Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows $x_{n/2+1},\dots,x_n$, and both parties know $\varphi$. The goal is to have Alice and Bob jointly write a PCP that $x$ satisfies $\varphi$, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of $x$. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of $2^{(\log n)^{1-o(1)}}$; only $(1+o(1))$-factor lower bounds (under SETH) were known before

The price of certainty: "waterslide curves" and the gap to capacity

The classical problem of reliable point-to-point digital communication is to achieve a low probability of error while keeping the rate high and the total power consumption small. Traditional information-theoretic analysis uses `waterfall' curves to convey the revolutionary idea that unboundedly low probabilities of bit-error are attainable using only finite transmit power. However, practitioners have long observed that the decoder complexity, and hence the total power consumption, goes up when attempting to use sophisticated codes that operate close to the waterfall curve. This paper gives an explicit model for power consumption at an idealized decoder that allows for extreme parallelism in implementation. The decoder architecture is in the spirit of message passing and iterative decoding for sparse-graph codes. Generalized sphere-packing arguments are used to derive lower bounds on the decoding power needed for any possible code given only the gap from the Shannon limit and the desired probability of error. As the gap goes to zero, the energy per bit spent in decoding is shown to go to infinity. This suggests that to optimize total power, the transmitter should operate at a power that is strictly above the minimum demanded by the Shannon capacity. The lower bound is plotted to show an unavoidable tradeoff between the average bit-error probability and the total power used in transmission and decoding. In the spirit of conventional waterfall curves, we call these `waterslide' curves.Comment: 37 pages, 13 figures. Submitted to IEEE Transactions on Information Theory. This version corrects a subtle bug in the proofs of the original submission and improves the bounds significantl

Identification of a reversible quantum gate: assessing the resources

We assess the resources needed to identify a reversible quantum gate among a finite set of alternatives, including in our analysis both deterministic and probabilistic strategies. Among the probabilistic strategies we consider unambiguous gate discrimination, where errors are not tolerated but inconclusive outcomes are allowed, and we prove that parallel strategies are sufficient to unambiguously identify the unknown gate with minimum number of queries. This result is used to provide upper and lower bounds on the query complexity and on the minimum ancilla dimension. In addition, we introduce the notion of generalized t-designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized t-design we give an explicit expression for the maximum probability of correct gate identification and we prove that there is no gap between the performances of deterministic strategies an those of probabilistic strategies. Hence, evaluating of the query complexity of perfect deterministic discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous discrimination. Finally, we consider discrimination strategies where the use of ancillas is forbidden, providing upper bounds on the number of additional queries needed to make up for the lack of entanglement with the ancillas.Comment: 24 + 8 pages, published versio

Memory lower bounds for deterministic self-stabilization

In the context of self-stabilization, a \emph{silent} algorithm guarantees that the register of every node does not change once the algorithm has stabilized. At the end of the 90's, Dolev et al. [Acta Inf. '99] showed that, for finding the centers of a graph, for electing a leader, or for constructing a spanning tree, every silent algorithm must use a memory of $\Omega(\log n)$ bits per register in $n$-node networks. Similarly, Korman et al. [Dist. Comp. '07] proved, using the notion of proof-labeling-scheme, that, for constructing a minimum-weight spanning trees (MST), every silent algorithm must use a memory of $\Omega(\log^2n)$ bits per register. It follows that requiring the algorithm to be silent has a cost in terms of memory space, while, in the context of self-stabilization, where every node constantly checks the states of its neighbors, the silence property can be of limited practical interest. In fact, it is known that relaxing this requirement results in algorithms with smaller space-complexity. In this paper, we are aiming at measuring how much gain in terms of memory can be expected by using arbitrary self-stabilizing algorithms, not necessarily silent. To our knowledge, the only known lower bound on the memory requirement for general algorithms, also established at the end of the 90's, is due to Beauquier et al.~[PODC '99] who proved that registers of constant size are not sufficient for leader election algorithms. We improve this result by establishing a tight lower bound of $\Theta(\log \Delta+\log \log n)$ bits per register for self-stabilizing algorithms solving $(\Delta+1)$-coloring or constructing a spanning tree in networks of maximum degree~$\Delta$. The lower bound $\Omega(\log \log n)$ bits per register also holds for leader election

From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity

The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527