This paper introduces basic concepts of annealing-based quantum computing, also known as adiabatic quantum computing (AQC) and quantum annealing (QA), and surveys what is known about this novel computing paradigm. Extensive empirical research on physical quantum annealing processers built by D-Wave Systems has exposed many interesting features and properties. However, because of longstanding differences between abstract and empirical approaches to the study of computational performance, much of this work may not be considered relevant to questions of interest to complexity theory; by the same token, several theoretical results in quantum computing may be considered irrelevant to practical experience. To address this communication gap, this paper proposes models of computation and of algorithms that lie on a scale of instantiation between pencil-and-paper abstraction and physical device. Models at intermediate points on these scales can provide a common language, allowing researchers from both ends to communicate and share their results. The paper also gives several examples of common terms that have different technical meanings in different regions of this highly multidisciplinary field, which can create barriers to effective communication across disciplines.

We discuss how mathematical semantics has evolved, and suggest some new directions for future work. As an example, we discuss some recent work on encapsulating model comparison games as comonads, in the context of finite model theory.

In resource allocation games, selfish players share resources that are needed in order to fulfill their objectives. The cost of using a resource depends on the load on it. In the traditional setting, the players make their choices concurrently and in one-shot. That is, a strategy for a player is a subset of the resources. We introduce and study dynamic resource allocation games. In this setting, the game proceeds in phases. In each phase each player chooses one resource. A scheduler dictates the order in which the players proceed in a phase, possibly scheduling several players to proceed concurrently. The game ends when each player has collected a set of resources that fulfills his objective. The cost for each player then depends on this set as well as on the load on the resources in it – we consider both congestion and cost-sharing games. We argue that the dynamic setting is the suitable setting for many applications in practice. We study the stability of dynamic resource allocation games, where the appropriate notion of stability is that of subgame perfect equilibrium, study the inefficiency incurred due to selfish behavior, and also study problems that are particular to the dynamic setting, like constraints on the order in which resources can be chosen or the problem of finding a scheduler that achieves stability.

Recent work by the authors equips Petri occurrence nets (PN) with probability distributions which fully replace nondeterminism. To avoid the so-called confusion problem, the construction imposes additional causal dependencies which restrict choices within certain subnets called structural branching cells (s-cells). Bayesian nets (BN) are usually structured as partial orders where nodes define conditional probability distributions. In the paper, we unify the two structures in terms of Symmetric Monoidal Categories (SMC), so that we can apply to PN ordinary analysis techniques developed for BN. Interestingly, it turns out that PN which cannot be SMC-decomposed are exactly s-cells. This result confirms the importance for Petri nets of both SMC and s-cells.

This paper investigates a new form of delegation for multiparty session calculi. Usually, delegation allows a session participant to appoint a participant in another session to act on her behalf. This means that delegation is inherently an inter-session mechanism, which requires session interleaving. Hence delegation falls outside the descriptive power of global types, which specify single sessions. As a consequence, properties such as deadlock-freedom or lock-freedom are difficult to ensure in the presence of delegation. Here we adopt a different view of delegation, by allowing participants to delegate tasks to each other within the same multiparty session. This way, delegation occurs within a single session (internal delegation) and may be captured by its global type. To increase flexibility in the use of delegation, our calculus uses connecting communications, which allow optional participants in the branches of choices. By these means, we are able to express conditional delegation. We present a session type system based on global types with internal delegation, and show that it ensures the usual safety properties of multiparty sessions, together with a progress property.

Proving behavioural equivalences in higher-order languages is a difficult task, because interactions involve complex values, namely terms of the language. In coinductive (i.e., bisimulation-like) techniques for these languages, a useful enhancement is the ‘up-to context’ reasoning, whereby common pieces of context in related terms are factorised out and erased. In higher-order process languages, however, such techniques are rare, as their soundness is usually delicate and difficult to establish. In this paper we adapt the technique of unique solution of equations, that implicitly captures ‘up-to context’ reasoning, to the setting of the Higher-order π-calculus. Equations are written and solved with respect to normal bisimilarity, chosen both because of its efficiency — its clauses do not require universal quantifications on terms supplied by the external observer — and because of the challenges it poses on the ‘up-to context’ reasoning and that already show up when proving its congruence properties.

CCS-like calculi can be viewed as an extension of classical automata with communication primitives. We are interested here to follow this principle, applied to tree-automata. It naturally yields a calculus of branching processes (CBP), where the continuations of communications are allowed to branch according to the arity of the communication channel. After introducing the calculus with a reduction semantics we show that CBP can be “implemented” by a fully compositional LTS semantics. We argue that CBP offers an interesting tradeoff between calculi with a fixed communication topology à la CCS and calculi with dynamic connectivity such as the π-calculus.

We consider the problem of approximating the smallest 2-vertex connected spanning subgraph (2VCSS) of a 2-vertex connected directed graph, and provide new efficient algorithms. We provide two linear-time algorithms, the first based on a linear-time test for 2-vertex connectivity and divergent spanning trees, and the second based on low-high orders, that correspondingly give 3- and 2-approximations. Then we show that these linear-time algorithms can be combined with an algorithm of Cheriyan and Thurimella that achieves a 3/2-approximation. The combined algorithms preserve the 3/2 approximation guarantee of the Cheriyan-Thurimella algorithm and improve its running time from O(m2) to O(mn+n2), for a digraph with n vertices and m edges. Finally, we present an experimental evaluation of the above algorithms for a variety of input data. The experimental results show that our linear-time algorithms perform very well in practice. Furthermore, the experiments show that the combined algorithms not only improve the running time of the Cheriyan-Thurimella algorithm, but it may also compute a better solution.

This paper studies the average complexity on the number of comparisons for sorting algorithms. Its information-theoretic lower bound is nlg⁡n−1.4427n+O(log⁡n). For many efficient algorithms, the first nlg⁡n term is easy to achieve and our focus is on the (negative) constant factor of the linear term. The current best value is −1.3999 for the MergeInsertion sort. Our new value is −1.4106, narrowing the gap by some 25%. An important building block of our algorithm is “two-element insertion,” which inserts two elements A and B, A

In this paper, we show that every Schnyder drawing is a greedy embedding. Schnyder drawings are used to represent planar (maximal) graphs. It is a way of getting coordinates in R2 given a graph G=(V,E) such that the representation is planar. The Schnyder technique leads to a family of representations and previous results show that a particular representation may be chosen such that the drawing has additional properties like being greedy or monotone. In this article, we relax the definition of greediness to a definition that does not rely on the geometry and the Euclidean distance in R2, but rather on the combinatorial graph G. The construction of greedy paths valid for all Schnyder representations shows that, provided the relaxed definition, every Schnyder drawing is a greedy embedding.

We investigate the least number of distinct palindromic sub-arrays in two-dimensional words over a finite alphabet Σ={a1,a2⋯,aq} for a given alphabet size q. We discuss the case for both periodic as well as aperiodic words.

The Riemann Hypothesis is reformulated as the statement that particular explicitly presented register machine with 29 registers and 130 instructions never halts.

Conditional Value-at-Risk, denoted as CVaRα, is becoming the prevailing measure of risk over two paramount economic domains: the insurance domain and the financial domain; α∈(0,1) is the confidence level. In this work, we study the strategic equilibria for an economic system modeled as a game, where risk-averse players seek to minimize the Conditional Value-at-Risk of their costs. Concretely, in a CVaRα-equilibrium, the mixed strategy of each player is a best-response. We establish two significant properties of CVaRα at equilibrium: (1) The Optimal-Value property: For any best-response of a player, each mixed strategy in the support gives the same cost to the player. This follows directly from the concavity of CVaRα in the involved probabilities, which we establish. (2) The Crawford property: For every α, there is a 2-player game with no CVaRα-equilibrium. The property is established using the Optimal-Value property and a new functional property of CVaRα, called Weak-Equilibrium-for-VaRα, we establish. On top of these properties, we show, as one of our two main results, that deciding the existence of a CVaRα-equilibrium is strongly NP-hard even for 2-player games. As our other main result, we show the strong NP-hardness of deciding the existence of a V-equilibrium, over 2-player minimization games, for any valuation V with the Optimal-Value and the Crawford properties. This result has a rich potential since we prove that the very significant and broad class of strictly quasiconcave valuations has the Optimal-Value property.

Domain theory has a long history of applications in theoretical computer science and mathematics. In this article, we explore the relation of domain theory to probability theory and stochastic processes. The goal is to establish a theory in which Polish spaces are replaced by domains, and measurable maps are replaced by Scott-continuous functions. We illustrate the approach by recasting one of the fundamental results of stochastic process theory – Skorohod's Representation Theorem – in domain-theoretic terms. We anticipate the domain-theoretic version of results like Skorohod's Theorem will improve our understanding of probabilistic choice in computational models, and help devise models of probabilistic programming, with its focus on programming languages that support sampling from distributions where the results are applied to Bayesian reasoning.

Despite the improved accuracy of deep neural networks, the discovery of adversarial examples has raised serious safety concerns. In this paper, we study two variants of pointwise robustness, the maximum safe radius problem, which for a given input sample computes the minimum distance to an adversarial example, and the feature robustness problem, which aims to quantify the robustness of individual features to adversarial perturbations. We demonstrate that, under the assumption of Lipschitz continuity, both problems can be approximated using finite optimisation by discretising the input space, and the approximation has provable guarantees, i.e., the error is bounded. We then show that the resulting optimisation problems can be reduced to the solution of two-player turn-based games, where the first player selects features and the second perturbs the image within the feature. While the second player aims to minimise the distance to an adversarial example, depending on the optimisation objective the first player can be cooperative or competitive. We employ an anytime approach to solve the games, in the sense of approximating the value of a game by monotonically improving its upper and lower bounds. The Monte Carlo tree search algorithm is applied to compute upper bounds for both games, and the Admissible A⁎ and the Alpha-Beta Pruning algorithms are, respectively, used to compute lower bounds for the maximum safety radius and feature robustness games. When working on the upper bound of the maximum safe radius problem, our tool demonstrates competitive performance against existing adversarial example crafting algorithms. Furthermore, we show how our framework can be deployed to evaluate pointwise robustness of neural networks in safety-critical applications such as traffic sign recognition in self-driving cars.

We provide a proofs-as-concurrent-programs interpretation for a large class of intermediate logics that can be formalized by cut-free hypersequent calculi. Obtained by adding classical disjunctive tautologies to intuitionistic logic, these logics are used to type concurrent λ-calculi by Curry–Howard correspondence; each of the calculi features a specific communication mechanism, enhanced expressive power when compared to the λ-calculus, and implements forms of code mobility. We thus confirm Avron's 1991 thesis that intermediate logics formalizable by hypersequent calculi can serve as basis for concurrent λ-calculi.

An instance of the Connected Maximum Cut problem consists of an undirected graph G=(V,E) and the goal is to find a subset of vertices S⊆V that maximizes the number of edges in the cut δ(S) such that the induced graph G[S] is connected. We present the first non-trivial Ω(1log⁡n) approximation algorithm for the Connected Maximum Cut problem in general graphs using novel techniques. We then extend our algorithm to edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in contrast to the classical Max-Cut problem that can be solved in polynomial time on planar graphs, we show that the Connected Maximum Cut problem remains NP-hard on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the Connected Maximum Cut problem on planar graphs and more generally on bounded genus graphs.

The hardness of discrete logarithm problem over finite fields is the security foundation of many cryptographic protocols. When the characteristic of the finite field is medium or large, the state-of-art algorithms for solving the corresponding problem are the number field sieve and its variants. In 2016, Kim and Barbulescu presented the extended tower number field sieve, which achieves a new complexity in the medium prime case and imposes a new estimation of the security of concrete parameters in certain cryptosystems such as pairing-based cryptosystems. In this paper, a refined analysis to this algorithm is given as follows. – Firstly, a uniform formula is given for the total complexity of the extended tower number field sieve. For a given polynomial selection method, this formula can directly give the complexity in this case. – Then, a method is proposed to improve the computation in the smoothing phase by exploring subfield structures when the extension degree is composite. – At last, the complexity of the descent phase is analyzed when sieving over degree-one polynomials and high-degree polynomials respectively and it is shown still negligible compared to the improved smoothing phase.

We introduce a hybrid metaphor for the visualization of the reconciliations of co-phylogenetic trees, that are mappings among the nodes of two trees with constraints on the leaves. The typical application is the visualization of the co-evolution of hosts and parasites in biology. Our strategy combines a space-filling and a node-link approach. Differently from traditional methods, it guarantees an unambiguous and downward representation whenever the reconciliation is time-consistent (i.e., biologically-feasible). We address the problem of the minimization of the number of crossings in the representation, by giving a characterization of planar instances and by establishing the complexity of the problem. Finally, we propose heuristics for computing representations with few crossings.

Given a set B of d-dimensional boxes (i.e., axis-aligned hyperrectangles), a minimum coverage kernel is a subset of B of minimum size covering the same region as B. Computing it is NP-hard, but as for many similar NP-hard problems (e.g., Box Cover, and Orthogonal Polygon Covering), the problem becomes solvable in polynomial time under restrictions on B. We show that computing minimum coverage kernels remains NP-hard even when restricting the graph induced by the input to a highly constrained class of graphs. Alternatively, we present two polynomial-time approximation algorithms for this problem: one deterministic with an approximation ratio within O(log⁡n), and one randomized with an improved approximation ratio within O(lg⁡OPT) (with high probability).

Let A be an alphabet and SP⋄(A) denote the class of all countable N-free partially ordered sets labeled by A, in which chains are scattered linear orderings and antichains are finite. We characterize the rational languages of SP⋄(A) by means of logic. We define an extension of monadic second-order logic by Presburger arithmetic, named P-MSO, such that a language L of SP⋄(A) is rational if and only if L is the language of a sentence of P-MSO, with effective constructions from one formalism to the other. As a corollary, the P-MSO theory of SP⋄(A) is decidable.

The k-abelian equivalence of words, counting the numbers of occurrences of factors of length at most k, has been analyzed in recent years from several different directions. We continue this analysis. The k-abelian equivalence classes are known to constitute a monoid. Hence, equations over these monoids are well defined. We show that these monoids satisfy a compactness property: each system of equations with a finite number of unknowns is equivalent to some of its finite subsystems. We give two proofs for this compactness result. One is based the fact that the monoid can be embedded into the (multiplicative) monoid of matrices, and the other directly on linear algebra. The former method allows the application of Hilbert's basis theorem. The latter one, in turn, allows to conclude an upper bound for the size of the finite subsystem.

We study the minimum connected sensor cover problem (MIN-CSC) and the budgeted connected sensor cover (Budgeted-CSC) problem, both motivated by important applications (e.g., reduce the communication cost among sensors) in wireless sensor networks. In both problems, we are given a set of sensors and a set of target points in the Euclidean plane. In MIN-CSC, our goal is to find a set of sensors of minimum cardinality, such that all target points are covered, and all sensors can communicate with each other (i.e., the communication graph is connected). We obtain a constant factor approximation algorithm, assuming that the ratio between the sensor radius and communication radius is bounded. In Budgeted-CSC problem, our goal is to choose a set of B sensors, such that the number of targets covered by the chosen sensors is maximized and the communication graph is connected. We also obtain a constant approximation under the same assumption.

Rearrangeable Clos networks have been studied for a long time due to their many applications, as well as their theoretical interest. In a rearrangeable Clos network, a newly requested connection may be blocked by existing connections. However, this blocking is eliminated by adequately rearranging some other existing connections. In this operation, an interesting topic is the sufficient number of rearrangements required to eliminate the blocking. Despite previous studies, the sufficient number of rearrangements has only been found for limited cases and has not been completely determined for generic parameter values. This paper analyses the number of rearrangements using the connection chain concept, which clearly and efficiently represents a sequence of connections to be rearranged. The analysis assumes the employment of a rearrangement algorithm, which eliminates the blocking using the shortest connection chain. The usage of the shortest connection chain results in the minimum number of rearrangements. As a result, this paper determines a new bound on the number of rearrangements for a parameter range that has not been considered in any previous studies. In addition, this paper examines the condition for which the system is unblocked via one rearrangement.

A binary decision diagram (BDDs) is a compact data structure used to represent a Boolean function, which facilitates scalable constructions of Boolean functions using reversible logic gates. Motivated by the scalable synthesis approach, this paper proposes an effective scheme for transforming the BDD representation of a Boolean function into a reversible circuit composed by reversible logic elements. Unlike a logic gate, a reversible logic element carries a memory to record a finite number of states. Especially, logic circuits composed by reversible elements can operate in asynchronous mode, thereby no need of a clock signal to drive all elements operating simultaneously.

In this work we determine the metric dimension of Zn×Zn×Zn as ⌊3n/2⌋ for all n≥2. We prove this result by investigating a variant of Mastermind. Mastermind is a famous two-player game that has attracted much attention in the literature in recent years. In particular we consider the static (also called non-adaptive) black-peg variant of Mastermind. The game is played by a codemaker and a codebreaker. Given c colors and p pegs, the principal rule is that the codemaker has to choose a secret by assigning colors to the pegs, i.e., the secret is a p-tuple of colors, and the codebreaker asks a number of questions all at once. Like the secret, a question is a p-tuple of colors chosen from the c available colors. The codemaker then answers all of those questions by telling the codebreaker how many pegs in each question are correctly colored. The goal is to find the minimal number of questions that allows the codebreaker to determine the secret from the received answers. We present such a strategy for this game for p=3 pegs and an arbitrary number c≥2 of colors using ⌊3c/2⌋+1 questions, which we prove to be both feasible and optimal. The minimal number of questions required for p pegs and c colors is easily seen to be equal to the metric dimension of Zcp plus 1 which proves our main result.

We present a uniform method for translating an arbitrary nondeterministic finite automaton (NFA) into a deterministic mass action input/output chemical reaction network (I/O CRN) that simulates it. The I/O CRN receives its input as a continuous time signal consisting of concentrations of chemical species that vary to represent the NFA's input string in a natural way. The I/O CRN exploits the inherent parallelism of chemical kinetics to simulate the NFA in real time with a number of chemical species that is linear in the size of the NFA. We prove that the simulation is correct and that it is robust with respect to perturbations of the input signal, the initial concentrations of species, the output (decision), and the rate constants of the reactions of the I/O CRN.

While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat, that is, the unnecessary growth of solution lengths, which may slow down the optimization process. So far, the mathematical runtime analysis could not deal well with bloat and required explicit assumptions limiting bloat. In this paper, we provide the first mathematical runtime analysis of a GP algorithm that does not require any assumptions on the bloat. Previous performance guarantees were only proven conditionally for runs in which no strong bloat occurs. Together with improved analyses for the case with bloat restrictions our results show that such assumptions on the bloat are not necessary and that the algorithm is efficient without explicit bloat control mechanism. More specifically, we analyzed the performance of the (1+1) GP on the two benchmark functions Order and Majority. When using lexicographic parsimony pressure as bloat control, we show a tight runtime estimate of O(Tinit+nlog⁡n) iterations both for Order and Majority. For the case without bloat control, the bounds O(Tinitlog⁡Tinit+n(log⁡n)3) and Ω(Tinit+nlog⁡n) (and Ω(Tinitlog⁡Tinit) for n=1) hold for Majority.1

The purpose of this paper is to answer two questions left open in [B. Durand, A. Shen, and N. Vereshchagin, Descriptive Complexity of Computable Sequences, Theoretical Computer Science 171 (2001), pp. 47–58]. Namely, we consider the following two complexities of an infinite computable 0-1-sequence α: C0′(α), defined as the minimal length of a program with oracle 0′ that prints α, and M∞(α), defined as limsupC(α1:n|n), where α1:n denotes the length-n prefix of α and C(x|y) stands for conditional Kolmogorov complexity. We show that C0′(α)⩽M∞(α)+O(1) and M∞(α) is not bounded by any computable function of C0′(α), even on the domain of computable sequences.

A normal form system (NFS) for representing Boolean functions is thought of as a set of stratified terms over a fixed set of connectives. For a fixed NFS A, the complexity of a Boolean function f with respect to A is the minimum of the sizes of terms in A that represent f. This induces a preordering of NFSs: an NFS A is polynomially as efficient as an NFS B if there is a polynomial P with nonnegative integer coefficients such that the complexity of any Boolean function f with respect to A is at most the value of P in the complexity of f with respect to B. In this paper we study monotonic NFSs, i.e., NFSs whose connectives are increasing or decreasing in each argument. We describe the monotonic NFSs that are optimal, i.e., that are minimal with respect to the latter preorder. We show that these minimal monotonic NFSs are all equivalent. Moreover, we address some natural questions, e.g.: does optimality depend on the arity of connectives? Does it depend on the number of connectives used? We show that optimal monotonic NFSs are exactly those that use a single connective or one connective and the negation. Finally, we show that optimality does not depend on the arity of the connectives.

The weak 2-linkage problem for digraphs asks for a given digraph and vertices s1,s2,t1,t2 whether D contains a pair of arc-disjoint paths P1,P2 such that Pi is an (si,ti)-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs [8]. Recently it was shown [3] that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair of constants k1,k2, there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P1,P2 such that Pi is an (si,ti)-path of length no more than d(si,ti)+ki, for i=1,2, where d(si,ti) denotes the length of the shortest (si,ti)-path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P1,P2 to the weak 2-linkage problem where each path Pi has length at most d(si,ti)+clog1+ϵ⁡n for some constant c.

In recent years, the string datasets have increased exponentially, so is the need to process them. Most of these datasets have been deeply rooted in the field of bioinformatics since the entire characteristics of any living organism is encoded in their genes. Genes consist of nucleic bases which will, therefore, makeup the entire genome. A genome is made of a concatenation of different types of nucleic bases. To efficiently extract the information encrypted in these sequences there is a need to use algorithms to decrypt it. Most available methods use the data structure commonly referred to as the suffix tree. They have tremendously evolved over the years, and the on-line construction of the suffix tree is deemed as the best data structure, however, it is not optimal when it comes to finding repeated sequences because of many traversals algorithm will have to do when identifying repeats. To improve the speed and of finding repeats we developed a counter based suffix tree algorithm. Our work presents a novel algorithm of constructing a counter based suffix tree without losing its properties. The counter based suffix tree time complexity is θ(n) where n represents the length of a string. Which is the same as the fastest suffix tree implementation. We have shown that the counter based suffix tree will reduce the search time when identifying repeats. We have proved that a counter based suffix tree can be developed during construction.

The article describes the structural and algorithmic relations between Cartesian trees and Lyndon trees. This leads to a uniform presentation of the Lyndon table of a word corresponding to the Next Nearest Smaller table of a sequence of numbers. It shows how to efficiently compute runs, that is, maximal periodicities occurring in a word.

In our earlier paper [Peltomäki and Whiteland (2017) [5]], we introduced a symbolic square root map. Every optimal squareful infinite word s contains exactly six minimal squares and can be written as a product of these squares: s=X12X22⋯. The square root s of s is the infinite word X1X2⋯ obtained by deleting half of each square. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). The dynamics of the square root map on a Sturmian subshift are well understood. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The main results are characterizations of periodic points and the limit set. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case.

Let A be an arbitrary alphabet and let θ be an (anti-)automorphism of A⁎ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under θ (θ-invariant for short) that is, languages L satisfying θ(L)⊆L. We establish an extension of the famous defect theorem. With regard to the so-called notion of completeness, we provide a series of examples of finite complete θ-invariant codes. Moreover, we establish a formula which allows to embed any non-complete θ-invariant code into a complete one. As a consequence, in the family of the so-called thin θ-invariant codes, maximality and completeness are two equivalent notions.

As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph G=(V,E), a partition Π of V is said to be a k-partition generator of G if any pair of different vertices u,v∈V is distinguished by at least k vertex sets of Π, i.e., there exist at least k vertex sets S1,…,Sk∈Π such that d(u,Si)≠d(v,Si) for every i∈{1,…,k}. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for k∈{1,…,r}.

The notion of metric dimension, dim⁡(G), of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdimG(v), the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z∈S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdimG(v) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim(G), is min⁡{cdimG(v):v∈V(G)}. Noting that 1≤dim⁡(G)≤cdim(G)≤cdimG(v)≤|V(G)|−1 for any vertex v of G, we show the existence of a pair (G,v) such that cdimG(v) takes all positive integer values from dim⁡(G) to |V(G)|−1, as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdimG(v)∈{1,|V(G)|−1}. We show that cdim(G)=2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim⁡(H)=2. We also characterize trees and unicyclic graphs G satisfying cdim(G)=dim⁡(G). We show that cdim(G)−dim⁡(G) can be arbitrarily large. We determine cdim(G) and cdimG(v) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems.

We propose a simple idea for improving the randomized algorithm of Hertli (2014) [2] for the Unique 3SAT problem. Using recently developed techniques (Hertli (2015) [3] and Scheder–Steinberger (2017) [7]), we can derive from this algorithm currently the fastest randomized algorithm for the general 3SAT problem. Though the efficiency improvement is extremely small, we hope that this idea would lead to better improvements.

We propose a new approach to large-scale machine learning, learning over compressed data: First compress the training data somehow and then employ various machine learning algorithms on the compressed data, with the hope that the computation time is significantly reduced when the training data is well compressed. As a first step toward this approach, we consider a variant of the Zero-Suppressed Binary Decision Diagram (ZDD) as the data structure for representing the training data, which is a generalization of the ZDD by incorporating non-determinism. For the learning algorithm to be employed, we consider a boosting algorithm called AdaBoost⁎ and its precursor AdaBoost. In this paper, we give efficient implementations of the boosting algorithms whose running times (per iteration) are linear in the size of the given ZDD.

Graph compression is a data analysis technique that consists in the replacement of parts of a graph by more concise structural patterns in order to reduce its description length. It notably provides interesting exploration tools for the study of real, large-scale, and complex graphs which cannot be grasped at first glance. This article proposes a framework for the compression of temporal graphs, that is for the compression of graphs that evolve with time. This framework first builds on a simple and limited scheme, exploiting structural equivalence for the lossless compression of static graphs, then generalises it to the lossy compression of link streams, a recent formalism for the study of temporal graphs. Such generalisation builds on the natural extension of (bidimensional) relational data by the addition of a third temporal dimension. Moreover, we introduce an information-theoretic measure to quantify and to control the information that is lost during compression, as well as an algebraic characterisation of the space of possible compression patterns to enhance the expressiveness of the initial compression scheme. These contributions lead to the definition of a combinatorial optimisation problem, that is the Lossy Multistream Compression Problem, for which we provide an exact algorithm.

A shuffle of two words is formed by interleaving the characters into a new word, keeping the characters of each word in order. A word is a shuffle square if it is a shuffle of two identical words. Deciding whether a word is a shuffle square has been proved to be NP-complete independently by Buss and Soltys [5] and Rizzi and Vialette [20], the former proving the result for alphabets as small as 9 letters. We prove in this paper that deciding whether a binary word is a shuffle square is NP-complete.

A set of vertices W in a graph G is called resolving if for any two distinct x,y∈V(G), there is v∈W such that dG(v,x)≠dG(v,y), where dG(u,v) denotes the length of a shortest path between u and v in the graph G. The metric dimension md(G) of G is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether md(G)⩽k, is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by k was proved to be W[2]-hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether md(G)⩽n−k, where n is the order of G. We prove that the dual parameterization admits (a) a kernel with at most 6(k+1) vertices and (b) a randomized algorithm of runtime O⁎(4k+o(k)). Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number vc(G) of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by vc(G)+k, unless NP ⊆ coNP/poly. Our reduction also gives evidence for non-existence of polynomial Turing kernels. We also prove that Metric Dimension parameterized by bandwidth or cutwidth does not admit a polynomial kernel, unless NP ⊆ coNP/poly. Finally, using Eppstein's results (2015) we show that Metric Dimension parameterized by max-leaf number does admit a polynomial kernel.

A k-coloring of a graph G is a k-partition Π={S1,…,Sk} of V(G) into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u,v belonging to the same color Si, the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic number χNL(G) is the minimum cardinality of a neighbor-locating coloring of G. We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n≥5 with neighbor-locating chromatic number n or n−1. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs.

Resolving sets are designed to locate an object in a network by measuring the distances to the object. However, if there are more than one object present in the network, this can lead to wrong conclusions. To overcome this problem, we introduce the concept of solid-resolving sets. In this paper, we study the structure and constructions of solid-resolving sets. In particular, we classify the forced vertices with respect to a solid-resolving set. We also give bounds on the solid-metric dimension utilizing concepts like the Dilworth number, the boundary of a graph, and locating-dominating sets. It is also shown that deciding whether there exists a solid-resolving set with a certain number of elements is an NP-complete problem.

Broadcast domination models the idea of covering a network of cities by transmitters of varying powers while minimizing the total cost of the transmitters used to achieve full coverage. To be exact, let G be a connected graph of order at least two with vertex set V(G) and edge set E(G). Let d(x,y), e(v), and diam(G), respectively, denote the length of a shortest x−y path in G, the eccentricity of a vertex v in G, and the diameter of G. A function f:V(G)→{0,1,…,diam(G)} is called a broadcast if f(v)≤e(v) for each v∈V(G). A broadcast f is called a dominating broadcast of G if, for each vertex u∈V(G), there exists a vertex v∈V(G) such that f(v)>0 and d(u,v)≤f(v). The broadcast domination number, γb(G), of G is the minimum of ∑v∈V(G)f(v) over all dominating broadcasts f on G. Let G1 and G2 be two disjoint copies of a graph G, and let σ:V(G1)→V(G2) be a bijection. Then a permutation graph Gσ=(V,E) has vertex set V=V(G1)∪V(G2) and edge set E=E(G1)∪E(G2)∪{uv:v=σ(u)}. For a connected graph G of order at least two, we prove the sharp bounds 2≤γb(Gσ)≤2γb(G); we give an example showing that there is no function h such that γb(G)

A link stream is a sequence of pairs of the form (t,{u,v}), where t∈N represents a time instant and u≠v. Given an integer γ, the γ-edge between vertices u and v, starting at time t, is the set of temporally consecutive edges defined by {(t′,{u,v})|t′∈〚t,t+γ−1〛}. We introduce the notion of temporal matching of a link stream to be an independent γ-edge set belonging to the link stream. We show that the problem of computing a temporal matching of maximum size is NP-hard as soon as γ>1. We depict a kernelization algorithm parameterized by the solution size for the problem. As a byproduct we also give a 2-approximation algorithm. Both our 2-approximation and kernelization algorithms are implemented and confronted to link streams collected from real world graph data. We observe that finding temporal matchings is a sensitive question when mining our data from such a perspective as: managing peer-working when any pair of peers X and Y are to collaborate over a period of one month, at an average rate of at least two email exchanges every week. We furthermore design a link stream generating process by mimicking the behavior of a random moving group of particles under natural simulation, and confront our algorithms to these generated instances of link streams. All the implementations are open source.

We investigate for temporal graphs the computational complexity of separating two distinct vertices s and z by vertex deletion. In a temporal graph, the vertex set is fixed but the edges have (discrete) time labels. Since the corresponding Temporal (s,z)-Separation problem is NP-complete, it is natural to investigate whether relevant special cases exist that are computationally tractable. To this end, we study restrictions of the underlying (static) graph—there we observe polynomial-time solvability in the case of bounded treewidth—as well as restrictions concerning the “temporal evolution” along the time steps. Systematically studying partially novel concepts in this direction, we identify sharp borders between tractable and intractable cases.

We investigate the benefits of using multiple channels in wireless networks, under the full-duplex multi-packet reception model of communication. The main question we address is the following: Is a speedup linear in the number of channels available, for some interesting communication primitive? We provide a positive answer to this interrogative for the Information Exchange Problem, in which up to k arbitrary nodes have information they intend to share with the entire network. In particular, we give non-adaptive deterministic protocols for both the scenario in which the channels provide the transmitting stations with the feedback on whether their transmissions have been successful and for the scenario in which channels provide no such feedback. To this aim, we devise and exploit a new combinatorial structure that generalizes well known combinatorial tools, widely used in the area of data-exchange in multiple-access channels (i.e., strongly selective families, selectors, and related mathematical objects). For our new combinatorial structures we provide both existential results and randomized algorithms to generate them. We also prove non-existence results showing that our protocol for the model with feedback is optimal, whereas that for the no-feedback scenario uses a number of time slots that exceeds the lower bound by a log⁡k factor. Leveraging on properties of error correcting codes, we show that for an infinite set of the relevant parameters n and k, one can construct our combinatorial structure for the no-feedback scenario in polynomial time and of minimum length.

A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each robot can be in walking mode or in searching mode. It is assumed that each robot's maximum allowed searching speed is strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it in searching mode. The Beachcombers' Problem consists in developing efficient schedules (algorithms) for the robots which collectively search all the points of the given domain as fast as possible. We consider searching schedules in the following one-dimensional geometric domains: the cycle of a known circumference L, the finite straight line segment of a known length L, and the semi-infinite line [0,+∞). We first consider the online Beachcombers' Problem (i.e. the scenario when the robots do not know in advance the length of the segment to be searched), where the robots are initially collocated at the origin of a semi-infinite line. It is sought to design a schedule A with maximum speed S, defined as S=infℓ⁡ℓtA(ℓ), where tA(ℓ) denotes the time when the search of the segment [0,ℓ] is completed under A. We consider a discrete and a continuous version of the problem, depending on whether the infimum is taken over ℓ∈N⁎ or ℓ≥1. We prove that the LeapFrog algorithm, which was proposed in Czyzowicz et al. (2015) [12], is in fact optimal in the discrete case. This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general continuous online setting. For the offline version of the Beachcombers' Problem (i.e. the scenario when the robots know in advance the length of the segment to be searched), we consider the t-source Beachcombers' Problem (i.e. all robots start from a fixed number t≥1 of starting positions) on the cycle and on the finite segment. For the t-source Beachcombers' Problem on the cycle, we show that the structure of the optimal solutions is identical to the structure of the optimal solutions to the 2t-source Beachcombers' Problem on a finite segment. In consequence, by using results from Czyzowicz et al. (2014) [13], we prove that the 1-source Beachcombers' Problem on the cycle is NP-hard, and we derive approximation algorithms for the problem. For the t-source variant of the Beachcombers' Problem on the cycle and on the finite segment, we also derive efficient approximation algorithms. One important contribution of our work is that, in all variants of the offline Beachcombers' Problem that we discuss, we allow the robots to change direction of movement and search points of the domain on both sides of their respective starting positions. This represents a significant generalization compared to the model considered in Czyzowicz et al. (2014) [13], in which each robot had a fixed direction of movement that was specified as part of the solution to the problem. We manage to prove that changes of direction do not help the robots achieve optimality.

Computational biology is mainly concerned with discovering an object from a given set of observations that are supposed to be good approximations of the real object. Two important steps here are to define a way to measure the distance between different objects and to calculate the distance between two given objects. The main problem is then to find an object that has the minimum total distance to the given observations. We study two NP-hard problems formulated in computational biology. The minimum tree cut/paste distance problem asks for the minimum number of cut/paste operations we need to transform a tree to another tree. The minimum common integer partition problem asks for a minimum-cardinality integer partition of a number that refines two given integer partitions of the same number. We give parameterized algorithms for both problems.

Let G=(V,E,w) be a Δβ-metric graph with a distance function w(⋅,⋅) on V such that w(v,v)=0, w(u,v)=w(v,u), and w(u,v)≤β⋅(w(u,x)+w(x,v)) for all u,v,x∈V. Given a positive integer p, let H be a spanning subgraph of G satisfying the conditions that vertices (hubs) in C⊂V form a clique of size at most p in H, vertices (non-hubs) in V∖C form an independent set in H, and each non-hub v∈V∖C is adjacent to exactly one hub in C. Define dH(u,v)=w(u,f(u))+w(f(u),f(v))+w(v,f(v)) where f(u) and f(v) are hubs adjacent to u and v in H respectively. Notice that if u is a hub in H then w(u,f(u))=0. Let r(H)=∑u,v∈VdH(u,v) be the routing cost of H. The Single Allocation at most p-Hub Center Routing problem is to find a spanning subgraph H of G such that r(H) is minimized. In this paper, we show that the Single Allocation at most p-Hub Center Routing problem is NP-hard in Δβ-metric graphs for any β>1/2. Moreover, we give 2β-approximation algorithms running in time O(n2) for any β>1/2 where n is the number of vertices in the input graph. Finally, we show that the approximation ratio of our algorithms is at least Ω(β), and we examine the structure of any potential o(β)-approximation algorithm.

We extend the framework for complexity of operators in analysis devised by Kawamura and Cook (2012) to allow for the treatment of a wider class of representations. The main novelty is to endow represented spaces of interest with an additional function on names, called a parameter, which measures the complexity of a given name. This parameter generalises the size function which is usually used in second-order complexity theory and therefore also central to the framework of Kawamura and Cook. The complexity of an algorithm is measured in terms of its running time as a second-order function in the parameter, as well as in terms of how much it increases the complexity of a given name, as measured by the parameters on the input and output side. As an application we develop a rigorous computational complexity theory for interval computation. In the framework of Kawamura and Cook the representation of real numbers based on nested interval enclosures does not yield a reasonable complexity theory. In our new framework this representation is polytime equivalent to the usual Cauchy representation based on dyadic rational approximation. By contrast, the representation of continuous real functions based on interval enclosures is strictly smaller in the polytime reducibility lattice than the usual representation, which encodes a modulus of continuity. Furthermore, the function space representation based on interval enclosures is optimal in the sense that it contains the minimal amount of information amongst those representations which render evaluation polytime computable.

The locating-chromatic number of a graph G(V,E) is the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices not contained in the same partition class. Determining the locating-chromatic number of any tree is a difficult task. In this paper, we propose an algorithm to compute the upper bound on the locating-chromatic number of any tree. To do so, we decompose a tree into caterpillars and then compute the upper bound of the locating-chromatic number of this tree in terms of the ones for these caterpillars.

In this paper, a general framework for enumerating every element in a graph class is given. The main feature of this framework is that it is designed to enumerate only nonisomorphic graphs in a graph class. Applying this framework to the classes of interval graphs and permutation graphs, we give efficient enumeration algorithms for these graph classes such that each element in the class is output in a polynomial time delay. The experimental results are also given. The catalogs of graphs in these graph classes are also provided.

Suppose that we are given a graph whose each vertex is either a supply vertex or a demand vertex and is assigned a nonnegative integer supply or demand value. We consider partitioning G into connected components by removing edges from G so that each connected component has exactly one supply vertex and there exists a flow in each connected component satisfying the supply/demand constraints. The problem that determines the existence of such a partition is called the partition problem. Ito et al. (2005) showed that the partition problem is NP-complete in general and it can be solved in linear time if the given graph is a tree. When the graph does not have such a partition, we scale the demand values uniformly by scale factor r so that the obtained graph has a desired partition. The maximum supply rate problem is the problem that finds the maximum value of such r. Whereas the maximum supply rate problem is NP-hard in general in the same way as the partition problem, Morishita and Nishizeki (2015) gave a weakly polynomial-time algorithm for the problem on trees. In this paper, we give a first strongly polynomial-time algorithm for the maximum supply rate problem on trees. Our algorithm is based on the dynamic programming technique, in which we compute “surplus” and “deficit” of the supply in subproblems from leaves to the root. We use piecewise linear functions of r to represent them, and one of our important contributions is to bound the size of the representation of each function.

We consider the computational complexity of reconfiguration problems, in which one is given two combinatorial configurations satisfying some constraints, and is asked to transform one into the other using elementary operations, while satisfying the constraints at all times. Such problems appear naturally in many contexts, such as model checking, motion planning, enumeration, sampling, and recreational mathematics. We provide hardness results for problems in this family, in which the constraints and operations are particularly simple. More precisely, we prove the PSPACE-completeness of the following decision problems: • Given two satisfying assignments of a planar monotone instance of NAE 3-SAT, can one assignment be transformed into the other by a sequence of variable flips such that the formula remains satisfied at every step? • Given two subsets of a set S of integers with the same sum, can one subset be transformed into the other by adding or removing at most three elements of S at a time, such that the intermediate subsets also have the same sum? • Given two points in {0,1}n contained in a polytope P specified by a constant number of linear inequalities, is there a path in the n-hypercube connecting the two points and contained in P? These problems can be interpreted as reconfiguration analogues of standard problems in NP. Interestingly, the sets of instances that appear as input to the reconfiguration problems in our reductions lie in P. In particular, the elements of S and the coefficients of the inequalities defining P can be restricted to have logarithmic bit-length.

We study the time complexity of single and multi token broadcast in adversarial dynamic radio networks. Initially, k tokens (which are k pieces of information) are distributed among the n nodes of a network and all the tokens need to be disseminated to all the nodes in the network. We first consider the single-token broadcast problem (i.e., the case k=1). By presenting upper and lower bounds, we show that the time complexity of single-token broadcast depends on the amount of stability and connectivity of the dynamic network topology and on the adaptiveness of the adversary providing the dynamic topology. Then, we give two generic algorithms which allow to transform generalized forms of single-token broadcast algorithms into multi-token broadcast (k-token broadcast) algorithms. Based on these generic algorithms, we obtain k-token broadcast algorithms for a number of different dynamic network settings. For one of the modeling assumptions, our algorithm is complemented by a lower bound which shows that the upper bound is close to optimal.

We consider the problem of locating a set of k sinks on a path network with general edge capacities that minimizes the sum of the evacuation times of all evacuees. We first present an O(knlog4⁡n) time algorithm when the edge capacities are non-uniform, where n is the number of vertices. We then present an O(knlog3⁡n) time algorithm when the edge capacities are uniform. We also present an O(nlog⁡n) time algorithm for the special case where k=1 and the edge capacities are non-uniform.

We study a parameter of bipartite graphs called readability, introduced by Chikhi et al. (Discrete Applied Mathematics, 2016) and motivated by applications of overlap graphs in bioinformatics. The behavior of the parameter is poorly understood. The complexity of computing it is open and it is not known whether the decision version of the problem is in NP. The only known upper bound on the readability of a bipartite graph (following from a work of Braga and Meidanis, LATIN 2002) is exponential in the maximum degree of the graph. Graphs that arise in bioinformatics applications have low readability. In this paper, we focus on graph families with readability o(n), where n is the number of vertices. We show that the readability of n-vertex bipartite chain graphs is between Ω(log⁡n) and O(n). We give an efficiently testable characterization of bipartite graphs of readability at most 2 and completely determine the readability of grids, showing in particular that their readability never exceeds 3. As a consequence, we obtain a polynomial time algorithm to determine the readability of induced subgraphs of grids. One of the highlights of our techniques is the appearance of Euler's totient function in the analysis of the readability of bipartite chain graphs. We also develop a new technique for proving lower bounds on readability, which is applicable to dense graphs with a large number of distinct degrees.

In a recent breakthrough STOC 2015 paper, a continuous diffusion process was considered on hypergraphs (which has been refined in a recent JACM 2018 paper) to define a Laplacian operator, whose spectral properties satisfy the celebrated Cheeger's inequality. However, one peculiar aspect of this diffusion process is that each hyperedge directs flow only from vertices with the maximum density to those with the minimum density, while ignoring vertices having strict in-between densities. In this work, we consider a generalized diffusion process, in which vertices in a hyperedge can act as mediators to receive flow from vertices with maximum density and deliver flow to those with minimum density. We show that the resulting Laplacian operator still has a second eigenvalue satisfying the Cheeger's inequality. Our generalized diffusion model shows that there is a family of operators whose spectral properties are related to hypergraph conductance, and provides a powerful tool to enhance the development of spectral hypergraph theory. Moreover, since every vertex can participate in the new diffusion model at every instant, this can potentially have wider practical applications.