Tight Trade-off Research Articles

Special Soundness Revisited

We generalize and abstract the problem of extracting a witness from a prover of a special sound protocol into a combinatorial problem induced by a sequence of matroids and a predicate, and present a parametrized algorithm for solving this problem. The parametrization provides a tight tradeoff between the running time and the extraction error of the algorithm, which allows optimizing the parameters to minimize: the soundness error for interactive proofs, or the extraction time for proofs of knowledge. In contrast to previous work we bound the distribution of the running time and not only the expected running time. Tail bounds give a tighter analysis when applied recursively and a concentrated running time.

On Optimal Tradeoffs between EFX and Nash Welfare

A major problem in fair division is how to allocate a set of indivisible resources among agents fairly and efficiently. The goal of this work is to characterize the tradeoffs between two well-studied measures of fairness and efficiency --- envy freeness up to any item (EFX) for fairness, and Nash welfare for efficiency --- by saying, for given constants α and β, whether there exists an α-EFX allocation that guarantees a β-fraction of the maximum Nash welfare (β-MNW). For additive valuations, we show that for any α ∈ [0,1], there exists a partial allocation that is α-EFX and 1/(α+1)-MNW. This tradeoff turns out to be tight (for every α) as demonstrated by an impossibility result that we give. We also show that for α ∈ [0, φ-1 ≃ 0.618] these partial allocations can be turned into complete allocations where all items are assigned. Furthermore, for any α ∈ [0, 1/2], we show that the tight tradeoff of α-EFX and 1/(α+1)-MNW with complete allocations holds for the more general setting of subadditive valuations. Our results improve upon the current state of the art, for both additive and subadditive valuations, and match the best-known approximations of EFX under complete allocations, regardless of Nash welfare guarantees. Notably, our constructions for additive valuations also provide EF1 and constant approximations for maximin share guarantees.

A constant charging-current relaxation oscillator with a duty-cycled main comparator and an adaptive auxiliary comparator

A low-power constant charging-current relaxation oscillator that outputs a frequency-tunable sawtooth signal and a pulse signal is proposed and implemented in 0.5-μm CMOS process. At the architecture level, two comparators with 10 times difference in power consumption are proposed to break up the tight tradeoff between power consumption and oscillation performance. Most of the time, only the low-power auxiliary comparator works to save power. The high-power main comparator is controlled by the auxiliary comparator output and only works when a ramping voltage is close to a reference voltage. At the circuit level, an adaptive current control technique that makes current adversely proportional to input voltage differences is used to further reduce the power consumed in the low-power comparator. The measured clock phase noise at the offset frequency of 100 kHz is −115.07 dBc/Hz at the output frequency of 1 MHz. The measured output frequency variations are 124 ppm/°C over temperatures from −55 °C to 125 °C, and −0.13%–0.18% over power supplies from 3 V to 5.5 V. The frequency range from 200 kHz to 1350 kHz is achieved. The power consumption at the 1 MHz output frequency is 9.5 μW, which is comparable with state-of-the-art designs in the literature.

Do mechanisms matter? Comparing cancer treatment strategies across mathematical models and outcome objectives.

When eradication is impossible, cancer treatment aims to delay the emergence of resistance while minimizing cancer burden and treatment. Adaptive therapies may achieve these aims, with success based on three assumptions: resistance is costly, sensitive cells compete with resistant cells, and therapy reduces the population of sensitive cells. We use a range of mathematical models and treatment strategies to investigate the tradeoff between controlling cell populations and delaying the emergence of resistance. These models extend game theoretic and competition models with four additional components: 1) an Allee effect where cell populations grow more slowly at low population sizes, 2) healthy cells that compete with cancer cells, 3) immune cells that suppress cancer cells, and 4) resource competition for a growth factor like androgen. In comparing maximum tolerable dose, intermittent treatment, and adaptive therapy strategies, no therapeutic choice robustly breaks the three-way tradeoff among the three therapeutic aims. Almost all models show a tight tradeoff between time to emergence of resistant cells and cancer cell burden, with intermittent and adaptive therapies following identical curves. For most models, some adaptive therapies delay overall tumor growth more than intermittent therapies, but at the cost of higher cell populations. The Allee effect breaks these relationships, with some adaptive therapies performing poorly due to their failure to treat sufficiently to drive populations below the threshold. When eradication is impossible, no treatment can simultaneously delay emergence of resistance, limit total cancer cell numbers, and minimize treatment. Simple mathematical models can play a role in designing the next generation of therapies that balance these competing objectives.

Joint Subcarrier and Power Allocation in NOMA: Optimal and Approximate Algorithms

Non-orthogonal multiple access (NOMA) is a promising technology to increase the spectral efficiency and enable massive connectivity in 5G and future wireless networks. In contrast to orthogonal schemes, such as OFDMA, NOMA multiplexes several users on the same frequency and time resource. Joint subcarrier and power allocation problems (JSPA) in NOMA are NP-hard to solve in general. In this family of problems, we consider the weighted sum-rate (WSR) objective function as it can achieve various tradeoffs between sum-rate performance and user fairness. Because of JSPA's intractability, a common approach in the literature is to solve separately the power control and subcarrier allocation (also known as user selection) problems, therefore achieving sub-optimal result. In this work, we first improve the computational complexity of existing single-carrier power control and user selection schemes. These improved procedures are then used as basic building blocks to design new algorithms, namely Opt-JSPA, $\varepsilon$-JSPA and Grad-JSPA. Opt-JSPA computes an optimal solution with lower complexity than current optimal schemes in the literature. It can be used as a benchmark for optimal WSR performance in simulations. However, its pseudo-polynomial time complexity remains impractical for real-world systems with low latency requirements. To further reduce the complexity, we propose a fully polynomial-time approximation scheme called $\varepsilon$-JSPA. Since, no approximation has been studied in the literature, $\varepsilon$-JSPA stands out by allowing to control a tight trade-off between performance guarantee and complexity. Finally, Grad-JSPA is a heuristic based on gradient descent. Numerical results show that it achieves near-optimal WSR with much lower complexity than existing optimal methods.

Trade-offs in multiparty Bell-inequality violations in qubit networks

Two overlapping bipartite binary input Bell inequalities cannot be simultaneously violated as this would contradict the usual no-signalling principle. This property is known as monogamy of Bell inequality violations and generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. It turns out that multipartite Bell inequalities admit weaker forms of monogamies that allow for violations of a few inequalities at once. Here we systematically study monogamy relations between correlation Bell inequalities both within quantum theory and under the sole assumption of no signalling. We first investigate the trade-offs in Bell violations arising from the uncertainty relation for complementary binary observables, and exhibit several network configurations in which a tight trade-off arises in this fashion. We then derive a tight trade-off relation which cannot be obtained from the uncertainty relation showing that it does not capture monogamy entirely. The results are extended to Bell inequalities involving different number of parties and find applications in device-independent secret sharing and device-independent randomness extraction. Although two multipartite Bell inequalities may be violated simultaneously, we show that genuine multi-party non-locality, as evidenced by a generalised Svetlichny inequality, does exhibit monogamy property. Finally, using the relations derived we reveal the existence of flat regions in the set of quantum correlations.

Semi-Streaming Set Cover

This article studies the set cover problem under the semi-streaming model. The underlying set system is formalized in terms of a hypergraph G = ( V , E ) whose edges arrive one by one, and the goal is to construct an edge cover F ⊆ E with the objective of minimizing the cardinality (or cost in the weighted case) of F . We further consider a parameterized relaxation of this problem, where, given some 0 ⩽ ϵ < 1, the goal is to construct an edge (1 − ϵ)-cover, namely, a subset of edges incident to all but an ϵ-fraction of the vertices (or their benefit in the weighted case). The key limitation imposed on the algorithm is that its space is limited to (poly)logarithmically many bits per vertex. Our main result is an asymptotically tight tradeoff between ϵ and the approximation ratio: We design a semi-streaming algorithm that on input hypergraph G constructs a succinct data structure D such that for every 0 ⩽ ϵ < 1, an edge (1 − ϵ)-cover that approximates the optimal edge (1-)cover within a factor of f (ϵ, n ) can be extracted from D (efficiently and with no additional space requirements), where f (ϵ, n ) = { O (1/ ϵ ), if ϵ > 1/√ n O (√ n ), otherwise . In particular, for the traditional set cover problem, we obtain an O (√ n -approximation. This algorithm is proved to be best possible by establishing a family (parameterized by ϵ) of matching lower bounds.

Design and Experimental Study of 4H-SiC Trenched Junction Barrier Schottky Diodes

This paper presents the design, fabrication, and experimental analysis of 1200 V 4H-SiC trenched junction barrier Schottky (TJBS) diodes. Design considerations and device performances of the TJBS devices are compared with those of the conventional planar JBS diodes via numerical simulation, analytical modeling, and experiments. It was found that, for conventional planar JBS diodes, due to its limited P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> implantation depth, there is a tight tradeoff between forward ON-resistance and blocking voltage. This does not only put stringent requirement on obtaining a narrow photolithography line width ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(\sim 1.5\) </tex-math></inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(\mu \) </tex-math></inline-formula> m), but also makes the device design window narrow. The TJBS diodes can substantially alleviate such tradeoff and obtain a larger design window that enables good reverse blocking and forward conduction capabilities at the same time. As a result, this structure demands less restriction on line width control (2.2–3.2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(\mu \) </tex-math></inline-formula> m) of the fabrication process and hence can improve the device yield.

Time versus space trade-offs for rendezvous in trees

Two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. We consider deterministic algorithms for this rendezvous task. The main result of this paper is a tight trade-off between the optimal time of completing rendezvous and the size of memory of the agents. For agents with $$k$$ memory bits, we show that optimal rendezvous time is $$\Theta (n+n^2/k)$$ in $$n$$ -node trees. More precisely, if $$k \ge c\log n$$ , for some constant $$c$$ , we design agents accomplishing rendezvous in arbitrary trees of size $$n$$ (unknown to the agents) in time $$O(n+n^2/k)$$ , starting with arbitrary delay. We also show that no pair of agents can accomplish rendezvous in time $$o(n+n^2/k)$$ , even in the class of lines of known length and even with simultaneous start. Finally, we prove that at least logarithmic memory is necessary for rendezvous, even for agents starting simultaneously in a $$n$$ -node line.

Multipartite Entanglement Verification Resistant against Dishonest Parties

Future quantum information networks will consist of quantum and classical agents, who have the ability to communicate in a variety of ways with trusted and untrusted parties and securely delegate computational tasks to untrusted large-scale quantum computing servers. Multipartite quantum entanglement is a fundamental resource for such a network and, hence, it is imperative to study the possibility of verifying a multipartite entanglement source in a way that is efficient and provides strong guarantees even in the presence of multiple dishonest parties. In this Letter, we show how an agent of a quantum network can perform a distributed verification of a source creating multipartite Greenberger-Horne-Zeilinger (GHZ) states with minimal resources, which is, nevertheless, resistant against any number of dishonest parties. Moreover, we provide a tight tradeoff between the level of security and the distance between the state produced by the source and the ideal GHZ state. Last, by adding the resource of a trusted common random source, we can further provide security guarantees for all honest parties in the quantum network simultaneously.

High-accuracy Comparator-Based Switched-Capacitor structure

This paper analysis the conventional Comparator-Based Switched-Capacitor (CBSC) structure and presents a modified structure that uses interposing middle-phase in charge-transfer mode to solve the tight trade-off between accuracy, speed, and input-swing at no extra power. To meet the accuracy, speed and input-swing requirements, appropriate values of design parameters are analytically discussed and a procedure for choosing optimum values for design is presented. Moreover, digital controller for current sources and comparator are given for the new structure. The effectiveness of the proposed structure is verified through simulation of a gain-stage using a 0.13-um CMOS-technology with 1V power-supply and comparing with conventional CBSC.

A Theorem on Sensitivity and Applications in Private Computation

In this paper we prove a theorem that gives an (almost) tight upper bound on the sensitivity of a multiple-output Boolean function in terms of the sensitivity of its coordinates and the size of the range of the function. We apply this theorem to get improved lower bounds on the time (number of rounds) to compute Boolean functions by private protocols. These bounds are given in terms of the sensitivity of the function being computed and the amount of randomness used by the private protocol. These lower bounds are tight (up to constant factors) for the case of the xor function and together with the results in [E. Kushilevitz and A. Rosen, SIAM J. Discrete Math., 11 (1998), pp. 61--80.] establish a tight (up to constant factors) tradeoff between randomness and time in private computation.

Optimal Time–Space Tradeoff for Shared Memory Leader Election

Though it is common practice to treat synchronization primitives for multiprocessors as abstract data types, they are in reality machine instructions on registers. A crucial theoretical question with practical implications is the relationship between the size of the register and its computational power. We wish to study this question and choose as a first target the popularcompare&swapoperation (which is the basis for many modern multiprocessor architectures). Our main results are:1.We show that leader election amongnprocesses can be solved using acompare&swapregister that can hold ;≈logn/loglognvalues andnread/write registers of size 2lognbits. That is,n=(k−1)! wherekis the number of values in thecompare&swapregister, so thecompare&swapregister has onlyO(loglogn) bits.2.We prove a tight tradeoff between the space and time necessary to solve leader election with acompare&swapregister. Specifically, we show that any algorithm for leader election amongnprocesses with akvalue compare&swap register, wherek≥logn/loglogn, must have a run that accesses thecompare&swapregister Θ(logkn) times. We conjecture that fork<logn/loglognthere is no solution.The results of this paper suggest that a complexity hierarchy for multiprocessor synchronization operations should be based on the space complexity of synchronization registers and their type.

DO IMPERFECT TRADE-OFFS AFFECT THE EXTINCTION DEBT PHENOMENON?

A tight trade-off between competitive and colonization abilities is a key assumption of so-called “extinction debt models.” This trade-off implies that species ranked high in competitive ability will be ranked low with respect to dispersal. Although such a strict trade-off is plausible as a general trend, there are many examples in nature of deviations from perfect trade-offs. To determine how robust the “extinction debt” phenomenon is to imperfect trade-off relationships, I reexamined the extinction debt model of Tilman et al. in light of imperfect trade-offs. Numerically solving a system of differential equations, I found that, although the extinction debt phenomenon is not eliminated by the presence of imperfect trade-offs, the loss of species occurs earlier (with less habitat destruction) and more linearly than with perfect trade-offs.

Lower Bounds for Shellsort

We show lower bounds on the worst-case complexity of Shellsort. In particular, we give a fairly simple proof of an Ω(n(lg2n)/(lglgn)2) lower bound for the size of Shellsort sorting networks for arbitrary increment sequences. We also show an identical lower bound for the running time of Shellsort algorithms, again for arbitrary increment sequences. Our lower bounds establish an almost tight trade-off between the running time of a Shellsort algorithm and the length of the underlying increment sequence.

A Communication-Randomness Tradeoff for Two-Processor Systems

We present a tight tradeoff between the expected communication complexity C (for a two-processor system) and the number R of random bits used by any Las Vegas protocol for the list-nondisjointness function of two lists of n numbers of n bits each. This function evaluates to 1 if and only if the two lists correspond in at least one position. We show a log( n 2 / C ) lower bound on the number of random bits used by any Las Vegas protocol, Ω ( n ) ≤ C ≤ O ( n 2 ). We also show that expected communication complexity C , Ω ( n log n ) ≤ C ≤ O ( n 2 ), can be achieved using no more than log( n 2 / C ) + ⌈log(2 + log( n 2 / C ))⌉ + 6 random bits.

A Time-Randomness Trade-Off for Oblivious Routing

Three parameters characterize the performance of a probabilistic algorithm: T, the run-time of the algorithm; Q, the probability that the algorithm fails to complete the computation in the first T steps; and R, the amount of randomness used by the algorithm, measured by the entropy of its random source. A tight trade-off between these three parameters for the problem of oblivious packet routing on N-vertex bounded-degree networks is presented. A $(1 - Q) \log ({N / T}) - \log Q - O(1)$ lower bound for the entropy of a random source of any oblivious packet routing algorithm that routes an arbitrary permutation in T steps with probability $1 - Q$ is proved. It is shown that this lower bound is almost optimal by proving the existence, for every $e^{3} \log N \leqq T \leqq N^{{1 / 2}}$, of an oblivious algorithm that terminates in T steps with probability $1 - Q$ and uses $(1- Q + o(1)) \log ({N / T}) - \log Q$ independent random bits. This result is complemented with an explicit construction of a family of o...