Computer Science

In this paper, I take a step toward answering the following question: for m different small circuits that compute m orthogonal n qubit states, is there a small circuit that will map m computational basis states to these m states without any input leaving any auxiliary bits changed. While this may seem simple, the constraint that auxiliary bits always be returned to 0 on any input (even ones besides the m we care about) led me to use sophisticated techniques. I give an approximation of such a unitary in the m = 2 case that has size polynomial in the approximation error, and the number of qubits n.

We prove the first, even super-polynomial, lower bounds on the size of tropical (min,+) and (max,+) circuits approximating given optimization problems. Many classical dynamic programming (DP) algorithms for optimization problems are pure in that they only use the basic min, max, + operations in their recursion equations. Tropical circuits constitute a rigorous mathematical model for this class of algorithms. An algorithmic consequence of our lower bounds for tropical circuits is that the approximation powers of pure DP algorithms and greedy algorithms are incomparable. That pure DP algorithms can hardly beat greedy in approximation, is long known. New in this consequence is that also the converse holds.

In this paper we show that the approximating the Kolmogorov complexity of a set of numbers is equivalent to having common information with the halting sequence. The more precise the approximations are, and the greater the number of approximations, the more information is shared with the halting sequence. An encoding of the 2^N unique numbers and their Kolmogorov complexities contains at least >N mutual information with the halting sequence. We also provide a generalization of the "Sets have Simple Members" theorem to conditional complexity.

When can n given numbers be combined using arithmetic operators from a given subset of {+,−,×,÷} to obtain a given target number? We study three variations of this problem of Arithmetic Expression Construction: when the expression (1) is unconstrained; (2) has a specified pattern of parentheses and operators (and only the numbers need to be assigned to blanks); or (3) must match a specified ordering of the numbers (but the operators and parenthesization are free). For each of these variants, and many of the subsets of {+,−,×,÷} , we prove the problem NP-complete, sometimes in the weak sense and sometimes in the strong sense. Most of these proofs make use of a "rational function framework" which proves equivalence of these problems for values in rational functions with values in positive integers.

Transportation Network Companies employ dynamic pricing methods at periods of peak travel to incentivise driver participation and balance supply and demand for rides. Surge pricing multipliers are commonly used and are applied following demand and estimates of customer and driver trip valuations. Combinatorial double auctions have been identified as a suitable alternative, as they can achieve maximum social welfare in the allocation by relying on customers and drivers stating their valuations. A shortcoming of current models, however, is that they fail to account for the effects of trip detours that take place in shared trips and their impact on the accuracy of pricing estimates. To resolve this, we formulate a new shared-ride assignment and pricing algorithm using combinatorial double auctions. We demonstrate that this model is reduced to a maximum weighted independent set model, which is known to be APX-hard. A fast local search heuristic is also presented, which is capable of producing results that lie within 10% of the exact approach for practical implementations. Our proposed algorithm could be used as a fast and reliable assignment and pricing mechanism of ride-sharing requests to vehicles during peak travel times.

We show that Cutting Planes (CP) proofs are hard to find: Given an unsatisfiable formula F , 1) It is NP-hard to find a CP refutation of F in time polynomial in the length of the shortest such refutation; and 2)unless Gap-Hitting-Set admits a nontrivial algorithm, one cannot find a tree-like CP refutation of F in time polynomial in the length of the shortest such refutation. The first result extends the recent breakthrough of Atserias and Müller (FOCS 2019) that established an analogous result for Resolution. Our proofs rely on two new lifting theorems: (1) Dag-like lifting for gadgets with many output bits. (2) Tree-like lifting that simulates an r -round protocol with gadgets of query complexity O(logr) independent of input length.

This paper develops several average-case reduction techniques to show new hardness results for three central high-dimensional statistics problems, implying a statistical-computational gap induced by robustness, a detection-recovery gap and a universality principle for these gaps. A main feature of our approach is to map to these problems via a common intermediate problem that we introduce, which we call Imbalanced Sparse Gaussian Mixtures. We assume the planted clique conjecture for a version of the planted clique problem where the position of the planted clique is mildly constrained, and from this obtain the following computational lower bounds: (1) a k -to- k 2 statistical-computational gap for robust sparse mean estimation, providing the first average-case evidence for a conjecture of Li (2017) and Balakrishnan et al. (2017); (2) a tight lower bound for semirandom planted dense subgraph, which shows that a semirandom adversary shifts the detection threshold in planted dense subgraph to the conjectured recovery threshold; and (3) a universality principle for k -to- k 2 gaps in a broad class of sparse mixture problems that includes many natural formulations such as the spiked covariance model. Our main approach is to introduce several average-case techniques to produce structured and Gaussianized versions of an input graph problem, and then to rotate these high-dimensional Gaussians by matrices carefully constructed from hyperplanes in F t r . For our universality result, we introduce a new method to perform an algorithmic change of measure tailored to sparse mixtures. We also provide evidence that the mild promise in our variant of planted clique does not change the complexity of the problem.

The paper explores the correspondence between balanced incomplete block designs (BIBD) and certain linear CNF formulas by identifying the points of a block design with the clauses of the Boolean formula and blocks with Boolean variables. Parallel classes in BIBDs correspond to XSAT solutions in the corresponding formula. This correspondence allows for transfers of results from one field to the other. As a new result we deduce from known satisfiability theorems that the problem of finding a parallel class in a partially balanced incomplete block design is NP-complete.

We study a class of optimization problems including matrix scaling, matrix balancing, multidimensional array scaling, operator scaling, and tensor scaling that arise frequently in theory and in practice. Some of these problems, such as matrix and array scaling, are convex in the Euclidean sense, but others such as operator scaling and tensor scaling are geodesically convex on a different Riemannian manifold. Trust region methods, which include box-constrained Newton's method, are known to produce high precision solutions very quickly for matrix scaling and matrix balancing (Cohen et. al., FOCS 2017, Allen-Zhu et. al. FOCS 2017), and result in polynomial time algorithms for some geodesically convex problems like operator scaling (Garg et. al. STOC 2018, Bürgisser et. al. FOCS 2019). One is led to ask whether these guarantees also hold for multidimensional array scaling and tensor scaling. We show that this is not the case by exhibiting instances with exponential diameter bound: we construct polynomial-size instances of 3-dimensional array scaling and 3-tensor scaling whose approximate solutions all have doubly exponential condition number. Moreover, we study convex-geometric notions of complexity known as margin and gap, which are used to bound the running times of all existing optimization algorithms for such problems. We show that margin and gap are exponentially small for several problems including array scaling, tensor scaling and polynomial scaling. Our results suggest that it is impossible to prove polynomial running time bounds for tensor scaling based on diameter bounds alone. Therefore, our work motivates the search for analogues of more sophisticated algorithms, such as interior point methods, for geodesically convex optimization that do not rely on polynomial diameter bounds.

We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give optimal algorithms. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously known barriers, both in the numerical sense, as well as in its generality. We prove our result using the asymptotic spectrum of tensors. More precisely, we crucially make use of two families of real tensor parameters with special algebraic properties: the quantum functionals and the support functionals. In particular, we prove that any lower bound on the dual exponent of matrix multiplication α via the big Coppersmith-Winograd tensors cannot exceed 0.625.