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1. Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization

   https://doi.org/10.1007/s10107-022-01875-8  

   Zhang, Shixuan ; Sun, Xu Andy ( August 2022 , Mathematical Programming)

   In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization withnon-Lipschitzianvalue functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a$$(T+1)$$$(T+1)$-stage stochastic MINLP satisfyingL-exact Lipschitz regularization withd-dimensional state spaces, to obtain an$$\varepsilon $$$\epsilon$-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by$${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$$O\left({\left(\frac{2LT}{\epsilon }\right)}^d\right)$, and is lower bounded by$${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$$O\left({\left(\frac{\mathrm{LT}}{4\epsilon }\right)}^d\right)$for the general case or by$${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$$O\left({\left(\frac{\mathrm{LT}}{8\epsilon }\right)}^{d/2-1}\right)$for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity dependspolynomiallyon the number of stages. We further show that the iteration complexity dependslinearlyonT, if all the state spaces are finite sets, or if we seek a$$(T\varepsilon )$$$\left(T\epsilon \right)$-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale withT. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.

    

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2. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

   https://doi.org/10.1007/s00224-022-10106-8  

   Vyas, Nikhil ; Williams, R_Ryan ( November 2022 , Theory of Computing Systems)

   We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[n^{{\left(\log n\right)}^{O\left(1\right)}}\right]$and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathcal { C}$$C$, by showing that$\mathcal { C}$$C$admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class${\mathcal C}$$C$. Say that a symmetric Boolean functionf(x₁,…,x_n) issparseif it outputs 1 onO(1) values of${\sum }_{i} x_{i}$${\sum }_i{x}_i$. We show that for every sparsef, and for all “typical”$\mathcal { C}$$C$, faster#SAT algorithms for$\mathcal { C}$$C$circuits imply lower bounds against the circuit class$f \circ \mathcal { C}$$f\circ C$, which may bestrongerthan$\mathcal { C}$$C$itself. In particular:

   #SAT algorithms forn^k-size$\mathcal { C}$$C$-circuits running in 2ⁿ/n^ktime (for allk) implyNEXPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

   #SAT algorithms for$2^{n^{{\varepsilon }}}$$2^{n^{\epsilon }}$-size$\mathcal { C}$$C$-circuits running in$2^{n-n^{{\varepsilon }}}$$2^{n-n^{\epsilon }}$time (for someε> 0) implyQuasi-NPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

   Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJ∘ACC⁰∘THRcircuits of polynomial size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC⁰[Williams JACM’14] andACC⁰∘THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

    

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3. Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength

   https://doi.org/10.1007/s00454-023-00523-y  

   Tao, Terence ( July 2023 , Discrete & Computational Geometry)

   A well-known open problem of Meir and Moser asks if the squares of sidelength 1/nfor$$n\ge 2$$$n\ge 2$can be packed perfectly into a rectangle of area$$\sum _{n=2}^\infty n^{-2}=\pi ^2/6-1$$${\sum }_{n=2}^{\infty }{n}^{-2}={\pi }^2/6-1$. In this paper we show that for any$$1/2$1/2<t<1$, and any$$n_0$$$n_0$that is sufficiently large depending on t, the squares of sidelength$$n^{-t}$$$n^{-t}$for$$n\ge n_0$$$n\ge n_0$can be packed perfectly into a square of area$$\sum _{n=n_0}^\infty n^{-2t}$$${\sum }_{n=n_0}^{\infty }{n}^{-2t}$. This was previously known (if one packs a rectangle instead of a square) for$$1/2$1/2<t\le 2/3$(in which case one can take$$n_0=1$$$n_0=1$).

    

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4. Efficient Computation of Sequence Mappability

   https://doi.org/10.1007/s00453-022-00934-y  

   Charalampopoulos, Panagiotis ; Iliopoulos, Costas S. ; Kociumaka, Tomasz ; Pissis, Solon P. ; Radoszewski, Jakub ; Straszyński, Juliusz ( February 2022 , Algorithmica)

   Sequence mappability is an important task in genome resequencing. In the (k, m)-mappability problem, for a given sequenceTof lengthn, the goal is to compute a table whoseith entry is the number of indices$$j \ne i$$$j\ne i$such that the length-msubstrings ofTstarting at positionsiandjhave at mostkmismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of$$k=1$$$k=1$. We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for$$k=O(1)$$$k=O\left(1\right)$, works in$$O(n)$$$O\left(n\right)$space and, with high probability, in$$O(n \cdot \min \{m^k,\log ^k n\})$$$O(n·min\{m^k,{log}^{k}n\})$time. Our algorithm requires a careful adaptation of thek-errata trees of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. Our technique can also be applied to solve the all-pairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop$$O(n^2)$$$O\left(n^2\right)$-time algorithms to computeall(k, m)-mappability tables for a fixedmand all$$k\in \{0,\ldots ,m\}$$$k\in \{0,\dots ,m\}$or a fixedkand all$$m\in \{k,\ldots ,n\}$$$m\in \{k,\dots ,n\}$. Finally, we show that, for$$k,m = \Theta (\log n)$$$k,m=\Theta (logn)$, the (k, m)-mappability problem cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails. This is an improved and extended version of a paper presented at SPIRE 2018.

    

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5. Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates

   https://doi.org/10.1007/s00220-023-04675-z  

   Harrow, Aram W. ; Mehraban, Saeed ( May 2023 , Communications in Mathematical Physics)

   We prove that$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$$\text{poly}\left(t\right)·n^{1/D}$-depth local random quantum circuits with two qudit nearest-neighbor gates on aD-dimensional lattice withnqudits are approximatet-designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was$${{\,\textrm{poly}\,}}(t)\cdot n$$$\text{poly}\left(t\right)·n$due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$$D=1$$$D=1$. We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($${{\,\mathrm{\textsf{PH}}\,}}$$$\mathrm{PH}$) is infinite and that certain counting problems are$$\#{\textsf{P}}$$$\#P$-hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that$$O(\sqrt{n})$$$O\left(\sqrt{n}\right)$depth suffices for anti-concentration. The proof is based on a previous construction oft-designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size$$O(n\ln ^2 n)$$$O\left(n{ln}^{2}n\right)$corresponding to depth$$O(\ln ^3 n)$$$O\left({ln}^{3}n\right)$. We also show a lower bound of$$\Omega (n \ln n)$$$\Omega (nlnn)$for the size of such circuit in this case. We also prove that anti-concentration is possible in depth$$O(\ln n \ln \ln n)$$$O(lnnlnlnn)$(size$$O(n \ln n \ln \ln n)$$$O(nlnnlnlnn)$) using a different model.

    

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