Complexity-theoretic Assumptions Research Articles

Quantum Algorithm for Linear Systems of Equations

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b(-->), find a vector x(-->) such that Ax(-->) = b(-->). We consider the case where one does not need to know the solution x(-->) itself, but rather an approximation of the expectation value of some operator associated with x(-->), e.g., x(-->)(dagger) Mx(-->) for some matrix M. In this case, when A is sparse, N x N and has condition number kappa, the fastest known classical algorithms can find x(-->) and estimate x(-->)(dagger) Mx(-->) in time scaling roughly as N square root(kappa). Here, we exhibit a quantum algorithm for estimating x(-->)(dagger) Mx(-->) whose runtime is a polynomial of log(N) and kappa. Indeed, for small values of kappa [i.e., poly log(N)], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.

Tractable cases of the extended global cardinality constraint

We study the consistency and domain consistency problem for extended global cardinality (EGC) constraints. An EGC constraint consists of a set X of variables, a set D of values, a domain $D(x) \subseteq D$ for each variable x, and a "cardinality set" K(d) of non-negative integers for each value d. The problem is to instantiate each variable x with a value in D(x) such that for each value d, the number of variables instantiated with d belongs to the cardinality set K(d). It is known that this problem is NP-complete in general, but solvable in polynomial time if all cardinality sets are intervals. First we pinpoint connections between EGC constraints and general factors in graphs. This allows us to extend the known polynomial-time case to certain non-interval cardinality sets. Second we consider EGC constraints under restrictions in terms of the treewidth of the value graph (the bipartite graph representing variable-value pairs) and the cardinality-width (the largest integer occurring in the cardinality sets). We show that EGC constraints can be solved in polynomial time for instances of bounded treewidth, where the order of the polynomial depends on the treewidth. We show that (subject to the complexity theoretic assumption FPT???W[1]) this dependency cannot be avoided without imposing additional restrictions. If, however, also the cardinality-width is bounded, this dependency gets removed and EGC constraints can be solved in linear time.

Probabilistic Field Development in the Presence of Uncertainty

This article, written by Technology Editor Dennis Denney, contains highlights of paper IPTC 11294, "Probabilistic Field Development in Presence of Uncertainty," by Sara Passone, SPE, BP plc, and Gregory J. McRae, SPE, Massachusetts Institute of Technology, prepared for the 2007 International Petroleum Technology Conference, Dubai, 4-6 December. The paper has not been peer reviewed. Field developments are characterized by high levels of uncertainty and dynamic interconnected decisions having complex value descriptions. A unified fully Bayesian design-optimization methodology helps with the complexity of decision making in field development. The proposed scheme can be customized with respect to the decision components: the metrics chosen, objective function, and associated uncertainties. Introduction Various methods are used in decision making in the oil industry. Methods include worst-case/best-case scenarios, tornado graphs, Boston grid, expected net present value, decision trees, Monte Carlo simulation, and real options. These methods have different degrees of complexity and specific theoretical assumptions. For example, the decision-tree technique is a standard tool, but it is affected by important limitations. The number of decisions considered complicates the tree very quickly, especially when the decision tree is used to represent the whole-project design over time. Also, because decision trees provide only the maximum expected net present value of a project, they do not offer a way to quantify the volatility of the estimate. Some decision-making methods model the decision process only partially. Monte Carlo simulation, for example, is a formal probabilistic technique in which different types of knowledge can be combined and uncertainty can be propagated. Various problems characterize the application. For example, correlation between uncertain variables can be modeled with Monte Carlo simulation, but it requires complicated mathematical algorithms. Monte Carlo simulation is not suitable for incorporating management flexibility or analyzing the value added by new available information. Overall, current decision-making methods do not integrate the ability to represent uncertain factors efficiently, their contribution to uncertainty propagation and their correlation effect, or the possibility of managing flexibility in the presence of project irreversibility and proactive information gathering. The decision-making approach presented in this paper aims to overcome such limitations by providing a way of framing and designing decisions and objectives that can be used in a variety of contexts, from the design of experiments to exploratory wells to a company's portfolio.

On the Complexity of Succinct Zero-Sum Games

We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i,j) = C(i,j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approximating the value of a succinct zero-sum game to within an additive error is complete for the class promise- $$S^{p}_{2}$$ , the "promise" version of $$S^{p}_{2}$$ . To the best of our knowledge, it is the first natural problem shown complete for this class. (2) We describe a ZPP NP algorithm for constructing approximately optimal strategies, and hence for approximating the value, of a given succinct zero-sum game. As a corollary, we obtain, in a uniform fashion, several complexity-theoretic results, e.g., a ZPP NP algorithm for learning circuits for SAT (Bshouty et al., JCSS, 1996) and a recent result by Cai (JCSS, 2007) that $$S^{p}_{2} \subseteq$$ ZPP NP . (3) We observe that approximating the value of a succinct zero-sum game to within a multiplicative factor is in PSPACE, and that it cannot be in promise- $$S^{p}_{2}$$ unless the polynomial-time hierarchy collapses. Thus, under a reasonable complexity-theoretic assumption, multiplicative-factor approximation of succinct zero-sum games is strictly harder than additive-error approximation.

Closest Substring Problems with Small Distances

We study two pattern matching problems that are motivated by applications in computational biology. In the Closest Substring problem k strings $s_1,\dots, s_k$ are given, and the task is to find a string s of length L such that each string $s_i$ has a consecutive substring of length L whose distance is at most d from s. We present two algorithms that aim to be efficient for small fixed values of d and k: for some functions f and g, the algorithms have running time $f(d)\cdot n^{O(\log d)}$ and $g(d,k)\cdot n^{O(\log\log k)}$, respectively. The second algorithm is based on connections with the extremal combinatorics of hypergraphs. The Closest Substring problem is also investigated from the parameterized complexity point of view. Answering an open question from [P. A. Evans, A. D. Smith, and H. T. Wareham, Theoret. Comput. Sci., 306 (2003), pp. 407–430, M. R. Fellows, J. Gramm, and R. Niedermeier, Combinatorica, 26 (2006), pp. 141–167, J. Gramm, J. Guo, and R. Niedermeier, Lecture Notes in Comput. Sci. 2751, Springer, Berlin, 2003, pp. 195–209, J. Gramm, R. Niedermeier, and P. Rossmanith, Algorithmica, 37 (2003), pp. 25–42], we show that the problem is W[1]-hard even if both d and k are parameters. It follows as a consequence of this hardness result that our algorithms are optimal in the sense that the exponent of n in the running time cannot be improved to $o(\log d)$ or to $o(\log \log k)$ (modulo some complexity-theoretic assumptions). Consensus Patterns is the variant of the problem where, instead of the requirement that each $s_i$ has a substring that is of distance at most d from s, we have to select the substrings in such a way that the average of these k distances is at most $\delta$. By giving an $f(\delta)\cdot n^9$ time algorithm, we show that the problem is fixed-parameter tractable. This answers an open question from [M. R. Fellows, J. Gramm, and R. Niedermeier, Combinatorica, 26 (2006), pp. 141–167].

A Note on Problem Difficulty Measures in Black-Box Optimization: Classification, Realizations and Predictability

Various methods have been defined to measure the hardness of a fitness function for evolutionary algorithms and other black-box heuristics. Examples include fitness landscape analysis, epistasis, fitness-distance correlations etc., all of which are relatively easy to describe. However, they do not always correctly specify the hardness of the function. Some measures are easy to implement, others are more intuitive and hard to formalize. This paper rigorously defines difficulty measures in black-box optimization and proposes a classification. Different types of realizations of such measures are studied, namely exact and approximate ones. For both types of realizations, it is proven that predictive versions that run in polynomial time in general do not exist unless certain complexity-theoretical assumptions are wrong.

The complexity of homomorphism and constraint satisfaction problems seen from the other side

We give a complexity theoretic classification of homomorphism problems for graphs and, more generally, relational structures obtained by restricting the left hand side structure in a homomorphism. For every class C of structures, let HOM(C,−) be the problem of deciding whether a given structure A ∈C has a homomorphism to a given (arbitrary) structure ß. We prove that, under some complexity theoretic assumption from parameterized complexity theory, HOM(C,−) is in polynomial time if and only if C has bounded tree width modulo homomorphic equivalence. Translated into the language of constraint satisfaction problems, our result yields a characterization of the tractable structural restrictions of constraint satisfaction problems. Translated into the language of database theory, it implies a characterization of the tractable instances of the evaluation problem for conjunctive queries over relational databases.

Searching the k-change neighborhood for TSP is W[1]-hard

We show that searching the k-change neighborhood is W[1]-hard for metric TSP, which means that finding the best tour in the k-change neighborhood essentially requires complete search (modulo some complexity-theoretic assumptions).

Self-improved gaps almost everywhere for the agnostic approximation of monomials

Given a learning sample, we focus on the hardness of finding monomials having low error, inside the interval bounded below by the smallest error achieved by a monomial (the best rule), and bounded above by the error of the default class (the poorest rule). It is well-known that when its lower bound is zero, it is an easy task to find, in linear time, a monomial with zero error. What we prove is that when this bound is not zero, regardless of the location of the default class in ( 0 , 1 / 2 ) , it becomes a huge complexity burden to beat significantly the default class. In fact, under some complexity-theoretical assumptions, it may already be hard to beat the trivial approximation ratios, even when relaxing the time complexity constraint to be quasi-polynomial or sub-exponential. Our results also hold with uniform weights over the examples.

On the complexity of nonrecursive XQuery and functional query languages on complex values

This article studies the complexity of evaluating functional query languages for complex values such as monad algebra and the recursion-free fragment of XQuery. We show that monad algebra, with equality restricted to atomic values, is complete for the class TA[2 O ( n ) , O ( n )] of problems solvable in linear exponential time with a linear number of alternations if the query is assumed to be part of the input. The monotone fragment of monad algebra with atomic value equality but without negation is NEXPTIME-complete. For monad algebra with deep value equality, that is, equality of complex values, we establish TA[2 O ( n ) , O ( n )] lower and exponential-space upper bounds. We also study a fragment of XQuery, Core XQuery, that seems to incorporate all the features of a query language on complex values that are traditionally deemed essential. A close connection between monad algebra on lists and Core XQuery (with “child” as the only axis) is exhibited. The two languages are shown expressively equivalent up to representation issues. We show that Core XQuery is just as hard as monad algebra with respect to query and combined complexity. As Core XQuery is NEXPTIME-hard, the best-known techniques for processing such problems require exponential amounts of working memory and doubly exponential time in the worst case. We present a property of queries---the lack of a certain form of composition---that virtually all real-world XQueries have and that allows for query evaluation in PSPACE and thus singly exponential time. Still, we are able to show for an important special case---Core XQuery with equality testing restricted to atomic values---that the composition-free language is just as expressive as the language with composition. Thus, under widely-held complexity-theoretic assumptions, the language with composition is an exponentially more succinct version of the composition-free language.

How to fool an unbounded adversary with a short key

The symmetric encryption problem which manifests itself when two parties must securely transmit a message m with a short shared secret key is considered in conjunction with a computationally unbounded adversary. As the adversary is unbounded, any encryption scheme must leak information about m; in particular, the mutual information between m and its ciphertext cannot be zero. Despite this, a family of encryption schemes is presented that guarantee that for any message space in {0,1}/sup n/ with minimum entropy n-/spl lscr/ and for any Boolean function h:{0,1}/sup n/ /spl rarr/ {0,1}, no adversary can predict h(m) from the ciphertext of m with more than 1/n/sup /spl omega/(1)/ advantage; this is achieved with keys of length /spl lscr/+/spl omega/(logn). In general, keys of length /spl lscr/+s yield a bound of 2/sup -/spl Theta/(s)/ on the advantage. These encryption schemes rely on no unproven assumptions and can be implemented efficiently. Applications of this to cryptosystems based on complexity-theoretic assumptions are discussed and, in addition, a simplified proof of a fundamental "elision lemma" of Goldwasser and Micali is provided.

Exact learning of DNF formulas using DNF hypotheses

We show the following: (a) For any ε > 0 , log ( 3 + ε ) n -term DNF cannot be polynomial-query learned with membership and strongly proper equivalence queries. (b) For sufficiently large t, t-term DNF formulas cannot be polynomial-query learned with membership and equivalence queries that use t 1 + ε -term DNF formulas as hypotheses, for some ε < 1 (c) Read-thrice DNF formulas are not polynomial-query learnable with membership and proper equivalence queries. (d) log n -term DNF formulas can be polynomial-query learned with membership and proper equivalence queries. (This complements a result of Bshouty, Goldman, Hancock, and Matar that log n -term DNF can be so learned in polynomial time.) Versions of (a)–(c) were known previously, but the previous versions applied to polynomial- time learning and used complexity theoretic assumptions. In contrast, (a)–(c) apply to polynomial- query learning, imply the results for polynomial-time learning, and do not use any complexity-theoretic assumptions.

Upper and Lower Bounds for Randomized Search Heuristics in Black-Box Optimization

Randomized search heuristics like local search, tabu search, simulated annealing, or all kinds of evolutionary algorithms have many applications. However, for most problems the best worst-case expected run times are achieved by more problem-specific algorithms. This raises the question about the limits of general randomized search heuristics. Here a framework called black-box optimization is developed. The essential issue is that the problem but not the problem instance is knownto the algorithm which can collect information about the instance only by asking for the value of points in the search space. All known randomized search heuristics fit into this scenario. Lower bounds on the black-box complexity of problems are derived without complexity theoretical assumptions and are compared with upper bounds in this scenario.

Comparing the succinctness of monadic query languages over finite trees

We study the succinctness of monadic second-order logic and a variety of monadic fixed point logics on trees. All these languages are known to have the same expressive power on trees, but some can express the same queries much more succinctly than others. For example, we show that, under some complexity theoretic assumption, monadic second-order logic is non-elementarily more succinct than monadic least fixed point logic, which in turn is non-elementarily more succinct than monadic datalog. Succinctness of the languages is closely related to the combined and parameterised complexity of query evaluation for these languages.

The complexity of counting homomorphisms seen from the other side

For every class of relational structures C, let HOM(C,_) be the problem of deciding whether a structure A∈C has a homomorphism to a given arbitrary structure B. Grohe has proved that, under a certain complexity-theoretic assumption, HOM(C,_) is solvable in polynomial time if and only if the cores of all structures in C have bounded tree-width. We prove (under a weaker complexity-theoretic assumption) that the corresponding counting problem #HOM(C,_) is solvable in polynomial time if and only if all structures in C have bounded tree-width. This answers an open question posed by Grohe.

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.

On Interpolation and Automatization for Frege Systems

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC0-Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC0-Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-sized TC0-Frege. As a corollary, we obtain that TC0-Frege, as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integers is hard). We also show under the same hardness assumption that the k-provability problem for Frege systems is hard.

Creating Strong, Total, Commutative, Associative One-Way Functions from Any One-Way Function in Complexity Theory

Rabi and Sherman presented novel digital signature and unauthenticated secret-key agreement protocols, developed by themselves and by Rivest and Sherman. These protocols use strong, total, commutative (in the case of multiparty secret-key agreement), associative one-way functions as their key building blocks. Although Rabi and Sherman did prove that associative one-way functions exist if P≠NP, they left as an open question whether any natural complexity-theoretic assumption is sufficient to ensure the existence of strong, total, commutative, associative one-way functions. In this paper, we prove that if P≠NP then strong, total, commutative, associative one-way functions exist.

L-Printable Sets

A language is if there is a logspace algorithm which, on input 1n, prints all members in the language of length n. Following the work of Allender and Rubinstein [SIAM J. Comput., 17 (1988), pp. 1193--1202] on P-printable sets, we present some simple properties of the sets. This definition of L-printable is robust and allows us to give alternate characterizations of the sets in terms of tally sets and Kolmogorov complexity. In addition, we show that a regular or context-free language is if and only if it is sparse, and we investigate the relationship between sets, L-rankable sets (i.e., sets A having a logspace algorithm that, on input x, outputs the number of elements of A that precede x in the standard lexicographic ordering of strings), and the sparse sets in L. We prove that under reasonable complexity-theoretic assumptions, these three classes of sets are all different. We also show that the class of sets of small generalized Kolmogorov space complexity is exactly the class of sets that are L-isomorphic to tally languages.