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It is a fundamental problem to understand the complexity of high-accuracy sampling from a strongly log-concave density π on (mathbb {R}^d ). Indeed, in practice, high-accuracy samplers such as the Metropolis-adjusted Langevin algorithm (MALA) remain the de facto gold standard; and in theory, via the proximal sampler reduction, it is understood that such samplers are key for sampling even beyond log-concavity (in particular, for sampling under isoperimetric assumptions). This paper improves the dimension dependence of this sampling problem to (widetilde{O}(d^{1/2}) ). The previous best result for MALA was (widetilde{O}(d) ). This closes the long line of work on the complexity of MALA, and moreover leads to state-of-the-art guarantees for high-accuracy sampling under strong log-concavity and beyond (thanks to the aforementioned reduction). Our starting point is that the complexity of MALA improves to (widetilde{O}(d^{1/2}) ), but only under a warm start (an initialization with constant Rényi divergence w.r.t. π). Previous algorithms for finding a warm start took O(d) time and thus dominated the computational effort of sampling. Our main technical contribution resolves this gap by establishing the first (widetilde{O}(d^{1/2}) ) Rényi mixing rates for the discretized underdamped Langevin diffusion. For this, we develop new differential-privacy-inspired techniques based on Rényi divergences with Orlicz–Wasserstein shifts, which allow us to sidestep longstanding challenges for proving fast convergence of hypocoercive differential equations.

An average-case variant of the k-SUM conjecture asserts that finding k numbers that sum to 0 in a list of r random numbers, each of the order r^k, cannot be done in much less than r^⌈k/2⌉ time. On the other hand, in the dense regime of parameters, where the list contains more numbers and many solutions exist, the complexity of finding one of them can be significantly improved by Wagner’s k-tree algorithm. Such algorithms for k-SUM in the dense regime have many applications, notably in cryptanalysis.

In this paper, assuming the average-case k-SUM conjecture, we prove that known algorithms are essentially optimal for k = 3, 4, 5. For k > 5, we prove the optimality of the k-tree algorithm for a limited range of parameters. We also prove similar results for k-XOR, where the sum is replaced with exclusive or.

Our results are obtained by a self-reduction that, given an instance of k-SUM which has a few solutions, produces from it many instances in the dense regime. We solve each of these instances using the dense k-SUM oracle, and hope that a solution to a dense instance also solves the original problem. We deal with potentially malicious oracles (that repeatedly output correlated useless solutions) by an obfuscation process that adds noise to the dense instances. Using discrete Fourier analysis, we show that the obfuscation eliminates correlations among the oracle’s solutions, even though its inputs are highly correlated.

We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory.

This has several consequences. First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh, Bollobás, and Morris [Eur. J. Comb. ’06] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width. Finally, it settles the small conjecture [SODA ’21] in the case of ordered graphs.

In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form (bigwedge _{i=1}^n mathbf {G}mathbf {F} varphi _i vee mathbf {F}mathbf {G} psi _i ), where φ_i and ψ_i contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalization procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present direct and purely syntactic normalization procedures for LTL, yielding a normal form very similar to the one by Chang, Manna, and Pnueli, that exhibit only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalizes the formula, translates it into a special very weak alternating automaton, and applies a simple determinization procedure, valid only for these special automata.

A constraint satisfaction problem (CSP), (mathsf {Max-CSP}(mathcal {F}) ), is specified by a finite set of constraints (mathcal {F} subseteq lbrace [q]^k rightarrow lbrace 0,1rbrace rbrace ) for positive integers q and k. An instance of the problem on n variables is given by m applications of constraints from (mathcal {F} ) to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ, β)-approximation version of the problem for parameters 0 ≤ β < γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied.

In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family (mathcal {F} ) and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in (o(sqrt {n}) ) space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems.

We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of q = k = 2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of (mathcal {F} ) support a distribution on [q]^k with uniform marginals.

Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables q = 2, binary constraints k = 2, singleton families (|mathcal {F}|=1 ) and only considered the setting where constraints are placed on literals rather than variables.

Our positive results show wide applicability of bias-based algorithms used previously by [47] and [41], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [56], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general q-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

We extend Choiceless Polynomial Time (CPT), the currently only remaining promising candidate in the quest for a logic capturing Ptime, so that this extended logic has the following property: for every class of structures for which isomorphism is definable, the logic automatically captures Ptime.

For the construction of this logic we extend CPT by a witnessed symmetric choice operator. This operator allows for choices from definable orbits. But, to ensure polynomial-time evaluation, automorphisms have to be provided to certify that the choice set is indeed an orbit.

We argue that, in this logic, definable isomorphism implies definable canonization. Thereby, our construction removes the non-trivial step of extending isomorphism definability results to canonization. This step was a part of proofs that show that CPT or other logics capture Ptime on a particular class of structures. The step typically required substantial extra effort.

Recursive queries have been traditionally studied in the framework of datalog, a language that restricts recursion to monotone queries over sets, which is guaranteed to converge in polynomial time in the size of the input. But modern big data systems require recursive computations beyond the Boolean space. In this paper we study the convergence of datalog when it is interpreted over an arbitrary semiring. We consider an ordered semiring, define the semantics of a datalog program as a least fixpoint in this semiring, and study the number of steps required to reach that fixpoint, if ever. We identify algebraic properties of the semiring that correspond to certain convergence properties of datalog programs. Finally, we describe a class of ordered semirings on which one can use the semi-naïve evaluation algorithm on any datalog program.

Compositionality is at the core of programming languages research and has become an important goal toward scalable verification of large systems. Despite that, there is no compositional account of linearizability, the gold standard of correctness for concurrent objects.

In this paper, we develop a compositional semantics for linearizable concurrent objects. We start by showcasing a common issue, which is independent of linearizability, in the construction of compositional models of concurrent computation: interaction with the neutral element for composition can lead to emergent behaviors, a hindrance to compositionality. Category theory provides a solution for the issue in the form of the Karoubi envelope. Surprisingly, and this is the main discovery of our work, this abstract construction is deeply related to linearizability and leads to a novel formulation of it. Notably, this new formulation neither relies on atomicity nor directly upon happens-before ordering and is only possible because of compositionality, revealing that linearizability and compositionality are intrinsically related to each other.

We use this new, and compositional, understanding of linearizability to revisit much of the theory of linearizability, providing novel, simple, algebraic proofs of the locality property and of an analogue of the equivalence with observational refinement. We show our techniques can be used in practice by connecting our semantics with a simple program logic that is nonetheless sound concerning this generalized linearizability.

We consider the numerical taxonomy problem of fitting a positive distance function ({mathcal {D}:{Schoose 2}rightarrow mathbb {R}_{gt 0}} ) by a tree metric. We want a tree T with positive edge weights and including S among the vertices so that their distances in T match those in (mathcal {D} ). A nice application is in evolutionary biology where the tree T aims to approximate the branching process leading to the observed distances in (mathcal {D} ) [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees, and for the special case of ultrametrics with a root having the same distance to all vertices in S.

The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was O((log n)(log log n)) by Ailon and Charikar [2005] who wrote “Determining whether an O(1) approximation can be obtained is a fascinating question”.

Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 asynchronously, even against an omniscient, computationally unbounded adversary that can adaptively corrupt up to f < n/3 parties. Moreover, the problem is insoluble with f ≥ n/3 corruptions. However, Bracha’s [13] 1984 protocol (see also Ben-Or [8]) achieved f < n/3 resilience at the cost of exponential expected latency 2^Θ(n), a bound that has never been improved in this model with f = ⌊(n − 1)/3⌋ corruptions.

In this paper, we prove that Byzantine Agreement in the asynchronous, full information model can be solved with probability 1 against an adaptive adversary that can corrupt f < n/3 parties, while incurring only polynomial latency with high probability. Our protocol follows an earlier polynomial latency protocol of King and Saia [33,34], which had suboptimalresilience, namely f ≈ n/10⁹ [33,34].

Resilience f = (n − 1)/3 is uniquely difficult, as this is the point at which the influence of the Byzantine and honest players are of roughly equal strength. The core technical problem we solve is to design a collective coin-flipping protocol that eventuallylets us flip a coin with an unambiguous outcome. In the beginning, the influence of the Byzantine players is too powerful to overcome, and they can essentially fix the coin’s behavior at will. We guarantee that after just a polynomial number of executions of the coin-flipping protocol, either (a) the Byzantine players fail to fix the behavior of the coin (thereby ending the game) or (b) we can “blacklist” players such that the blacklisting rate for Byzantine players is at least as large as the blacklisting rate for good players. The blacklisting criterion is based on a simple statistical test of fraud detection.