Perfect Completeness Research Articles

A Quantum Characterization Of NP

In this article, we introduce a new complexity class called PQMA log(2). Informally, this is the class of languages for which membership has a logarithmic-size quantum proof with perfect completeness and soundness, which is polynomially close to 1 in a context where the verifier is provided a proof with two unentangled parts. We then show that PQMA log(2) = NP. For this to be possible, it is important, when defining the class, not to give too much power to the verifier. This result, when compared to the fact that QMA log = BQP, gives us new insight into the power of quantum information and the impact of entanglement.

Entangled Games Are Hard to Approximate

We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness $1-1/\,\mathrm{poly}$ even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless $\mathrm{P}=\mathrm{NP}$, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.

Inapproximability of Edge-Disjoint Paths and low congestion routing on undirected graphs

In the undirected Edge-Disjoint Paths problem with Congestion (EDPwC), we are given an undirected graph with V nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c = 1, the problem is simply referred to as the Edge-Disjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs. Our main result is that for every ɛ > 0 there exists an α > 0 such that for 1 ⩽ c ⩽ $$ \frac{{\alpha log log V }} {{\log \log \log V}} $$ , it is hard to distinguish between instances where we can route all terminal pairs on edge-disjoint paths, and instances where we can route at most a $$ {1 \mathord{\left/ {\vphantom {1 {\left( {\log V} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\log V} \right)}}^{\frac{{1 - \varepsilon }} {{c + 2}}} $$ fraction of the terminal pairs, even if we allow congestion c. This implies a $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 2}}} $$ hardness of approximation for EDPwC and an Ω(log logV/log log logV) hardness of approximation for the undirected congestion minimization problem. These results hold assuming NP ⊊ ∪ d ZPTIME( $$ 2^{\log ^{d_n } } $$ ). In the case that we do not require perfect completeness, i.e. we do not require that all terminal pairs are routed for “yes-instances”, we can obtain a slightly better inapproximability ratio of $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 1}}} $$ . Note that by setting c=1 this implies that the regular EDP problem is $$ \left( {\log V} \right)^{\frac{1} {2} - \varepsilon } $$ hard to approximate. Using standard reductions, our results extend to the node-disjoint versions of these problems as well as to the directed setting. We also show a $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 1}}} $$ inapproximability ratio for the All-or-Nothing Flow with Congestion (ANFwC) problem, a relaxation of EDPwC, in which the flow unit routed between the source-sink pairs does not have to follow a single path, so the resulting flow is not necessarily integral.

Milk and milk product consumption in Poland

The milk and dairy products belong to most precious alimentary products, being characterized with the high food value. This food products thanks to their components (calcium, protein, exogenous amino acids, vitamins) compose the perfect completion of the diet in all age groups. The special attention should be paid to fermented milk products contained probiotic bacteria. The aim of the study was the assessment of the consumption frequency of the milk and milk‐products by different population groups (children, the young people, adults, pregnant women, patients, persons doing sport). The questionnaire method was used. The study comprised 14217 persons (3–83 year old), living in Poland in the period of 1995–2008 years. The level of education was diverse. Subjects lived both in the country as and in town. The highest consumption of the milk and milk‐products was found among pre‐school children. The decrease of the milk consumption was observed with the following age groups. The low consumption of milk was discover in the group consisted of pregnant women. The most of subjects examined consumed the most often sweet fruit‐yoghurts, and the most seldom the buttermilk, smoked cheeses and blue cheeses. The stronger attention on the nutrition education in Poland should be paid in spite of several organized campaigns in media, popularizing among children and their parents the milk and its product consumption.

NP vs QMA_\log(2)

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve NP-complete problems given a "short" quantum proof; more precisely, NP\subseteq QMA_{\log}(2) where QMA_{\log}(2) denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion NP\subseteq QMA_{\log}(2) has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 1/24n^6. Moreover, Aaronson et al. have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if QMA_{\log}(2) with a constant gap contains NP. In this paper, we show that 3-SAT admits a QMA_{\log}(2) protocol with the gap 1/n^{3+\epsilon}} for every constant \epsilon>0.

Using Entanglement in Quantum Multi-Prover Interactive Proofs

The central question in quantum multi-prover interactive proof systems is whether or not entanglement shared among provers affects the verification power of the proof system. We study for the first time positive aspects of prior entanglement and show how it can be used to parallelize any multi-prover quantum interactive proof system to a one-round system with perfect completeness, soundness bounded away from one by an inverse-polynomial in the input size, and one extra prover. Alternatively, we can also parallelize to a three-turn system with the same number of provers, where the verifier only broadcasts the outcome of a coin flip. This “public-coin” property is somewhat surprising, since in the classical case public-coin multi-prover interactive proofs are equivalent to single-prover ones.

Polynomial flow-cut gaps and hardness of directed cut problems

We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given ann-vertex graphGalong withksource-sink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all source-sink pairs. The sparsest cut problem has the same input, but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of source-sink pairs that are separated by this deletion. The natural linear programming relaxation for multicut corresponds, by LP-duality, to the well-studied maximum (fractional) multicommodity flow problem, while the standard LP-relaxation for sparsest cut corresponds to maximum concurrent flow. Therefore, the integrality gap of the linear programming relaxation for multicut/sparsest cut is also theflow-cut gap: the largest gap, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem.Our first result is that the flow-cut gap between maximum multicommodity flow and minimum multicut is Ω˜(n1/7) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs. These results improve upon a long-standing lower bound of Ω(logn) for both types of flow-cut gaps. We notice that these polynomially large flow-cut gaps are in a sharp contrast to the undirected setting where both these flow-cut gaps are known to be Θ(logn). Our second result is that both directed multicut and sparsest cut are hard to approximate to within a factor of 2Ω(log1-ϵn)for any constant ϵ > 0, unless NP ⊆ ZPP. This improves upon the recent Ω(logn/log logn)-hardness result for these problems. We also show that existence of PCP's for NP with perfect completeness, polynomially small soundness, and constant number of queries would imply a polynomial factor hardness of approximation for both these problems. All our results hold for directed acyclic graphs.

Gowers Uniformity, Influence of Variables, and PCPs

We study the relation of query complexity and soundness in probabilistically checkable proofs (PCPs). We present a PCP verifier for languages that are Unique-Games-Hard and such that the verifier makes q queries, has almost perfect completeness, and has soundness error at most $2q/2^q+\varepsilon$ for arbitrarily small $\varepsilon>0$. For values of q of the form $2^t-1$, the soundness error is $(q+1)/2^q+\varepsilon$. Charikar, Makarychev, and Makarychev show that there is a constant $\beta$ such that every language that has a verifier of query complexity q and a ratio of soundness error to completeness smaller than $\beta q/2^q$ is decidable in polynomial time. Up to the value of the multiplicative constant and to the validity of the Unique Games Conjecture, our result is therefore tight. As a corollary, we show that approximating the Maximum Independent Set problem in graphs of degree $\Delta$ within a factor better than $\Delta/(\log\Delta)^\alpha$ is Unique-Games-Hard for a certain constant $\alpha>0$. Our main technical results are (i) a connection between the Gowers uniformity of a boolean function and the influence of its variables and (ii) the proof that “Gowers uniform” functions pass the “hypergraph linearity test” approximately with the same probability of a random function. The connection between Gowers uniformity and influence might have other applications.

On perfect completeness for QMA

Whether the class QMA (Quantum Merlin Arthur)\ is equal to QMA_1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult,\ by using ideas from real analysis to give a "quantum" relative to which QMA \neq QMA_1. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such "trivial" containments as BQP \subseteq ZQEXP.

Chernoff-Type Direct Product Theorems

Consider a challenge-response protocol where the probability of a correct response is at least α for a legitimate user and at most β<α for an attacker. One example is a CAPTCHA challenge, where a human should have a significantly higher chance of answering a single challenge (e.g., uncovering a distorted letter) than an attacker; another example is an argument system without perfect completeness. A natural approach to boost the gap between legitimate users and attackers is to issue many challenges and accept if the response is correct for more than a threshold fraction, for the threshold chosen between α and β. We give the first proof that parallel repetition with thresholds improves the security of such protocols. We do this with a very general result about an attacker’s ability to solve a large fraction of many independent instances of a hard problem, showing a Chernoff-like convergence of the fraction solved incorrectly to the probability of failure for a single instance.

Multidimensional perfect reconstruction filter banks: an approach of algebraic geometry

In image compression and some other applications, multidimensional filter banks are gaining their popularity due to the decrease in implementation cost. In this article, we study these filter banks from a viewpoint of algebraic geometry, where some insights are emerging. Familiar properties, such as perfect reconstruction and linear phase, appear differently when studied from this new angle. For the sake of practicability and a better understanding of problems, our focus in this work is further restricted to the filter banks that can achieve linear phase and perfect reconstruction properties. According to different symmetry nature, the filter banks are categorized into two types. Filters in a multidimensional filter bank represented by multivariate polynomials have common zeros that are different in nature from their 1D counterparts. In this work, the relation between the above-mentioned properties and various zeros is investigated. A criterion to test such filter banks is proposed and this criterion is also interpreted in resultant theory. Based on these results, as well as a proposed conjecture, a factorization of lifting scheme is presented for one type of these filter banks. For the other type of filter banks, we propose a method on perfect reconstruction completion.

Feasibility trial of adjuvant chemotherapy with paclitaxel and carboplatin after surgical resection in Japanese patients with non-small cell lung cancer: report of the Lung Oncology Group in Kyushu (LOGIK) protocol 0501

The present study was designed to determine whether adjuvant chemotherapy with paclitaxel (TXL) and carboplatin (CBDCA) after surgical resection is feasible in Japanese patients with non-small cell lung cancer (NSCLC) in a multiinstitutional trial. From August 2005 to March 2006, 34 patients received the following regimen: TXL (175 mg/m2) and CBDCA (AUC = 5) on day 1, every 3 weeks. The primary endpoint of this trial was the completion rate of four cycles. The completion rate of four cycles was 79.4% [90% confidence interval (CI), 67.5%-91.3%]. Perfect completion rate of four cycles on schedule and full doses without delay was 50% (90% CI, 34.9%-65.1%). The reasons for incomplete cycles were hypersensitivity to TXL infusion during the first cycle in 3 patients, patients refusal in 2, and anemia and cerebral infarction in 1 patient each. As a consequence of delay and/or dose reductions, the relative dose intensity of TXL and CBDCA was 86.2% and 85.8%, respectively. Doublet chemotherapy with TXL and CBDCA in the planned doses and schedule was found to be a feasible treatment for Japanese patients following surgical resection for NSCLC:

Zero-Knowledge and Correlation Intractability

The notion of correlation intractable function ensembles (CIFEs) was introduced in an attempt to capture the unpredictability property of random oracles [12]: If O is a random oracle then it is infeasible to find an inputx such that the input-output pair (x,O(x)) has some desired property. In this paper, we observe relationships between zero-knowledge protocols and CIFEs. Specifically, we show that, in the non-uniform model, the existence of CIFEs implies that 3-round auxiliary-input zero-knowledge (AIZK) AM interactive proofs exist only for BPP languages. In the uniform model, we show that 3-round AIZK AM interactive proofs with perfect completeness exist only for easy-to-approximate languages. These conditional triviality results extend to constant-round AIZK AM interactive proofs assuming the existence of CIFEs, where multi-input means that the correlation intractability is satisfied with respect to multiple input-output pairs. Also, as a corollary, we show that any construction of uniform CIFEs from uniform one-way functions proves unconditionally that constant-round AIZK AM interactive proofs with perfect completeness only for easy-to-approximate languages.

An Unconditional Study of Computational Zero Knowledge

We prove a number of general theorems about ZK, the class of problems possessing (computational) zero‐knowledge proofs. Our results are unconditional, in contrast to most previous works on ZK, which rely on the assumption that one‐way functions exist. We establish several new characterizations of ZK and use these characterizations to prove results such as the following: 1. Honest‐verifier ZK equals general ZK. 2. Public‐coin ZK equals private‐coin ZK. 3. ZK is closed under union. 4. ZK with imperfect completeness equals ZK with perfect completeness. 5. Any problem in ${\bf ZK} \cap {\bf NP}$ can be proven in computational zero knowledge by a ${\bf BPP}^{{\bf NP}}$ prover. 6. ZK with black‐box simulators equals ZK with general, non–black‐box simulators. The above equalities refer to the resulting class of problems (and do not necessarily preserve other efficiency measures such as round complexity). Our approach is to combine the conditional techniques previously used in the study of ZK with the unconditional techniques developed in the study of SZK, the class of problems possessing statistical zero‐knowledge proofs. To enable this combination, we prove that every problem in ZK can be decomposed into a problem in SZK together with a set of instances from which a one‐way function can be constructed.

Lower bounds for non-black-box zero knowledge

We show new lower bounds and impossibility results for general (possibly non-black-box) zero-knowledge proofs and arguments. Our main results are that, under reasonable complexity assumptions: 1. There does not exist a two-round zero-knowledge proof system with perfect completeness for an NP -complete language. The previous impossibility result for two-round zero knowledge, by Goldreich and Oren [J. Cryptol. 7(1) (1994) 1–32] was only for the case of auxiliary-input zero-knowledge proofs and arguments. 2. There does not exist a constant-round zero-knowledge strong proof or argument of knowledge (as defined by Goldreich [Foundations of Cryptography: Basic Tools, Cambridge University Press, Cambridge, 2001.]) for a non-trivial language. 3. There does not exist a constant-round public-coin proof system for a non-trivial language that is resettable zero knowledge. This result also extends to bounded-resettable zero knowledge, in which the number of resets is a priori bounded by a polynomial in the input length and prover-to-verifier communication. In contrast, we show that under reasonable assumptions, there does exist such a (computationally sound) argument system that is bounded-resettable zero knowledge. The complexity assumptions we use are not commonly used in cryptography. However, in all cases, we show that assumptions similar to ours are necessary for the above results. Most previously known lower bounds, such as those of Goldreich and Krawczyk [SIAM J. Comput. 25(1) (1996) 169–192], were only for black-box zero knowledge. However, a result of Barak (Proceedings of the 42nd FOCS, IEEE, 2001, pp. 106–115) shows that many (or even most) of these black-box lower bounds do not extend to the case of general zero knowledge.

Three‐query PCPs with perfect completeness over non‐Boolean domains

AbstractWe study non‐Boolean PCPs that have perfect completeness and query three positions in the proof. For the case when the proof consists of values from a domain of size d for some integer constant d ≥ 2, we construct a nonadaptive PCP with perfect completeness and soundness d−1 + d−2 + ϵ, for any constant ϵ &gt; 0, and an adaptive PCP with perfect completeness and soundness d−1 + ϵ, for any constant ϵ &gt; 0. The latter PCP can be converted into a nonadaptive PCP with perfect completeness and soundness d−1 + ϵ, for any constant ϵ &gt; 0, where four positions are read from the proof. These results match the best known constructions for the case d = 2 and our proofs also show that the particular predicates we use in our PCPs are nonapproximable beyond the random assignment threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005

Query efficient PCPs with perfect completeness

For every integer k >0, and an arbitrarily small constante> 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, non-adaptively queries 4 k +k2 bits in the proof, accepts a correct proof with probability 1, i. e., it has perfect completeness, and accepts any supposed proof of a false statement with probability at most 2 −k2 +e. In particular, the verifier achieves optimal amortized query complexity of 1 +dfor arbitrarily small constantd> 0. Such a characterization was already proved by Samorodnitsky and Trevisan (STOC 2000), but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier, we can decrease the number of query bits to 2 k +k2 , equal to the number obtained by Samorodnitsky and Trevisan. Finally we extend some of the results to PCPs over non-Boolean alphabets.

On interactive proofs with a laconic prover

We continue the investigation of interactive proofs with bounded communication, as initiated by Goldreich & Håstad (1998). Let L be a language that has an interactive proof in which the prover sends few (say b) bits to the verifier. We prove that the complement $\bar L$ has a constant-round interactive proof of complexity that depends only exponentially on b. This provides the first evidence that for NP-complete languages, we cannot expect interactive provers to be much more “laconic” than the standard NP proof. When the proof system is further restricted (e.g., when b = 1, or when we have perfect completeness), we get significantly better upper bounds on the complexity of $\bar L$.

Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups

A black-box secret sharing scheme for the threshold access structure T_t,n is one which works over any finite Abelian group G. Briefly, such a scheme differs from an ordinary linear secret sharing scheme (over, say, a given finite field) in that distribution matrix and reconstruction vectors are defined over Z and are designed independently of the group G from which the secret and the shares are sampled. This means that perfect completeness and perfect privacy are guaranteed regardless of which group G is chosen. We define the black-box secret sharing problem as the problem of devising, for an arbitrary given T_t,n, a scheme with minimal expansion factor, i.e., where the length of the full vector of shares divided by the number of players, n, is minimal. &lt;br /&gt; Such schemes are relevant for instance in the context of distributed cryptosystems based on groups with secret or hard to compute group order. A recent example is secure general multi-party computation over black-box rings. &lt;br /&gt; In 1994 Desmedt and Frankel have proposed an elegant approach to the black-box secret sharing problem based in part on polynomial interpolation over cyclotomic number fields. For arbitrary given T_t,n with 0 &lt; t &lt; n-1, the expansion factor of their scheme is O(n). This is the best previous general approach to the problem. &lt;br /&gt; Using low degree integral extensions of Z over which there exists a pair of sufficiently large Vandermonde matrices with co-prime determinants, we construct, for arbitrary given T_t,n with 0 &lt; t &lt; n-1, a black-box secret sharing scheme with expansion factor O(log n), which we show is minimal.

Approximating Satisfiable Satisfiability Problems

We study the approximability of the Maximum Satisfiability Problem (MAX SAT) and of the boolean k -ary Constraint Satisfaction Problem (MAX k CSP) restricted to satisfiable instances. For both problems we improve on the performance ratios of known algorithms for the unrestricted case. Our approximation for satisfiable MAX 3CSP instances is better than any possible approximation for the unrestricted version of the problem (unless P=NP). This result implies that the requirement of perfect completeness weakens the acceptance power of non-adaptive PCP verifiers that read 3 bits. We also present the first non-trivial results about PCP classes defined in terms of free bits that collapse to P.