| 1. |
- De Rezende, Susanna F., et al.
(författare)
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Automating algebraic proof systems is NP-hard
- 2021
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Ingår i: STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. - New York, NY, USA : ACM. - 0737-8017. - 9781450380539 ; , s. 209-222
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Konferensbidrag (refereegranskat)abstract
- We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula F, it is NP-hard to find a refutation of F in the Nullstellensatz, Polynomial Calculus, or Sherali-Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Müller (JACM 2020) that established an analogous result for Resolution.
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| 2. |
- Arora, Atul Singh, et al.
(författare)
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Quantum Depth in the Random Oracle Model
- 2023
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Ingår i: Proceedings of the Annual ACM Symposium on Theory of Computing. - 0737-8017. ; , s. 1111-1124
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Konferensbidrag (refereegranskat)abstract
- We give a comprehensive characterisation of the computational power of shallow quantum circuits combined with classical computation. Specifically, for classes of search problems, we show that the following statements hold, relative to a random oracle: (a) BPPQNCBPP BQP. This refutes Jozsa's conjecture in the random oracle model. As a result, this gives the first instantiatable separation between the classes by replacing the oracle with a cryptographic hash function, yielding a resolution to one of Aaronson's ten semi-grand challenges in quantum computing. (b) BPPQNC QNCBPP and QNCBPP BPPQNC. This shows that there is a subtle interplay between classical computation and shallow quantum computation. In fact, for the second separation, we establish that, for some problems, the ability to perform adaptive measurements in a single shallow quantum circuit, is more useful than the ability to perform polynomially many shallow quantum circuits without adaptive measurements. We also show that BPPQNC and BPPQNC are both strictly contained in BPPQNCBPP. (c) There exists a 2-message proof of quantum depth protocol. Such a protocol allows a classical verifier to efficiently certify that a prover must be performing a computation of some minimum quantum depth. Our proof of quantum depth can be instantiated using the recent proof of quantumness construction by Yamakawa and Zhandry.
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| 3. |
- Björklund, Andreas, et al.
(författare)
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The shortest even cycle problem is tractable
- 2022
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Ingår i: STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. - New York, NY, USA : ACM. - 0737-8017. - 9781450392648 ; , s. 117-130
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Konferensbidrag (refereegranskat)abstract
- Given a directed graph as input, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997) gave an efficiently testable structural characterisation of even-cycle-free directed graphs. Methodologically, our algorithm relies on the standard framework of algebraic fingerprinting and randomized polynomial identity testing over a finite field, and in fact relies on a generating polynomial implicit in a paper of Vazirani and Yannakakis (Discrete Appl. Math. 1989) that enumerates weighted cycle covers by the parity of their number of cycles as a difference of a permanent and a determinant polynomial. The need to work with the permanent-known to be #P-hard apart from a very restricted choice of coefficient rings (Valiant, Theoret. Comput. Sci. 1979)-is where our main technical contribution occurs. We design a family of finite commutative rings of characteristic 4 that simultaneously (i) give a nondegenerate representation for the generating polynomial identity via the permanent and the determinant, (ii) support efficient permanent computations by extension of Valiant's techniques, and (iii) enable emulation of finite-field arithmetic in characteristic 2. Here our work is foreshadowed by that of Björklund and Husfeldt (SIAM J. Comput. 2019), who used a considerably less efficient commutative ring design-in particular, one lacking finite-field emulation-to obtain a polynomial-time algorithm for the shortest two disjoint paths problem in undirected graphs. Building on work of Gilbert and Tarjan (Numer. Math. 1978) as well as Alon and Yuster (J. ACM 2013), we also show how ideas from the nested dissection technique for solving linear equation systems-introduced by George (SIAM J. Numer. Anal. 1973) for symmetric positive definite real matrices-leads to faster algorithm designs in our present finite-ring randomized context when we have control on the separator structure of the input graph; for example, this happens when the input has bounded genus.
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