Shuichi Hirahara

Limits of minimum circuit size problem as oracle

The Minimum Circuit Size Problem (MCSP) is known to be hard for statistical zero knowledge via a BPP-Turing reduction (Allender and Das, 2014), whereas establishing NP-hardness of MCSP via a polynomial-time many-one reduction is difficult (Murray and Williams, 2015) in the sense that it implies ZPP ≠ EXP, which is a major open problem in computational complexity. In this paper, we provide strong evidence that current techniques cannot establish NP-hardness of MCSP, even under polynomial-time Turing reductions or randomized reductions: Specifically, we introduce the notion of oracle-independent reduction to MCSP, which captures all the currently known reductions. We say that a reduction to MCSP is oracle-independent if the reduction can be generalized to a reduction to MCSPA for any oracle A, where MCSPA denotes an oracle version of MCSP. We prove that no language outside P is reducible to MCSP via an oracle-independent polynomial-time Turing reduction. We also show that the class of languages reducible to MCSP via an oracle-independent randomized reduction that makes at most one query is contained in AM ∩ coAM. Thus, NP-hardness of MCSP cannot be established via such oracle-independent reductions unless the polynomial hierarchy collapses. We also extend the previous results to the case of more general reductions: We prove that establishing NP-hardness of MCSP via a polynomial-time nonadaptive reduction implies ZPP ≠ EXP, and that establishing NP-hardness of approximating circuit complexity via a polynomial-time Turing reduction also implies ZPP ≠ EXP. Along the way, we prove that approximating Levins Kolmogorov complexity is provably not EXP-hard under polynomial-time Turing reductions, which is of independent interest.

On the Average-Case Complexity of MCSP and Its Variants

We prove various results on the complexity of MCSP (Minimum Circuit Size Problem) and the related MKTP (Minimum Kolmogorov Time-Bounded Complexity Problem): * We observe that under standard cryptographic assumptions, MCSP has a pseudorandom self-reduction. This is a new notion we define by relaxing the notion of a random self-reduction to allow queries to be pseudorandom rather than uniformly random. As a consequence we derive a weak form of a worst-case to average-case reduction for (a promise version of) MCSP. Our result also distinguishes MCSP from natural NP-complete problems, which are not known to have worst-case to average-case reductions. Indeed, it is known that strong forms of worst-case to average-case reductions for NP-complete problems collapse the Polynomial Hierarchy. * We prove the first non-trivial formula size lower bounds for MCSP by showing that MCSP requires nearly quadratic-size De Morgan formulas. * We show average-case superpolynomial size lower bounds for MKTP against AC0[p] for any prime p. * We show the hardness of MKTP on average under assumptions that have been used in much recent work, such as Feiges assumptions, Alekhnovichs assumption and the Planted Clique conjecture. In addition, MCSP is hard under Alekhnovichs assumption. Using a version of Feiges assumption against co-nondeterministic algorithms that has been conjectured recently, we provide evidence for the first time that MKTP is not in coNP. Our results suggest that it might worthwhile to focus on the average-case hardness of MKTP and MCSP when approaching the question of whether these problems are NP-hard.

Virtual machine placement for minimizing connection cost in data center networks

Virtualization is a key technology for the efficient operation of massive data centers. To minimize the communication costs among virtual machines (VMs) in a data center network, we formulate an optimization problem for finding efficient VM placements. In this problem, a set of requests is received from customers, where each request is defined as the required number of VMs. The problem seeks to determine those physical machines in the network that host the requested VMs under a capacity constraint such that the number of VMs placed on each physical machine does not exceed that of the available slots. To minimize the load of the networks, for each request, we consider the connection cost of the VM placements, which is defined as the minimum length of networks connecting all physical host machines and the root node. The objective in the problem is to minimize the total connection costs. We present an approximation algorithm for this optimization problem.

On Characterizations of Randomized Computation Using Plain Kolmogorov Complexity

Allender, Friedman, and Gasarch recently proved an upper bound of pspace for the class DTTR K of decidable languages that are polynomial-time truth-table reducible to the set of prefix-free Kolmogorov-random strings regardless of the universal machine used in the definition of Kolmogorov complexity. It is conjectured that DTTR K in fact lies closer to its lower bound BPP established earlier by Buhrman, Fortnow, Koucký, and Loff. It is also conjectured that we have similar bounds for the analogous class DTTR C defined by plain Kolmogorov randomness. In this paper, we provide further evidence for these conjectures. First, we show that the time-bounded analogue of DTTR C sits between BPP and pspace ∩ P/poly. Next, we show that the class DTTR C, α obtained from DTTR C by imposing a restriction on the reduction lies between BPP and pspace. Finally, we show that the class P/R\(^{=log}_{c}\) obtained by further restricting the reduction to ask queries of logarithmic length lies between BPP and \(\Sigma^{p}_{2} \cap\) P/poly.

NP-hardness of minimum circuit size problem for OR-AND-MOD circuits

The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have demonstrated the central role of this problem and its variations in diverse areas such as cryptography, derandomization, proof complexity, learning theory, and circuit lower bounds. The NP-hardness of computing the minimum numbers of terms in a DNF formula consistent with a given truth table was proved by W. Masek [31] in 1979. In this work, we make the first progress in showing NP-hardness for more expressive classes of circuits, and establish an analogous result for the MCSP problem for depth-3 circuits of the form OR-AND-MOD2. Our techniques extend to an NP-hardness result for MODm gates at the bottom layer under inputs from (Z/mZ)n.

Identifying an honest EXP NP oracle among many

We provide a general framework to remove short advice by formulating the following computational task for a function f : given two oracles at least one of which is honest (i.e. correctly computes f on all inputs) as well as an input, the task is to compute f on the input with the help of the oracles by a probabilistic polynomial-time machine, which we shall call a selector. We characterize the languages for which short advice can be removed by the notion of selector: a paddable language has a selector if and only if short advice of a probabilistic machine that accepts the language can be removed under any relativized world. Previously, instance checkers have served as a useful tool to remove short advice of probabilistic computation. We indicate that existence of instance checkers is a property stronger than that of removing short advice: although no instance checker for EXPNP-complete languages exists unless EXPNP = NEXP, we prove that there exists a selector for any EXPNP-complete language, by building on the proof of MIP = NEXP by Babai, Fortnow, and Lund (1991).