Lower Bounds for Non-binary Constraint Optimization Problems

Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where $k \geq 2$ machines jointly perform computations on graphs with n nodes (typically, $n \gg k$). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication \em rounds of the computation. Our main contribution is the \em General Lower Bound Theorem, a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. The General Lower Bound Theorem is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic and this theorem can be used in a cookbook fashion to show distributed lower bounds in the context of several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds for the round complexity of two fundamental graph problems, namely \em PageRank computation and \em triangle enumeration. Our approach, as demonstrated in the case of PageRank, can yield tight lower bounds for problems (including, and especially, under a stochastic partition of the input) where communication complexity techniques are not obvious. Our approach, as demonstrated in the case of triangle enumeration, can yield stronger round lower bounds as well as message-round tradeoffs compared to approaches that use communication complexity techniques. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity which scales superlinearly in k , improving significantly over previous results for these problems [Klauck et al., SODA 2015]. Specifically, we show the following results: \beginitemize item \em PageRank: We show a lower bound of $\tildeOmega (n/k^2)$ rounds, and present a distributed algorithm that computes the PageRank of all the nodes of a graph in $\tildeO (n/k^2)$ rounds. item \em Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires $\tildeOmega (m/k^5/3 )$ rounds. This result also implies the first non-trivial lower bound of $\tildeOmega(n^1/3 )$ rounds for the \em congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in $\tildeO (m/k^5/3 + n/k^4/3 )$ rounds. \enditemize

Derivation of the optimum detection operators for the mutual information is one of the most important topics to establish the quantum information theory. For information-optimum detection, Holevo has shown a necessary condition of the information-optimum detection operators [1]. However, the explicit representation of the information-optimum detection operators has not been given except for a few cases [24]. So far there were researches about the upper or lower bound of the accessible information which is the optimum (maximum) mutual information for given signal quantum states with coding theory. In the general case, the upper bound is given by Holevo, called “Holevo’s bound” [5] and the lower bound is the “subentropy” defined by Jozsa [6]. In some cases of specified signal quantum states and detection process, Ban and Schumacher showed tighter upper bounds, respectively [7, 8].

We investigate the right quotient and the reversal operations on the class of prefix-free languages. We get the tight bounds n − 1 and 2 n − 2 + 1 on the state complexity of right quotient and reversal, respectively. To prove the tightness of the bound for reversal, we use a ternary alphabet. Moreover, we prove that this bound cannot be met by any binary language. In the binary case, we get a lower bound 2 n − 2 − 7 infinitely often. Our calculations show that this lower bound cannot be exceeded.

\n Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a small circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erds and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called robust sunflowers, to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossmans bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain the most significant progress on the sunflower conjecture since its inception in 1960. In this work, we will show how the recent progresses in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, thus breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes.\n

Cumulative memory—the sum of space used per step over the duration of a computation—is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least 1/poly( n ) requires cumulative memory \(\tilde{\Omega }(n^2) \) , any classical matrix multiplication algorithm requires cumulative memory Ω ( n 6 / T ), any quantum sorting circuit requires cumulative memory Ω ( n 3 / T ), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω ( k 3 n / T 2 ).

On the lower bound for the length of minimal codes

We prove that the tight bound on the nondeterministic state complexity of complementation on prefix-free and suffix-free languages is 2 n − 1. To prove tightness, we use a ternary alphabet, and we show that this bound cannot be met by any binary prefix-free language. On non-returning languages, the upper bound is 2 n − 1 + 1, and it is tight already in the binary case. We also study the unary case in all three classes.

Characterizing the impact of model error in hydrologic time series recovery inverse problems

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal ’17, Aravind et al. ’16, Choudhari and Reddy ’18, Misra and Reddy ’17. Main focus of our work is lower bounds on the computational complexity of these problems. Established lower bounds can be divided into the following groups: NP-hardness of the above guarantee parameterization, kernelization lower bounds (answering questions of Misra and Reddy ’17), exponential lower bounds under the Set Cover Conjecture and the Exponential Time Hypothesis, and inapproximability results. Moreover, we present an \({\mathcal {O}}^*(\ell ^k)\) randomized algorithm for MHV and an \({\mathcal {O}}^*(2^k)\) algorithm for MHE, where \(\ell \) is the number of colors used and k is the number of required happy vertices or edges. These algorithms cannot be improved to subexponential taking proved lower bounds into account.

Every lower bound for treewidth can be extended by taking the maximum of the lower bound over all subgraphs or minors. This extension is shown to be a very vital idea for improving treewidth lower bounds. In this paper, we investigate a total of nine graph parameters, providing lower bounds for treewidth. The parameters have in common that they all are the vertex-degree of some vertex in a subgraph or minor of the input graph. We show relations between these graph parameters and study their computational complexity. To allow a practical comparison of the bounds, we developed heuristic algorithms for those parameters that are N P-hard to compute. Computational experiments show that combining the treewidth lower bounds with minors can considerably improve the lower bounds.

Orthogonal time frequency space (OTFS) modulations are robust to time-varying frequency-selective fading channels. OTFS modulations operate in the delay-Doppler domain whereas two-dimensional (2D) orthogonal frequency division multiplexing (OFDM) modulations operate in the time-frequency domain. For 2D OFDM modulations in time-varying frequency-selective fading channels, we investigate the pilot pattern design problem, which minimizes the mean square error (MSE) of channel estimation. The MSE lower bound (LB) is theoretically derived to provide the design criterion. Based on the criterion, we show that the LB achieving design can be found by exhaustive 2D pilot location search. Exhaustive 2D search has high computational complexity. To reduce the complexity, sufficient conditions on the existence of LB achieving design are provided to decouple the 2D problem into two one-dimensional (1D) problems. For the 1D problem, we propose the LB achieving design when the number of possible pilot locations is divisible by the number of pilots. Simulation results illustrate that our proposed LB achieving design is superior to the random pilot pattern design and the bilinear interpolation channel estimation method.

Given a Boolean formula in conjunctive normal form with \(n\) variables and \(m=rn\) clauses, if there exists a truth assignment satisfying \((1-2^{-k}-q(1-2^{-k}))m\) clauses, call the formula \(q\)-satisfiable. The Minimum Satisfiability Problem (MinSAT) is a special case of \(q\)-satisfiable, which asks for an assignment to minimize the number of satisfied clauses. When each clause contains \(k\) literals, it is called Min\(k\)SAT. If each clause is independently and randomly selected from all possible clauses over the \(n\) variables, it is called random MinSAT. In this paper, we give upper and lower bounds of \(r\) (the ratio of clauses to variables) for random \(k\)-CNF formula with \(q\)-satisfiable. The upper bound is proved by the first moment argument, while the proof of lower bound is the second moment with weighted scheme. Interestingly, our experimental results about MinSAT demonstrate that the lower and upper bounds are very tight. Moreover, these results give a partial explanation for the excellent performance of MinSatz, the state-of-the-art MinSAT solver, from the perspective of pruning effects. Finally, we give a conjecture about the relationship between the minimum number and the maximum number of satisfied clauses on random SAT instances.KeywordsMinSATUpper boundsLower bounds

In the range emptiness searching problem, we are given a set P of n points in Rd, and wish to preprocess them into a data structure that supports efficient range emptiness queries, in which we specify a range σ, which is a semi-algebraic set in Rd of some fixed kind, and wish to determine whether P ∩ σ = θ. Range emptiness searching arises in many applications, and has been treated by Matousek [15] in the special case where the ranges are halfspaces bounded by hyperplanes. In this paper we extend the analysis to the general semi-algebraic case, and show how to adapt Matousek's technique to this case, without the need to linearize the ranges into a higher-dimensional space. This yields more efficient solutions to several interesting problems, and we demonstrate the new technique in two applications:(i) An algorithm for ray shooting amid balls in R3, which uses O* (n) storage and preprocessing, and answers a query in O* (n2/3) time, improving the previous bound of O* (n3/47).(ii) An algorithm that preprocesses, in O*(n) time, a set P of n points in R3 into a data structure with O*(n) storage, so that, for any query line l or segment e, the point of P farthest from l or from e can be computed in O* (n1/2) time.Our technique is closely related to the notions of nearest- or farthest-neighbor generalized Voronoi diagrams, and of the union or intersection of geometric objects, where sharper bounds on the combinatorial complexity of these structures yield faster range emptiness searching algorithms. For example, in the case of ray shooting amid balls, the structure that arises in our algorithm is the Euclidean Voronoi diagram of lines in 3-space, and the performance of the algorithm depends on the complexity of such diagrams (for which tight bounds are still unknown).

There is an isomorphism between the matrices over the Boolean algebra of subsets of a k-element set and the k-tuples of Boolean binary (i.e. zero-one) matrices. This isomorphism allows many problems concerning nonbinary Boolean matrices to the referred to the binary ease. However, there are some features of the general (i.e. nonbinary) case that have not been mentioned, although they differ somewhat from the binary case. We exhibit characterizations of the linear operators that preserve several invariants of matrices over finite Boolean algebras to illustrate the differences and similarities of the general vs. the binary cases. We employ a canonical form that is useful in applying the isomorphism.

We show that if Arthur-Merlin protocols can be derandomized, then there is a Boolean function computable in deterministic exponential-time with access to an $\bm{\NP}$ oracle, that cannot be computed by Boolean circuits of {\em exponential} size. More formally, if $\bm{\prAM \subseteq \P^{\NP}}$ then there is a Boolean function in $\bm{\E^{\NP}}$ that requires circuits of size $\bm{2^{\Omega(n)}}$. $\bm{\prAM}$ is the class of promise problems that have Arthur-Merlin protocols, $\bm{\P^{\NP}}$ is the class of functions that can be computed in deterministic polynomial-time with an $\bm{\NP}$ oracle and $\bm{\E^{\NP}}$ is its exponential analogue. The lower bound in the conclusion of our theorem suffices to construct very strong pseudorandom generators. We also show that the same conclusion holds if the problem of approximate counting the number of accepting paths of a nondeterministic Turing machine up to multiplicative factors can be done in nondeterministic polynomial-time. In other words, showing nondeterministic fully polynomial-time approximation schemes for $\bm{\sharp \P}$-complete problems require proving exponential-size circuit lower bounds. A few works have already shown that if we can find efficient deterministic solutions to some specific tasks (or classes) that are known to be solvable efficiently by randomized algorithms (or proofs), then we obtain lower bounds against certain circuit models. These lower bounds were only with respect to polynomial-size circuits {\em even if full derandomization is assumed}. Thus they only implied fairly weak pseudorandom generators (if at all). A key ingredient in our proof is a connection between computational learning theory and exponential-size lower bounds. We show that the existence of deterministic learning algorithms with certain properties implies exponential-size lower bounds, where the complexity of the hard function is related to the complexity of the learning algorithm.