Exact Complexity Research Articles

Exact Complexity Certification of Active-Set Methods for Quadratic Programming

Active-set methods are recognized to often outperform other methods in terms of speed and solution accuracy when solving small-size quadratic programming (QP) problems, making them very attractive in embedded linear model predictive control (MPC) applications. A drawback of active-set methods is the lack of tight bounds on the worst-case number of iterations, a fundamental requirement for their implementation in a real-time system. Extensive simulation campaigns provide an indication of the expected worst-case computation load, but not a complete guarantee. This paper solves such a certification problem by proposing an algorithm to compute the exact bound on the maximum number of iterations and floating point operations required by a state-of-the-art dual active-set QP solver. The algorithm is applicable to a given QP problem whose linear term of the cost function and right-hand side of the constraints depend linearly on a vector of parameters, as in the case of linear MPC. In addition, a new solver is presented that combines explicit and implicit MPC ideas, guaranteeing improvements of the worst-case computation time. The ability of the approach to exactly quantify memory and worst-case computation requirements is tested on a few MPC examples, also highlighting when online optimization should be preferred to explicit MPC.

On the Complexity of Model Checking for Syntactically Maximal Fragments of the Interval Temporal Logic HS with Regular Expressions

In this paper, we investigate the model checking (MC) problem for Halpern and Shoham's interval temporal logic HS. In the last years, interval temporal logic MC has received an increasing attention as a viable alternative to the traditional (point-based) temporal logic MC, which can be recovered as a special case. Most results have been obtained under the homogeneity assumption, that constrains a proposition letter to hold over an interval if and only if it holds over each component state. Recently, Lomuscio and Michaliszyn proposed a way to relax such an assumption by exploiting regular expressions to define the behaviour of proposition letters over intervals in terms of their component states. When homogeneity is assumed, the exact complexity of MC is a difficult open question for full HS and for its two syntactically maximal fragments AA'BB'E' and AA'EB'E'. In this paper, we provide an asymptotically optimal bound to the complexity of these two fragments under the more expressive semantic variant based on regular expressions by showing that their MC problem is AEXP_pol-complete, where AEXP_pol denotes the complexity class of problems decided by exponential-time bounded alternating Turing Machines making a polynomially bounded number of alternations.

Separations in Query Complexity Based on Pointer Functions

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total Boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k , which achieves R 0 ( f ) = O ( D ( f ) 0.7537… ). We show that this is false by giving an example of a total Boolean function f on n bits whose deterministic query complexity is Ω( n ) while its zero-error randomized query complexity is Õ(√ n ). We further show that the quantum query complexity of the same function is Õ( n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total Boolean function g on n variables that has zero-error randomized query complexity Ω( n / log ( n )) and bounded-error randomized query complexity R ( g ) = Õ(√ n ). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E ( g ) = Õ(√ n ). These functions show that the relations D ( f ) = O ( R 1 ( f ) 2 ) and R 0 ( f ) = Õ( R ( f ) 2 ) are optimal, up to polylogarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R , and a 4th-power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson, which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

The exact autocorrelation distribution and 2-adic complexity of a class of binary sequences with almost optimal autocorrelation

Pseudo-random sequences with good statistical properties, such as low autocorrelation, high linear complexity and large 2-adic complexity, have been used in designing reliable stream ciphers. In this paper, we obtain the exact autocorrelation distribution of a class of binary sequences with three-level autocorrelation and analyze the 2-adic complexity of this class of sequences. Our results show that the 2-adic complexity of such a binary sequence with period N is at least (N + 1) − log2 (N + 1). We further show that it is maximal for infinitely many cases. This indicates that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).

Parameterized model checking of rendezvous systems

Parameterized model checking is the problem of deciding if a given formula holds irrespective of the number of participating processes. A standard approach for solving the parameterized model checking problem is to reduce it to model checking finitely many finite-state systems. This work considers the theoretical power and limitations of this technique. We focus on concurrent systems in which processes communicate via pairwise rendezvous, as well as the special cases of disjunctive guards and token passing; specifications are expressed in indexed temporal logic without the next operator; and the underlying network topologies are generated by suitable formulas and graph operations. First, we settle the exact computational complexity of the parameterized model checking problem for some of our concurrent systems, and establish new decidability results for others. Second, we consider the cases where model checking the parameterized system can be reduced to model checking some fixed number of processes, the number is known as a cutoff. We provide many cases for when such cutoffs can be computed, establish lower bounds on the size of such cutoffs, and identify cases where no cutoff exists. Third, we consider cases for which the parameterized system is equivalent to a single finite-state system (more precisely a Büchi word automaton), and establish tight bounds on the sizes of such automata.

DEGREES OF CATEGORICITY ON A CONE VIAη-SYSTEMS

AbstractWe investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is${\rm{\Delta }}_\alpha ^0 $-complete for someα. To prove this, we extend Montalbán’sη-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinalαand a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and another copy${\cal B}$in the cone is a c.e. degree in${\rm{\Delta }}_\alpha ^0\left( {\cal B} \right)$. In each of our theorems the cone in question is clearly described in the beginning of the proof, so it is easy to see how the theorems can be viewed as general theorems with certain effectiveness conditions.

Ordinal recursive complexity of Unordered Data Nets

Data Nets are a version of colored Petri nets in which tokens carry data from an infinite and linearly ordered domain. This is a very expressive class, though coverability and termination remain decidable. Those problems have recently been proven complete for the Fωωω class in the fast growing complexity hierarchy. We characterize the exact complexity of Unordered Data Nets (UDN), a subclass of Data Nets with unordered data. We bound the length of bad sequences in well-quasi orderings of multisets over tuples of naturals by adapting the analogous result by Schmitz and Schnoebelen for words over a finite alphabet. These bounds imply that both problems are in Fωω. We prove that this result is tight by constructing UDN that weakly compute fast-growing functions and their inverses. This is the first complete problem for Fωω with an underlying wqo not based on finite words over a finite alphabet.

On optimal approximability results for computing the strong metric dimension

The strong metric dimension of a graph was first introduced by Sebö and Tannier (2004) as an alternative to the (weak) metric dimension of graphs previously introduced independently by Slater (1975) and by Harary and Melter (1976), and has since been investigated in several research papers. However, the exact worst-case computational complexity of computing the strong metric dimension has remained open beyond being NP-complete. In this communication, we show that the problem of computing the strong metric dimension of a graph of n nodes admits a polynomial-time 2-approximation, admits a O∗(20.287n)-time exact computation algorithm, admits a O(1.2738k+nk)-time exact computation algorithm if the strong metric dimension is at most k, does not admit a polynomial time (2−ε)-approximation algorithm assuming the unique games conjecture is true, does not admit a polynomial time (105−21−ε)-approximation algorithm assuming P≠NP, does not admit a O∗(2o(n))-time exact computation algorithm assuming the exponential time hypothesis is true, and does not admit a O∗(no(k))-time exact computation algorithm if the strong metric dimension is at most k assuming the exponential time hypothesis is true.

On the orthogonal rank and impossibility of quantum round elimination

After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given Bob her two possible replies at once without listening to him at all, and have him select which one applies. Motivated by a conjecture stating that this form of “round elimination” is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number d for which one can label each vertex with a nonzero d-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp(n) lower bound on the orthogonal rank of the graph on {0, 1} n in which two strings are adjacent if they have Hamming distance at least n/2. In combination with previous work, this implies an affirmative answer to the above conjecture.

The exact information-based complexity of smooth convex minimization

We obtain a new lower bound on the information-based complexity of first-order minimization of smooth and convex functions. We show that the bound matches the worst-case performance of the recently introduced Optimized Gradient Method (Drori and Teboulle, 2013; Kim and Fessler, 2015), thereby establishing that the bound is tight and can be realized by an efficient algorithm. The proof is based on a novel construction technique of smooth and convex functions.

Decidability and Complexity of Fuzzy Description Logics

Fuzzy description logics (FDLs) have been introduced to represent concepts for which membership cannot be determined in a precise way, i.e., where instead of providing a strict border between being a member and not being a member, it is more appropriate to model a gradual change from membership to non-membership. First approaches for reasoning in FDLs where based either on a reduction to reasoning in classical description logics (DLs) or on adaptations of reasoning approaches for DLs to the fuzzy case. However, it turned out that these approaches in general do not work if expressive terminological axioms, called general concept inclusions (GCIs), are available in the FDL. The goal of this project was a comprehensive study of the border between decidability and undecidability for FDLs with GCIs, as well as determining the exact complexity of the decidable logics. As a result, we have provided an almost complete classification of the decidability and complexity of FDLs with GCIs.

On the complexity of bounded time and precision reachability for piecewise affine systems

Reachability for piecewise affine systems is known to be undecidable, starting from dimension 2. In this paper we investigate the exact complexity of several decidable variants of reachability and control questions for piecewise affine systems. We show in particular that the region-to-region bounded time versions leads to NP-complete or co-NP-complete problems, starting from dimension 2. We also prove that a bounded precision version leads to PSPACE-complete problems.

The exact complexity of projective image matching

Projective image matching is an important image processing task. Given digital images A and B, the challenge is to find a projective transformation f for A such that it most closely resembles B. Previous research led to a polynomial time algorithm that elaborately searches the so-called dictionary D(A) of all projective transformations of A using a preprocessed data structure. This paper shows that projective image matching is even TCO-complete by applying a much simpler parallel way of browsing D(A). This reveals further insight into the problem's structural properties and relates it to fundamental computer operations like integer multiplication and division.

Higher Randomness and Forcing with Closed Sets

Kechris showed in Kechris (Trans. Am. Math. Soc. 202, 259---297, 1975) that there exists a largest ź11${\Pi ^{1}_{1}}$ set of measure 0. An explicit construction of this largest ź11${\Pi ^{1}_{1}}$ nullset has later been given in Hjorth and Nies (J. Lond. Math. Soc. 75(2), 495---508, 2007). Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned in Chong and Yu (J. Symb. Log. 80(04), 1131---1148, 2015) and Yu (Fundam. Math. 215, 219---231, 2011)). In this paper, we refute this conjecture and show that this nullset is merely Σ30${\Sigma }^{0}_{3}$. Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is Σ30${\Sigma }^{0}_{3}$. To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties.

Computing abelian complexity of binary uniform morphic words

Although a lot of research has been done on the factor complexity (also called subword complexity) of morphic words obtained as fixed points of iterated morphisms, there has been little development in exploring algorithms that can efficiently compute their abelian complexity. The factor complexity counts the number of factors of a given length n, while the abelian complexity counts that number up to letter permutation. We propose and analyze a simple O(n) algorithm for quickly computing the exact abelian complexities for all indices from 1 up to n, when considering binary uniform morphisms. Using our algorithm we also analyze the structure in the abelian complexity for that class of morphisms. Our main result implies, in particular, that the infinite word over the alphabet {−1,0,1} constructed from the consecutive forward differences of the abelian complexity of a fixed point of a binary uniform morphism is in fact an automatic sequence with the same morphic length. Since the proof produces morphisms that typically contain many redundant letters, we present an efficient algorithm to eliminate them in order to simplify the morphisms and to see the patterns produced more clearly.

CNF and DNF succinct graph encodings

It is well-known that succinct encodings of computational problems – using circuits or formulas to encode large instances – generally result in an exponential complexity blow-up compared to their original complexity.We introduce a new way to encode graph problems, based on CNF or DNF formulas. We show that – contrary to the other existing succinct models – there are examples of problems whose complexity does not increase when encoded in the new form, or increases to an intermediate complexity class less powerful than the exponential blow up.We also study the complexity of the succinct versions of the Graph Isomorphism problem. We show that all the versions are hard for PSPACE. Although the exact complexity of these problems is still unknown, we show that under most existing succinct models the different versions of the problem are equivalent. We also give an algorithm for the DNF encoded version of GI whose running time depends mainly on the number of terms in the succinct representation.

The Exact Complexity of the First-Order Logic Definability Problem

We study the definability problem for first-order logic, denoted by FO-D ef . The input of FO-D ef is a relational database instance I and a relation R ; the question to answer is whether there exists a first-order query Q (or, equivalently, a relational algebra expression Q ) such that Q evaluated on I gives R as an answer. Although the study of FO-D ef dates back to 1978, when the decidability of this problem was shown, the exact complexity of FO-D ef remains as a fundamental open problem. In this article, we provide a polynomial-time algorithm for solving FO-D ef that uses calls to a graph-isomorphism subroutine (or oracle). As a consequence, the first-order definability problem is found to be complete for the class GI of all problems that are polynomial-time Turing reducible to the graph isomorphism problem, thus closing the open question about the exact complexity of this problem. The technique used is also applied to a generalized version of the problem that accepts a finite set of relation pairs, and whose exact complexity was also open; this version is also found to be GI -complete.

Unary NP-hardness of minimizing the number of tardy jobs with deadlines

For single machine scheduling to minimize the number of tardy jobs with deadlines, Lawler showed in 1983 that the problem is binary NP-hard. But the exact complexity (unary NP-hard or pseudo-polynomial-time solvability) is a long- standing open problem. We show in this paper that the problem is unary NP-hard. Our research also implies that the scheduling problem for finding an optimal schedule to minimize the number of tardy jobs that also satisfies the restriction of deadlines is unary NP-hard. As a consequence, some multi-agent scheduling problems related to minimizing the number of tardy jobs and maximum lateness are unary NP-hard.

Complexities of Some Problems on Multi-agent Scheduling on a Single Machine

We study the computational complexities of three problems on multi-agent scheduling on a single machine. Among the three problems, the computational complexities of the first two problems were still open and the last problem was shown to be unary NP-hard in the literature. We show in this paper that the first two problems are unary NP-hard. We also show that the unary NP-hardness proof for the last problem in the literature is invalid, and so, the exact complexity of the problem is still open.

Automata column

In this column, Sylvain Schmitz summarizes two recent advances on the complexity of the reachability problem for vector addition systems with states. This is one of the most celebrated decidable problems in theoretical computer science, and until now, there was absolutely no upper bound on the complexity. It was only known that the algorithms run in finite time. The first advance described in this column, by Leroux and Schmitz, is a computable upper bound on the running time of the (original) algorithm for the problem. Admittedly, the upper bound puts the Ackermann function to shame -- read the great column to see what it is --- but it's an upper bound. The second advance, by Blondin, Finkel, Göller, Haase and McKenzie, is finding the exact complexity, namely PSPACE complete, of the two dimensional case, something that has also been open for several decades.