithm that achieves a value of $1/2 + C/sqrtD - O (1/sqrtk)$ for $D$-regular graphs of $> 2k+1$, where
  arXiv  Detail & Related papers  (2021-06-10T16:28:23Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.