Standard Complexity-theoretic Assumptions Research Articles

Clifford Circuits can be Properly PAC Learned if and only if RP=NP

Given a dataset of input states, measurements, and probabilities, is it possible to efficiently predict the measurement probabilities associated with a quantum circuit? Recent work of Caro and Datta \cite{2020Caro} studied the problem of PAC learning quantum circuits in an information theoretic sense, leaving open questions of computational efficiency. In particular, one candidate class of circuits for which an efficient learner might have been possible was that of Clifford circuits, since the corresponding set of states generated by such circuits, called stabilizer states, are known to be efficiently PAC learnable \cite{rocchetto2018stabiliser}. Here we provide a negative result, showing that proper learning of CNOT circuits with 1/ poly(n) error is hard for classical learners unless RP = NP, ruling out the possibility of strong learners under standard complexity theoretic assumptions. As the classical analogue and subset of Clifford circuits, this naturally leads to a hardness result for Clifford circuits as well. Additionally, we show that if RP = NP then there would exist efficient proper learning algorithms for CNOT and Clifford circuits. By similar arguments, we also find that an efficient proper quantum learner for such circuits exists if and only if NP ⊆ RQP. We leave open the problem of hardness for improper learning or O(1) error to future work.

Polynomial combined first-order rewritings for linear and guarded existential rules

We consider the problem of ontological query answering, that is, the problem of answering a database query (typically a conjunctive query) in the presence of an ontology. This means that during the query answering process we also need to take into account the knowledge that can be inferred from the given database and ontology. Building, however, ontology-aware database systems from scratch, with sophisticated optimization techniques, is a highly non-trivial task that requires a great engineering effort. Therefore, exploiting conventional database systems is an important route towards efficient ontological query answering. Nevertheless, standard database systems are unaware of ontologies. An approach to ontological query answering that enables the use of standard database systems is the so-called polynomial combined query rewriting, originally introduced in the context of description logics: the conjunctive query q and the ontology Σ are rewritten in polynomial time into a first-order query qΣ (in a database-independent way), while the database D and the ontology Σ are rewritten in polynomial time into a new database DΣ (in a query-independent way), such that the answer to q in the presence of Σ over D coincides with the answer to qΣ over DΣ. The latter can then be computed by exploiting a conventional database system.In this work, we focus on linear and guarded existential rules, which form robust rule-based languages for modeling ontologies, and investigate the limits of polynomial combined query rewriting. In particular, we show that this type of rewriting can be successfully applied to (i) linear existential rules when the rewritten query can use the full power of first-order queries, (ii) linear existential rules when the arity of the underlying schema is fixed and the rewritten query is positive existential, namely it uses only existential quantification, conjunction, and disjunction, and (iii) guarded existential rules when the underlying schema is fixed and the rewritten query is positive existential. We can show that the above results reach the limits (under standard complexity-theoretic assumptions such as ▪) of polynomial combined query rewriting in the case of linear and guarded existential rules.

Nearly Optimal Pseudorandomness from Hardness

Existing proofs that deduce BPP = P from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown . Specifically, assuming exponential lower bounds against randomized NP ∩ coNP circuits, formally known as randomized SVN circuits, we convert any randomized algorithm over inputs of length n running in time t ≥ n into a deterministic one running in time t 2+α for an arbitrarily small constant α > 0. Such a slowdown is nearly optimal for t close to n , since under standard complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing). The latter derandomization result holds under weaker assumptions, of exponential lower bounds against deterministic SVN circuits. Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size s with seed length (1+α)log s , under the assumption that there exists a function f ∈ E that requires randomized SVN circuits of size at least 2 (1-α′) n , where α = O (α)′. The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.

Computational Short Cuts in Infinite Domain Constraint Satisfaction

A backdoor in a finite-domain CSP instance is a set of variables where each possible instantiation moves the instance into a polynomial-time solvable class. Backdoors have found many applications in artificial intelligence and elsewhere, and the algorithmic problem of finding such backdoors has consequently been intensively studied. Sioutis and Janhunen have proposed a generalised backdoor concept suitable for infinite-domain CSP instances over binary constraints. We generalise their concept into a large class of CSPs that allow for higher-arity constraints. We show that this kind of infinite-domain backdoors have many of the positive computational properties that finite-domain backdoors have: the associated computational problems are fixed parameter tractable whenever the underlying constraint language is finite. On the other hand, we show that infinite languages make the problems considerably harder: the general backdoor detection problem is W[2]-hard and fixed-parameter tractability is ruled out under standard complexity-theoretic assumptions. We demonstrate that backdoors may have suboptimal behaviour on binary constraints—this is detrimental from an AI perspective where binary constraints are predominant in, for instance, spatiotemporal applications. In response to this, we introduce sidedoors as an alternative to backdoors. The fundamental computational problems for sidedoors remain fixed-parameter tractable for finite constraint language (possibly also containing non-binary relations). Moreover, the sidedoor approach has appealing computational properties that sometimes leads to faster algorithms than the backdoor approach.

Reoptimization of parameterized problems

Parameterized complexity allows us to analyze the time complexity of problems with respect to a natural parameter depending on the problem. Reoptimization looks for solutions or approximations for problem instances when given solutions to neighboring instances. We combine both techniques, in order to better classify the complexity of problems in the parameterized setting. Specifically, we see that some problems in the class of compositional problems, which do not have polynomial kernels under standard complexity-theoretic assumptions, do have polynomial kernels under the reoptimization model for some local modifications. We also observe that, for some other local modifications, these same problems do not have polynomial kernels unless mathbf{NP}subseteq mathbf{coNP/poly}. We find examples of compositional problems, whose reoptimization versions do not have polynomial kernels under any of the considered local modifications. Finally, in another negative result, we prove that the reoptimization version of Connected Vertex Cover does not have a polynomial kernel unless Set Cover has a polynomial compression. In a different direction, looking at problems with polynomial kernels, we find that the reoptimization version of Vertex Cover has a polynomial kernel of size varvec{2k+1} using crown decompositions only, which improves the size of the kernel achievable with this technique in the classic problem.

Resolving Inconsistencies in Simple Temporal Problems: A Parameterized Approach

The simple temporal problem (STP) is one of the most influential reasoning formalisms for representing temporal information in AI. We study the problem of resolving inconsistency of data encoded in the STP. We prove that the problem of identifying a maximally large consistent subset of data is NP-hard. In practical instances, it is reasonable to assume that the amount of erroneous data is small. We therefore parameterize by the number of constraints that need to be removed to achieve consistency. Using tools from parameterized complexity we design fixed-parameter tractable algorithms for two large fragments of the STP. Our main algorithmic results employ reductions to the Directed Subset Feedback Arc Set problem and iterative compression combined with an efficient algorithm for the Edge Multicut problem. We complement our algorithmic results with hardness results that rule out fixed-parameter tractable algorithms for all remaining non-trivial fragments of the STP (under standard complexity-theoretic assumptions). Together, our results give a full classification of the classical and parameterized complexity of the problem.

On 2-Clubs in Graph-Based Data Clustering: Theory and Algorithm Engineering

Editing a graph into a disjoint union of clusters is a standard optimization task in graph-based data clustering. Here, complementing classical work where the clusters shall be cliques, we focus on clusters that shall be 2-clubs, that is, subgraphs of diameter at most two. This naturally leads to the two NP-hard problems $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ (the editing operations are edge insertions and edge deletions) and $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{V{\small ERTEX}}$ $\rm{D{\small ELETION}}$ (the editing operations are vertex deletions). Answering an open question from the literature, we show that $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ is W[2]-hard with respect to the number of edge modifications, thus contrasting the fixed-parameter tractability result for the classical $\rm{C{\small LUSTER}}$ $\rm{E{\small DITING}}$ problem (considering cliques instead of 2-clubs). Then, focusing on $\rm{2-C \small{LUB}}$ $\rm{C{\small LUSTER}}$ $\rm{V{\small ERTEX}}$ $\rm{D{\small ELETION}}$, which is easily seen to be fixed-parameter tractable, we show that under standard complexity-theoretic assumptions it does not have a polynomial-size problem kernel when parameterized by the number of vertex deletions. Nevertheless, we develop several effective data reduction and pruning rules, resulting in a competitive solver, outperforming a standard ILP formulation in most instances of an established biological test data set.

The Communication Complexity of Private Simultaneous Messages, Revisited

Private simultaneous message (PSM) protocols were introduced by Feige, Kilian, and Naor (STOC ’94) as a minimal non-interactive model for information theoretic three-party secure computation. While it is known that every function $$f:\{0,1\}^k\times \{0,1\}^k \rightarrow \{0,1\}$$ admits a PSM protocol with exponential communication of $$2^{k/2}$$ (Beimel et al., TCC ’14), the best known (non-explicit) lower-bound is $$3k-O(1)$$ bits. To prove this lower-bound, FKN identified a set of simple requirements, showed that any function that satisfies these requirements is subject to the $$3k-O(1)$$ lower-bound, and proved that a random function is likely to satisfy the requirements. We revisit the FKN lower-bound and prove the following results: (Counterexample) We construct a function that satisfies the FKN requirements but has a PSM protocol with communication of $$2k+O(1)$$ bits, revealing a gap in the FKN proof. (PSM lower-bounds) We show that by imposing additional requirements, the FKN argument can be fixed leading to a $$3k-O(\log k)$$ lower-bound for a random function. We also get a similar lower-bound for a function that can be computed by a polynomial-size circuit (or even polynomial-time Turing machine under standard complexity-theoretic assumptions). This yields the first non-trivial lower-bound for an explicit Boolean function partially resolving an open problem of Data, Prabhakaran, and Prabhakaran (Crypto ’14, IEEE Information Theory ’16). We further extend these results to the setting of imperfect PSM protocols which may have small correctness or privacy error. (CDS lower-bounds) We show that the original FKN argument applies (as is) to some weak form of PSM protocols which are strongly related to the setting of Conditional Disclosure of Secrets (CDS). This connection yields a simple combinatorial criterion for establishing linear $$\varOmega (k)$$-bit CDS lower-bounds. As a corollary, we settle the complexity of the inner-product predicate resolving an open problem of Gay, Kerenidis, and Wee (Crypto ’15).

On the complexity of inconsistency measurement

We survey a selection of inconsistency measures from the literature and investigate their computational complexity wrt. decision problems related to bounds on the inconsistency value and the functional problem of determining the actual value. Our findings show that those inconsistency measures can be partitioned into four classes related to their complexity. The first three classes contain measures whose complexities are located on the first three levels of the polynomial hierarchy, respectively. The final class is under standard complexity-theoretic assumptions located beyond the polynomial hierarchy. We provide membership results for all the investigated problems and completeness results for most of them. In addition, we undertake a preliminary study on the computational complexity of the measures on fragments of propositional logic.

Network Pollution Games

The problem of pollution control has been mainly studied in the environmental economics literature where the methodology of game theory is applied for the pollution control. To the best of our knowledge this is the first time this problem is studied from the computational point of view. We introduce a new network model for pollution control and present two applications of this model. On a high level, our model comprises a graph whose nodes represent the agents, which can be thought of as the sources of pollution in the network. The edges between agents represent the effect of spread of pollution. The government who is the regulator, is responsible for the maximization of the social welfare and sets bounds on the levels of emitted pollution in both local areas as well as globally in the whole network. We first prove that the above optimization problem is NP-hard even on some special cases of graphs such as trees. We then turn our attention on the classes of trees and planar graphs which model realistic scenarios of the emitted pollution in water and air, respectively. We derive approximation algorithms for these two kinds of networks and provide deterministic truthful and truthful in expectation mechanisms. In some settings of the problem that we study, we achieve the best possible approximation results under standard complexity theoretic assumptions. Our approximation algorithm on planar graphs is obtained by a novel decomposition technique to deal with constraints on vertices. We note that no known planar decomposition techniques can be used here and our technique can be of independent interest. For trees we design a two level dynamic programming approach to obtain an FPTAS. This approach is crucial to deal with the global pollution quota constraint. It uses a special multiple choice, multi-dimensional knapsack problem where coefficients of all constraints except one are bounded by a polynomial of the input size. We furthermore derive truthful in expectation mechanisms on general networks with bounded degree.

Improved integrality gap upper bounds for traveling salesperson problems with distances one and two

We prove new integrality gap upper bounds for the traveling salesperson problem with distances one and two ((1,2)-TSP).We obtain these bounds by investigating the structure of the support graph of optimal solutions to the subtour elimination linear programming relaxation, with one additional cutting plane inequality.For undirected (1,2)-TSP, our main results are as follows:•All instances have an integrality gap of at most 5/4.•Instances admitting half-integral solutions have integrality gap at most 7/6.•Instances admitting subcubic solutions of cost at most the order of the instance have integrality gap at most 10/9, even without the cutting plane. This bound is tight, and holds in particular for basic solutions in the fractional 2-matching polytope with cost at most the order of the instance.For directed (1,2)-TSP instances we show an integrality gap upper bound of 3/2 for general instances, and of 5/4 for instances admitting half-integral solutions.We prove these bounds by providing local search algorithms that, in polynomial time, find 2-matchings with few components in the support of the solution. We show that the run times of our algorithms cannot be considerably improved under standard complexity-theoretic assumptions: we show that finding improved TSP solutions via local search is intractable for edge change parameterized by the size of the neighborhoods even for instances with distances one and two; this strengthens a result of Dániel Marx.

Fréchet Distance Between a Line and Avatar Point Set

Frechet distance is an important geometric measure that captures the distance between two curves or more generally point sets. In this paper, we consider a natural variant of Frechet distance problem with multiple choice, provide an approximation algorithm and address its parameterized and kernelization complexity. A multiple choice problem consists of a set of color classes $$\mathcal {Q}=\{Q_1,Q_2,\ldots ,Q_n\}$$ , where each class $$Q_i$$ consists of a pair of points $$Q_i = \{q_i, \bar{q_i}\}$$ . We call a subset $$A\subset \{q_i , \bar{q_i}:1\le i\le n\}$$ conflict-free if A contains at most one point from each color class. The standard objective in multiple choice problem is to select a conflict-free subset that optimizes a given function. Given a line-segment $$\ell $$ and a set $$\mathcal {Q}$$ of a pair of points in $$\mathbb {R}^2$$ , our objective is to find a conflict-free subset that minimizes the Frechet distance between $$\ell $$ and the point set, where the minimum is taken over all possible conflict-free subsets. We first show that this problem is NP-hard, and provide a 3-approximation algorithm. Then we develop a simple randomized FPT algorithm for the problem when parametrized by the solution size, which is later derandomized using universal family of sets. We believe that our derandomization technique can be of independent interest, and can be used to solve other parameterized multiple choice problems. The randomized algorithm runs in $$\mathcal {O}(2^k n \log ^2 n)$$ time, and the derandomized deterministic algorithm runs in $$2^k k^{\mathcal {O}(\log k)} n \log ^2 n$$ time, where k, the parameter, is the number of elements in the conflict-free subset solution. Finally we present a simple branching algorithm for the problem running in $$\mathcal {O}(2^k n\log n)$$ time. We also show that the problem does not have a polynomial sized kernel under standard complexity theoretic assumptions.

Computational Aspects of the Gromov–Hausdorff Distance and its Application in Non-rigid Shape Matching

The Gromov---Hausdorff distance of two compact metric spaces is a measure for how far the spaces are from being isometric and has been extensively studied in the field of metric geometry. In recent years, a lot of attention has been devoted to computational aspects of the Gromov---Hausdorff distance. One of the most prominent applications is the problem of shape matching, where the goal is to decide whether two shapes given as polygonal meshes are equivalent under certain invariances. Therefore, many methods have been proposed which heuristically estimate the Gromov---Hausdorff distance of metric spaces induced by the shapes. However, the computational complexity of computing the Gromov---Hausdorff distance has not yet been thoroughly investigated. We show that--under standard complexity theoretic assumptions--determining the Gromov---Hausdorff distance of two finite metric spaces cannot be approximated within any reasonable bound in polynomial time. Furthermore, we discover attributes of metric spaces which have a major impact on the complexity of an instance. This enables us to develop an approximation algorithm which is fixed parameter tractable with respect to corresponding parameters.

Monodirectional P systems

We investigate the influence that the flow of information in membrane has on their computational complexity. In particular, we analyse the behaviour of P with active membranes where communication only happens from a membrane towards its parent, and never in the opposite direction. We prove that these monodirectional P systems are, when working in polynomial time and under standard complexity-theoretic assumptions, much less powerful than unrestricted ones: indeed, they characterise classes of problems defined by polynomial-time Turing machines with $${\mathbf{NP}}$$NP oracles, rather than the whole class $${\mathbf{PSPACE}}$$PSPACE of problems solvable in polynomial space.

Pricing lotteries

We examine optimal mechanisms for a multi-product monopolist selling n substitutable goods to a buyer of unknown type randomly sampled from a known distribution. The optimal selling strategy assigns posted prices to lotteries (probability distributions over items), and it is known that pricing lotteries can be strictly superior to pricing individual items. Our results quantify the magnitude of this phenomenon and shed light on the computational hardness of determining (or approximating) the optimal selling strategy. In particular, we show that when the number of items is at least three, there is no finite upper bound on the ratio between the maximum revenue attainable by pricing lotteries and that which can be attained by pricing items. Furthermore, the time required to compute the optimal selling strategy is polynomial in the support size of the buyer's type distribution. We also show that these two results are reversed in a model in which the buyer is allowed to buy any desired number of lotteries and receive an independent sample from each: the ratio between the revenue of the optimal selling strategy and that which can be attained by pricing items is only logarithmic in the number of goods n, and under standard complexity-theoretic assumptions there is no polynomial-time algorithm to compute or even approximate the optimal selling strategy.

Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most $$d$$ , throughout the uniqueness region of the Gibbs measure on the infinite $$d$$ -regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree $$d$$ for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.

One-way permutations and self-witnessing languages

A desirable property of one-way functions is that they be total, one-to-one, and onto—in other words, that they be permutations. We prove that one-way permutations exist exactly if P≠UP∩coUP. This provides the first characterization of the existence of one-way permutations based on a complexity-class separation and shows that their existence is equivalent to a number of previously studied complexity-theoretic hypotheses. We study permutations in the context of witness functions of nondeterministic Turing machines. A language is in PermUP if, relative to some unambiguous, nondeterministic, polynomial-time Turing machine accepting the language, the function mapping each string to its unique witness is a permutation of the members of the language. We show that, under standard complexity-theoretic assumptions, PermUP is a strict subset of UP. We study SelfNP, the set of all languages such that, relative to some nondeterministic, polynomial-time Turing machine that accepts the language, the set of all witnesses of strings in the language is identical to the language itself. We show that SAT∈SelfNP, and, under standard complexity-theoretic assumptions, SelfNP≠NP.

Counting Complexity Classes for Numeric Computations I: Semilinear Sets

We define a counting class ${\rm #P}_\add$ in the Blum--Shub--Smale setting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class $#{\sf P}$ introduced by Valiant. We also establish transfer theorems for both directions between the real additive and the discrete setting. Then we characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits. It turns out that the computation of the Euler characteristic is ${\rm FP}_{\rm add}^{{\rm #P}_{\rm add}}$-complete, while for fixed k the computation of the kth Betti number is ${\rm FPAR}_{\rm add}$-complete. Thus the latter is more difficult under standard complexity theoretic assumptions. We use all of the above to prove some analogous completeness results in the classical setting.

Classic learning

Description logics, also called terminological logics, are commonly used in knowledge-based systems to describe objects and their relationships. We investigate the learnability of a typical description logic, Classic, and show that Classic sentences are learnable in polynomial time in the exact learning model using equivalence queries and membership queries (which are in essence, “subsumption queries”—we show a prediction hardness result for the more traditional membership queries that convey information about specific individuals). We show that membership queries alone are insufficient for polynomial time learning of Classic sentences. Combined with earlier negative results (Cohen & Hirsh, 1994a) showing that, given standard complexity theoretic assumptions, equivalence queries alone are insufficient (or random examples alone in the PAC setting are insufficient), this shows that both sources of information are necessary for efficient learning in that neither type alone is sufficient. In addition, we show that a modification of the algorithm deals robustly with persistent malicious two-sided classification noise in the membership queries with the probability of a misclassification bounded below 1/2. Other extensions are considered.