Elementary net systems (ENS) are the most fundamental class of Petri nets. Given a labeled transition system (TS) A, feasibility is the NP-complete decision problem whether A can be synthesized into an ENS. We analyze the impact of state degree, the number of allowed successors and predecessors of states in A, and event manifoldness, the amount of occurrences that an event can have in A, on the computational complexity of feasibility and the related event state separation property (ESSP) and state separation property (SSP). Feasibility, ESSP and SSP are NP-complete for TSs with state degree one and event manifoldness not less than three and for TSs with state degree and event manifoldness at least two. As we also show that SSP becomes tractable for TSs with state degree one and event manifoldness two, the only cases left open are ESSP and feasibility for the same input restriction.

Similar to the satisfiability (SAT) problem, which can be seen to be the archetypical problem for NP, the quantified Boolean formula problem (QBF) is the archetypical problem for PSPACE. Recently, Atserias and Oliva (2014) showed that, unlike for SAT, many of the well-known decompositional parameters (such as treewidth and pathwidth) do not allow efficient algorithms for QBF. The main reason for this seems to be the lack of awareness of these parameters towards the dependencies between variables of a QBF formula. In this paper we extend the ordinary pathwidth to the QBF-setting by introducing prefix pathwidth, which takes into account the dependencies between variables in a QBF, and show that it leads to an efficient algorithm for QBF. We hope that our approach will help to initiate the study of novel tailor-made decompositional parameters for QBF and thereby help to lift the success of these decompositional parameters from SAT to QBF.

Edge connectivity is a crucial measure of the robustness of a network. Several edge connectivity variants have been proposed for measuring the reliability and fault tolerance of networks under various conditions. Let G be a connected graph, S be a subset of edges in G, and k be a positive integer. If G−S is disconnected and every component has at least k vertices, then S is a k-extra edge-cut of G. The k-extra edge-connectivity, denoted by λk(G), is the minimum cardinality over all k-extra edge-cuts of G. If λk(G) exists and at least one component of G−S contains exactly k vertices for any minimum k-extra edge-cut S, then G is super-λk. Moreover, when G is super-λk, the persistence of G, denoted by ρk(G), is the maximum integer m for which G−F is still super-λk for any set F⊆E(G) with |F|≤m. Previously, bounds of ρk(G) were provided only for k∈{1,2}. This study provides the bounds of ρk(G) for k≥2.

Stellar datacenter networks are a recent generic construction designed to transform a base-graph into a dual-port, server-centric datacenter network. We prove that the S-bisection width of any stellar datacenter network can be obtained from the solution of isoperimetric problems on the base-graph, provided that the base-graph is regular. We extend previous research on the stellar datacenter networks GQ⁎, instantiated with generalized hypercubes, and show that with respect to S-bisection width, GQ⁎ performs well in comparison with the dual-port datacenter network FiConn. Our work develops a strong combinatorial link between graph bisection width and throughput metrics for stellar datacenter networks.

We focus on the computational complexity of regular simple path queries (RSPQs). We consider the following problem RSPQ(L) for a regular language L: given an edge-labeled digraph G and two nodes x and y, is there a simple path from x to y that forms a word belonging to L? We fully characterize the frontier between tractability and intractability for RSPQ(L). More precisely, we prove RSPQ(L) is either , -complete or -complete depending on the language L. We also provide a simple characterization of the tractable fragment in terms of regular expressions. Finally, we also discuss the complexity of deciding whether a language L belongs to the fragment above. We consider several alternative representations of L: DFAs, NFAs or regular expressions, and prove that this problem is -complete for the first representation and -complete for the other two.

Relativizing computations of Turing machines to an oracle is a central concept in the theory of computation, both in complexity theory and in computability theory(!). Inspired by lowness notions from computability theory, Allender introduced the concept of “low for speed” oracles. An oracle A is low for speed if relativizing to A has essentially no effect on computational complexity, meaning that if a decidable language can be decided in time f(n) with access to oracle A, then it can be decided in time poly(f(n)) without any oracle. The existence of non-computable such A's was later proven by Bayer and Slaman, who even constructed a computably enumerable one, and exhibited a number of properties of these oracles. In this paper, we pursue this line of research, answering the questions left by Bayer and Slaman and give further evidence that the class of low for speed oracles is a very rich one.

We obtain a complete classification of graphs H for which the class of (triangle,H)-free graphs is well-quasi-ordered by the induced subgraph relation and an almost complete classification of graphs H for which the class of (triangle,H)-free graphs has bounded clique-width. In particular, we show that for these graph classes, well-quasi-orderability implies boundedness of clique-width. To obtain our results, we further refine a known method based on canonical decomposition. This leads to a new decomposition technique that is applicable to both notions, well-quasi-orderability and clique-width.

The 2-D Tucker search problem is shown to be PPA-hard under many-one reductions; therefore it is complete for PPA. The same holds for k-D Tucker for all k≥2. This corrects a claim in the literature that the Tucker search problem is in PPAD.

We study the derivational complexity of context-free and context-sensitive grammars by counting the maximal number of non-regular and non-context-free rules used in a derivation, respectively. The degree of non-regularity/non-context-freeness of a language is the minimum degree of non-regularity/non-context-freeness of context-free/context-sensitive grammars generating it. A language has finite degree of non-regularity iff it is regular. We give a condition for deciding whether the degree of non-regularity of a given unambiguous context-free grammar is finite. The problem becomes undecidable for arbitrary linear context-free grammars. The degree of non-regularity of unambiguous context-free grammars generating non-regular languages as well as that of grammars generating deterministic context-free languages that are not regular is of order Ω(n). Context-free non-regular languages of sublinear degree of non-regularity are presented. A language has finite degree of non-context-freeness iff it is context-free. Context-sensitive grammars with a quadratic degree of non-context-freeness are more powerful than those of a linear degree.

We present the first study of non-deterministic weighted automata under probabilistic semantics. In this semantics words are random events, generated by a Markov chain, and functions computed by weighted automata are random variables. We consider the probabilistic questions of computing the expected value and the cumulative distribution for such random variables. The exact answers to the probabilistic questions for non-deterministic automata can be irrational and are uncomputable in general. To overcome this limitation, we propose approximation algorithms for the probabilistic questions, which work in exponential time in the size of the automaton and polynomial time in the size of the Markov chain and the given precision. We apply this result to show that non-deterministic automata can be effectively determinised with respect to the standard deviation metric.

Truth Discovery is an important learning problem arising in data analytics related fields. It concerns about finding the most trustworthy information from a dataset acquired from a number of unreliable sources. The problem has been extensively studied and a number of techniques have already been proposed. However, all of them are of heuristic nature and do not have any quality guarantee. In this paper, we formulate the problem as a high dimensional geometric optimization problem, called Entropy based Geometric Variance. Relying on a number of novel geometric techniques, we further discover new insights to this problem. We show, for the first time, that the truth discovery problem can be solved with guaranteed quality of solution. Particularly, it is possible to achieve a (1+ϵ)-approximation within nearly linear time under some reasonable assumptions. We expect that our algorithm will be useful for other data related applications.

We consider subexponential algorithms finding weighted homomorphisms from intersection graphs of curves (string graphs) with n vertices to a fixed graph H. We provide a complete dichotomy: if H has no two vertices sharing two common neighbors, then the problem can be solved in time 2O(n2/3log⁡n), otherwise there is no subexponential algorithm, assuming the ETH. Then we consider locally constrained homomorphisms. We show that for each target graph H, the locally injective and locally bijective homomorphism problems can be solved in time 2O(nlog⁡n) in string graphs. For locally surjective homomorphisms we show a dichotomy for H being a path or a cycle. If H is P3 or C4, then the problem can be solved in time 2O(n2/3log3/2⁡n) in string graphs; otherwise, assuming the ETH, there is no subexponential algorithm. As corollaries, we obtain new results concerning the complexity of homomorphism problems in Pt-free graphs.

We analyse the complexity of approximate counting constraint satisfactions problems #CSP(F), where F is a set of nonnegative rational-valued functions of Boolean variables. A complete classification is known if F contains arbitrary unary functions. We strengthen this result by fixing any permissive strictly increasing unary function and any permissive strictly decreasing unary function, and requiring only those to be in F. The resulting classification is employed to characterise the complexity of a wide range of two-spin problems, fully classifying the ferromagnetic case. Furthermore, we also consider what happens if only the pinning functions are assumed to be in F. We show that any set of functions for which pinning is not sufficient to recover the two kinds of permissive unaries must either have a very simple range, or must satisfy a certain monotonicity condition. We exhibit a non-trivial example of a set of functions satisfying the monotonicity condition.

For a finite fixed collection of graphs F, the F-M-Deletion problem consists in, given a graph G and an integer k, decide whether there exists S⊆V(G) with |S|≤k such that G∖S does not contain any of the graphs in F as a minor. We provide lower bounds under the ETH on the smallest function fF such that F-M-Deletion can be solved in time fF(tw)⋅nO(1) on n-vertex graphs, where tw denotes the treewidth of G. We first prove that for any F containing connected graphs of size at least two, fF(tw)=2Ω(tw), even if G is planar. Our main result is that if F contains a single connected graph H that is either P5 or is not a minor of the banner, then fF(tw)=2Ω(tw⋅log⁡tw).

The problem if a given configuration of a pushdown automaton (PDA) is bisimilar with some (unspecified) finite-state process is shown to be decidable. The decidability is proven in the framework of first-order grammars, which are given by finite sets of labelled rules that rewrite roots of first-order terms. The framework is equivalent to PDA where also deterministic (i.e. alternative-free) epsilon-steps are allowed, hence to the model for which Sénizergues showed an involved procedure deciding bisimilarity (1998, 2005). Such a procedure is here used as a black-box part of the algorithm. The result extends the decidability of the regularity problem for deterministic PDA that was shown by Stearns (1967), and later improved by Valiant (1975) regarding the complexity. The decidability question for nondeterministic PDA, answered positively here, had been open (as indicated, e.g., by Broadbent and Göller, 2012).

Consider a finite set A and n≥1. We study complete simulation of transformations of An, also known as automata networks. For m≥n, a transformation of Am is n-complete of size m if it may simulate every transformation of An by updating one register at a time. Using tools from memoryless computation, we establish that there is no n-complete transformation of size n, but there is one of size n+1. By studying various constructions, we conjecture that the maximal time of simulation of any n-complete transformation is at least 2n. We also investigate the time and size of sequentially n-complete transformations, which may simulate every finite sequence of transformations of An. Finally, we show that there is no n-complete transformation updating all registers in parallel, but there exists one updating all but one register in parallel. This illustrates the strengths and weaknesses of sequential and parallel models of computation.

In the Graph bipartization (or Odd Cycle Transversal) problem, the objective is to decide whether a given graph G can be made bipartite by the deletion of k vertices for some given k. The parameterized complexity of Odd Cycle Transversal was resolved in the breakthrough paper of Reed, Smith and Vetta [Operations Research Letters, 2004], who developed an algorithm running in time O(3kkmn). The question of improving the dependence on the input size to linear, which was another long standing open problem in the area, was resolved by Iwata et al. [SICOMP 2016] and Ramanujan and Saurabh [TALG 2017], who presented O(4k(m+n)) and 4kkO(1)(m+n) algorithms respectively. In this paper, we obtain a faster algorithm that runs in time 3kkO(1)(m+n) and hence preserves the linear dependence on the input size while nearly matching the dependence on k incurred by the algorithm of Reed, Smith and Vetta.

Here we study cooperating distributed systems (CD-systems) of restarting automata that are very restricted: they are deterministic, they cannot rewrite, but only delete symbols, they restart immediately after performing a delete operation, they are stateless, and they have a read/write window of size 1 only, that is, these are stateless deterministic R(1)-automata. We study the expressive power of these systems by relating the class of languages that they accept by mode =1 computations to other well-studied language classes, showing in particular that this class only contains semi-linear languages. Our model can be viewed as a nondeterministic finite-state acceptor with translucent letters, that is, it processes its input in a different way than the usual left-to-right order. In this way all commutative semi-linear languages, and in fact all rational trace languages, can be accepted. In addition, we investigate the closure and non-closure properties of the class of languages accepted by our model and some of its algorithmic properties.

Given a boolean n × n matrix A we consider arithmetic circuits for computing the transformation x ↦ Ax over different semirings. Namely, we study three circuit models: monotone OR-circuits, monotone SUM-circuits (addition of non-negative integers), and non-monotone XOR-circuits (addition modulo 2). Our focus is on separating OR-circuits from the two other models in terms of circuit complexity: We show how to obtain matrices that admit OR-circuits of size O(n), but require SUM-circuits of size Ω(n3/2/log2n).We consider the task of rewriting a given OR-circuit as a XOR-circuit and prove that any subquadratic-time algorithm for this task violates the strong exponential time hypothesis.

We present the first implementation of a distributed clock generation scheme for Systems-on-Chip that recovers from an unbounded number of arbitrary transient faults despite a large number of arbitrary permanent faults. We devise self-stabilizing hardware building blocks and a hybrid synchronous/asynchronous state machine enabling metastability-free transitions of the algorithm's states. We provide a comprehensive modeling approach that permits to prove, given correctness of the constructed low-level building blocks, the high-level properties of the synchronization algorithm (which have been established in a more abstract model). We believe this approach to be of interest in its own right, since this is the first technique permitting to mathematically verify, at manageable complexity, high-level properties of a fault-prone system in terms of its very basic components. We evaluate a prototype implementation, which has been designed in VHDL, using the Petrify tool in conjunction with some extensions, and synthesized for an Altera Cyclone FPGA.

We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay forever in a given set of states, and its dual, the reachability objective to reach a given set of states. First, we present a simple proof of the fact that in concurrent reachability games, for all [Formula: see text], memoryless ε-optimal strategies exist. A memoryless strategy is independent of the history of plays, and an ε-optimal strategy achieves the objective with probability within ε of the value of the game. In contrast to previous proofs of this fact, our proof is more elementary and more combinatorial. Second, we present a strategy-improvement (a.k.a. policy-iteration) algorithm for concurrent games with reachability objectives. Finally, we present a strategy-improvement algorithm for turn-based stochastic games (where each player selects moves in turns) with safety objectives. Our algorithms yield sequences of player-1 strategies which ensure probabilities of winning that converge monotonically (from below) to the value of the game.