We present a general method for converting any family of unsatisfiable CNF formulas that is hard for one of the simplest proof systems, tree resolution, into formulas that require large rank in any proof system that manipulates polynomials or polynomial threshold functions of degree at most k (known as Th(k) proofs). Such systems include Lovasz-Schrijver and Cutting Planes proof systems as well as their high degree analogues.
These are based on analyzing two new proof systems, denoted by T^cc(k) and R^cc(k). The proof lines of T^cc(k) are arbitrary Boolean functions, each of which can be evaluated by an efficient k-party randomized communication protocol. They include Th{k-1} proofs as a special case. R^cc(k) proofs are stronger and only require that each inference be locally checkable by an efficient k-party randomized communication protocol.
Our main results are the following:
(1) When k is O(loglogn), for any unsatisfiable CNF formula F requiring resolution rank r, there is a related CNF formula G=Lift_k(F) requiring refutation rank r^Omega(1/k) log^O(1) n in all R^cc(k) systems.
(2) There are strict hierarchies for T^cc(k) and R^cc(k) systems with respect to k when k is O(loglogn in that there are unsatisfiable CNF formulas requiring large rank R^cc(k) refutations but having log^O(1) n rank Th(k) refutations.
(3) When k is O(loglogn) there are 2^(n^Omega(1/k)) lower bounds on the size of tree-like T^cc(k) refutations for large classes of lifted CNF formulas.
(4) A general method for producing integrality gaps for low rank R^cc(2) inference (and hence Cutting Planes and Th(1) inference) based on related gaps for low rank resolution. These gaps are optimal for MAX-2t-SAT.
General Hardness Amplification of Predicates and Puzzles
我们给出了新的证据，有效的采样谓词和弱验证的难题，推广到新的设置的硬度放大。更具体地说，在本文的第一部分中，我们给出了姚的XOR引理的一个新的证明，该引理也适用于密码学环境中的相关定理。我们的证明似乎比以前的简单，但立即推广到类似的精神陈述，如提取引理用于从单向函数获得伪随机生成器[Hastad，Impagliazzo，Levin，Luby，SIAM J. on Comp. 1999]。在本文的第二部分中，我们给出了弱可验证谜题的硬度放大的一个新的证明，它比以前的证明更一般，因为它给出了甚至对应用于底层谜题的检查电路的任意单调函数的右界。我们的证明是适用于许多设置的交互式密码协议，因为它们满足一个属性，我们称之为“非倒带”。特别是，我们表明，任何弱密码协议的安全性是由不可预测的单比特可以加强与自然的信息理论协议。作为一个例子，我们展示了这些定理如何解决[Halevi和Rabin，TCC 2008]中关于比特承诺的主要公开问题。
$\newcommand{\NP}{\mathsf{NP}}\newcommand{\GapSVP}{\textrm{GapSVP}}$We give a simple proof that the (approximate, decisional) Shortest Vector Problem is $\NP$-hard under a randomized reduction. Specifically, we show that for any $p \geq 1$ and any constant $γ< 2^{1/p}$, the $γ$-approximate problem in the $\ell_p$ norm ($γ$-$\GapSVP_p$) is not in $\mathsf{RP}$ unless $\NP \subseteq \mathsf{RP}$. Our proof follows an approach pioneered by Ajtai (STOC 1998), and strengthened by Micciancio (FOCS 1998 and SICOMP 2000), for showing hardness of $γ$-$\GapSVP_p$ using locally dense lattices. We constru…
A meta-complexity assumption, Feasible Chaitin Incompleteness (FCI), asserts the hardness of ruling out length $t$ proofs that string $x$ is Kolmogorov random (e.g. $x{\in}R$), by analogy to Chaitin's result that proving $x{\in}R$ is typically impossible. By assertion, efficiently ruling out short proofs requires, impossibly, ruling out any proof. FCI has strong implications: (i) randomly chosen $x$ typically yields tautologies hard with high probability for any given proof system, densely witnessing its nonoptimality; (ii) average-case impossibility of proving $x{\in}R$ implies average-case h…
A Note on NP-Hardness of Preemptive Mean Flow-Time Scheduling for Parallel Machines
Baptiste，Brucker，Chrobak，Dürr，Kravchenko和Sourd在论文“The complex of mean flow time scheduling problems with release times”中，证明了调度问题$P|pmtn,r_j|\sum C_j$的强NP-困难性，即以最小化平均流时间为目标的多处理器抢占式调度。指出了他们证明中的一个严重错误，并给出了该问题强NP-困难性的一个新证明。
Given a cryptographic group action, we show that the Group Action Inverse Problem (GAIP) and other related problems cannot be NP-hard unless the Polynomial Hierarchy collapses. We show this via random self-reductions and the design of interactive proofs. Since cryptographic group actions are the building block of many security protocols, this result serves both as an upper bound on the worst-case complexity of some cryptographic assumptions and as proof that the hardness in the worst and in the average case coincide. We also point out the link with Graph Isomorphism and other related NP interm…
We consolidate two widely believed conjectures about tautologies -- no optimal proof system exists, and most require superpolynomial size proofs in any system -- into a $p$-isomorphism-invariant condition satisfied by all paddable $\textbf{coNP}$-complete languages or none. The condition is: for any Turing machine (TM) $M$ accepting the language, $\textbf{P}$-uniform input families requiring superpolynomial time by $M$ exist (equivalent to the first conjecture) and appear with positive upper density in an enumeration of input families (implies the second). In that case, no such language is eas…
In this paper, we seek a natural problem and a natural distribution of instances such that any $O(n^{c-ε})$-time algorithm fails to solve most instances drawn from the distribution, while the problem admits an $n^{c+o(1)}$-time algorithm that correctly solves all instances. Specifically, we consider the $K_{a,b}$ counting problem in a random bipartite graph, where $K_{a,b}$ is a complete bipartite graph for constants $a$ and $b$. We proved that the $K_{a,b}$ counting problem admits an $n^{a+o(1)}$-time algorithm if $a\geq 8$, while any $n^{a-ε}$-time algorithm fails to solve it even on random …
The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Nu…
NP-hardness of the cluster minimization problem revisited
重新讨论了“簇极小化问题”的计算复杂性[L. T. Wille和J. Vennik，J.Phys.A18，L419（1985）]。有人认为，原始的NP-硬度证明不适用于物理意义上的成对势，例如那些取决于粒子之间几何距离的势。原问题的几何模拟制定，并提供了一个新的证明，这种潜力的多项式时间转换从单位盘图的独立集问题。这种提法的局限性，并承担更直接的后果集群的数值研究提出了新的子问题。
We provide several advances to the understanding of the class of Quantum Merlin-Arthur proof systems (QMA), the quantum analogue of NP. Our central contribution is proving a longstanding conjecture that the Consistency of Local Density Matrices (CLDM) problem is QMA-hard under Karp reductions. The input of CLDM consists of local reduced density matrices on sets of at most k qubits, and the problem asks if there is an n-qubit global quantum state that is consistent with all of the k-qubit local density matrices. The containment of this problem in QMA and the QMA-hardness under Turing reductions…
Gaussian boson sampling is a model of photonic quantum computing that has attracted attention as a platform for building quantum devices capable of performing tasks that are out of reach for classical devices. There is therefore significant interest, from the perspective of computational complexity theory, in solidifying the mathematical foundation for the hardness of simulating these devices. We show that, under the standard Anti-Concentration and Permanent-of-Gaussians conjectures, there is no efficient classical algorithm to sample from ideal Gaussian boson sampling distributions (even appr…
In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with $n$ nodes and maximum degree $Δ$ on which $Ω(\min\{\sqrt{\log n/\log \log n},\log Δ/\log \log Δ\})$ (expected) communication rounds are required to obtain polylogarithmic approximations to a minimum vertex cover, minimum dominating set, or maximum matching. Via reduction, this hardness extends to symmetry breaking tasks like finding maximal independent sets or maximal matchings. …
