Abstract We study the parameterized complexity of the problem to decide whether a given natural number n satisfies a given $\Delta _0$ -formula $\varphi (x)$ ; the parameter is the size of $\varphi $ . This parameterization focusses attention on instances where n is large compared to the size of $\varphi $ . We show unconditionally that this problem does not belong to the parameterized analogue of $\mathsf {AC}^0$ . From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameterized halting problem. Some of these upper bounds follow assuming that $I\Delta _0$ proves the MRDP theorem in a certain weak sense.

Abstract Let $\mathscr {C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś–Tarski Theorem from classical model theory implies that $\mathscr {C}$ is definable in first-order logic by a sentence $\varphi $ if and only if $\mathscr {C}$ has a finite set of forbidden induced finite subgraphs. This result provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from $\varphi $ the corresponding forbidden induced subgraphs. This machinery fails on finite graphs as shown by our results: – There is a class $\mathscr {C}$ of finite graphs that is definable in first-order logic and closed under induced subgraphs but has no finite set of forbidden induced subgraphs. – Even if we only consider classes $\mathscr {C}$ of finite graphs that can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from a first-order sentence $\varphi $ that defines $\mathscr {C}$ and the size of the characterization cannot be bounded by $f(|\varphi |)$ for any computable function f. Besides their importance in graph theory, the above results also significantly strengthen similar known theorems for arbitrary structures.

It is well known [Lovász, 67] that up to isomorphism a graph~$G$ is determined by the homomorphism counts $\hom(F, G)$, i.e., the number of homomorphisms from $F$ to $G$, where $F$ ranges over all graphs. Thus, in principle, we can answer any query concerning $G$ with only accessing the $\hom(\cdot,G)$'s instead of $G$ itself. In this paper, we deal with queries for which there is a hom algorithm, i.e., there are finitely many graphs $F_1, \ldots, F_k$ such that for any graph $G$ whether it is a Yes-instance of the query is already determined by the vector\[\overrightarrow{\hom}_{F_1,\ldots,F_k}(G):= \big(\hom(F_1,G),\ldots,\hom(F_k,G)\big),\]where the graphs $F_1, \ldots, F_k$ only depend on $φ$. We observe that planarity of graphs and 3-colorability of graphs, properties expressible in monadic second-order logic, have no hom algorithm. On the other hand, queries expressible as a Boolean combination of universal sentences in first-order logic FO have a hom algorithm. Even though it is not easy to find FO definable queries without a hom algorithm, we succeed to show this for the non-existence of an isolated vertex, a property expressible by the FO sentence $\forall x\exists y Exy$, somehow the ``simplest'' graph property not definable by a Boolean combination of universal sentences.These results provide a characterization of the prefix classes of first-order logic with the property that each query definable by a sentence of the prefix class has a hom algorithm. For adaptive query algorithms, i.e., algorithms that again access $\overrightarrow{\hom}_{F_1,\ldots,F_k}(G)$ but here $F_{i+1}$ might depend on $\hom(F_1,G),\ldots,\hom(F_i,G)$, we show that three homomorphism counts $\hom(\cdot,G)$ are both sufficient and in general necessary to determine the isomorphism type of $G$.

LetCbe a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś-Tarski Theorem from classical model theory implies thatCis definable in first-order logic (FO) by a sentence φ if and only ifChas a finite set of forbidden induced finite subgraphs. It provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from φ the corresponding forbidden induced subgraphs. Our results (a) and (b) show that this machinery fails on finite graphs. (a)There is a class of finite graphs that is definable in FO and closed under induced subgraphs but has no finite set of forbidden induced subgraphs. (b)Even if we only consider classesCof finite graphs that can be characterized by a finite set of forbidden induced subgraphs such a characterization cannot be computed from an FO-sentence φ that definesCand the size of the characterization cannot be bounded by f(|φ|) for any computable function f.Besides their importance in graph theory, our results also significantly strengthen similar known theorems for arbitrary structures.

Shrub-depth is a graph invariant often considered as an extension of tree-depth to dense graphs. We show that the model-checking problem of monadic second-order logic on a class of graphs of bounded shrub-depth can be decided by AC^0-circuits after a precomputation on the formula. This generalizes a similar result on graphs of bounded tree-depth [Y. Chen and J. Flum, 2018]. At the core of our proof is the definability in first-order logic of tree-models for graphs of bounded shrub-depth.

For every graph class {F}, let HomInd({F}) be the problem of deciding whether two given graphs are homomorphism-indistinguishable over {F}, i.e., for every graph F in {F}, the number hom(F, G) of homomorphisms from F to G equals the corresponding number hom(F, H) for H. For several natural graph classes (such as paths, trees, bounded treewidth graphs), homomorphism-indistinguishability over the class has an efficient structural characterization, resulting in polynomial time solvability [H. Dell et al., 2018]. In particular, it is known that two non-isomorphic graphs are homomorphism-indistinguishable over the class {T}_k of graphs of treewidth k if and only if they are not distinguished by k-dimensional Weisfeiler-Leman algorithm, a central heuristic for isomorphism testing: this characterization implies a polynomial time algorithm for HomInd({T}_k), for every fixed k in N. In this paper, we show that there is a polynomial-time-decidable class {F} of undirected graphs of bounded treewidth such that HomInd({F}) is undecidable. Our second hardness result concerns the class {K} of complete graphs. We show that HomInd({K}) is co-NP-hard, and in fact, we show completeness for the class C_=P (under P-time Turing reductions). On the algorithmic side, we show that HomInd({P}) can be solved in polynomial time for the class {P} of directed paths. We end with a brief study of two variants of the HomInd({F}) problem: (a) the problem of lexographic-comparison of homomorphism numbers of two graphs, and (b) the problem of computing certain distance-measures (defined via homomorphism numbers) between two graphs.

A set $D$ of vertices of a graph $G$ is a dominating set if every vertex of $G$ is contained in $D$ or adjacent to some vertex of $D$. The number of vertices in a smallest dominating set of $G$ is denoted by $\gamma(G)$. We prove that, under the assumption ${FPT}\ne {W}[1]$ from parameterized complexity, for any constant $c\in \mathbb{N}^+$ and computable function $f: \mathbb{N}\to \mathbb{N}$ there is no algorithm which on every input graph $G$ finds a dominating set of size at most $c\cdot \gamma(G)$ in $f(\gamma(G))\cdot |G|^{O(1)}$ time. In other words, any constant approximation of the parameterized dominating set problem is ${W}[1]$-hard. Furthermore, assuming the exponential time hypothesis (ETH) [R. Impagliazzo and R. Paturi, J. Comput. System Sci., 62 (2001), pp. 367--375], we can even rule out the existence of a $f(\gamma(G))\cdot |G|^{{({log}\;\gamma(G))}^{{\varepsilon}/{12}}}$-time algorithm which on every input graph $G$ outputs a dominating set of size at most $\sqrt[3+\varepsilon]{{log}\; (\gamma(G))} \cdot \gamma(G)$ for every $0<\varepsilon<1$. Our hardness reduction is built on the second author's recent ${W}[1]$-hardness proof of the biclique problem [B. Lin, The parameterized complexity of $k$-Biclique, in Proceedings of the 26th Annual ACM--SIAM Symposium on Discrete Algorithms, SODA 2015 (San Diego, CA), ACM, New York, SIAM, Philadelphia, 2015, pp. 605--615]. This yields, among other things, a proof without the probabilistically checkable proof (PCP) machinery that the classic dominating set problem has no polynomial time constant approximation under ETH.

The classical view of epistemic logic is that an agent knows all the logical consequences of their knowledge base. This assumption of logical omniscience is often unrealistic and makes reasoning computationally intractable. One approach to avoid logical omniscience is to limit reasoning to a certain belief level, which intuitively measures the reasoning "depth".This paper investigates the computational complexity of reasoning with belief levels. First we show that while reasoning remains tractable if the level is constant, the complexity jumps to PSPACE-complete -- that is, beyond classical reasoning -- when the belief level is part of the input. Then we further refine the picture using parameterized complexity theory to investigate how the belief level and the number of non-logical symbols affect the complexity.

For a class K of finite graphs we consider the following three statements. (i) K has bounded tree-depth. (ii) First-order logic FO has an effective generalized quantifier elimination on K. (iii) The parameterized model checking for FO on K is in para-AC0. We prove that (i) ⟹ (ii) and (ii) ⟺ (iii). All three statements are equivalent if K is closed under taking subgraphs, but not in general.

The complexity of the parameterized halting problem for nondeterministic Turing machines p-Halt is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-Halt is in para-AC0, the parameterized version of the circuit complexity class AC0, then AC0, or equivalently, (+, x)-invariant FO, has a logic. Although it is widely believed that p-Halt ∉. para-AC0, we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊈ LINH. Here, LINH denotes the linear time hierarchy.

For every natural number q let FO_q denote the class of sentences of first-order logic FO of quantifier rank at most q. If a graph property can be defined in FO_q, then it can be decided in time O(n^q). Thus, minimizing q has favorable algorithmic consequences. Many graph properties amount to the existence of a certain set of vertices of size k. Usually this can only be expressed by a sentence of quantifier rank at least k. We use the color coding method to demonstrate that some (hyper)graph problems can be defined in FO_q where q is independent of k. This property of a graph problem is equivalent to the question of whether the corresponding parameterized problem is in the class para-AC^0. It is crucial for our results that the FO-sentences have access to built-in addition and multiplication (and constants for an initial segment of natural numbers whose length depends only on k). It is known that then FO corresponds to the circuit complexity class uniform AC^0. We explore the connection between the quantifier rank of FO-sentences and the depth of AC^0-circuits, and prove that FO_q is strictly contained in FO_{q+1} for structures with built-in addition and multiplication.

We prove that there is no fpt-algorithm that can approximate the dominating set problem with any constant ratio, unless FPT = W[1]. Our hardness reduction is built on the second author's recent W[1]-hardness proof of the biclique problem [25]. This yields, among other things, a proof without the PCP machinery that the classical dominating set problem has no polynomial time constant approximation under the exponential time hypothesis.

We demonstrate some lower bounds for parameterized problems via parameterized classes corresponding to the classical AC^0. Among others, we derive such a lower bound for all fpt-approximations of the parameterized clique problem and for a parameterized halting problem, which recently turned out to link problems of computational complexity, descriptive complexity, and proof theory. To show the first lower bound, we prove a strong AC^0 version of the planted clique conjecture: AC^0-circuits asymptotically almost surely can not distinguish between a random graph and this graph with a randomly planted clique of any size <= n^xi (where 0 <= xi < 1).

It is well-known that least fixed-point logic LFP captures the complexity class PTIME on ordered structures. The ordered conjecture claims that LFP is more expressive than first-order logic FO on every infinite class O of finite ordered structures. We present two methods which yield that LFP is more expressive than FO on various types of classes of ordered structures. The first method, the model-checking method, among others, can be applied for all such classes O of bounded cliquewidth. By the second method, the padding method, we show that for classes O of ``bounded treewidth,'' more precisely, for classes O such that there is a bound for the treewidth of the successor structures associated with the members of O, even DTC is more expressive than FO on O, where DTC denotes the deterministic transitive closure logic, a logic that captures the complexity class L on ordered structures. Furthermore, with the padding method we get that for every infinite class of ordered structures O we have that DTC is more expressive than FO on the class of all ordered sums of pairs of structures in O. Under some complexity theoretic assumption, we prove the existence of a class O of ordered structures such that on O not only LFP is more expressive than FO, but also LFP has the expressive power of existential second-order logic. Furthermore, we characterize those classes of structures whose corresponding class of all ordered versions has bounded treewidth.

Let C denote one of the complexity classes “polynomial time,” “logspace,” or “nondeterministic logspace.” We introduce a logic L (C) inv and show generalizations and variants of the equivalence ( L (C) inv captures C if and only if there is an almost C-optimal algorithm in C for the set Taut of tautologies of propositional logic). These statements are also equivalent to the existence of a listing of subsets in C of Taut by corresponding Turing machines and equivalent to the fact that a certain parameterized halting problem is in the parameterized complexity class XC uni .

There are standard logics DTC, TC, and LFP capturing the complexity classes L, NL, and P on ordered structures, respectively. In we have shown that LFPinv, the "order-invariant least fixed-point logic LFP," captures P (on all finite structures) if and only if there is a listing of the P subsets of the set TAUT of propositional tautologies. We are able to extend the result to listings of the L-subsets (NL-subsets) of TAUT and the logic DTCinv(TCinv). As a byproduct we get that LFPinvcaptures P if DTCinvcaptures L. Furthermore, we show that the existence of a listing of the L-subsets of TAUT is equivalent to the existence of an almost space optimal algorithm for TAUT. To obtain this result we have to derive a space version of a theorem of Levin on optimal inverters.

Abstract We give the first systematic study of strong isomorphism reductions, a notion of reduction more appropriate than polynomial time reduction when, for example, comparing the computational complexity of the isomorphim problem for different classes of structures. We show that the partial ordering of its degrees is quite rich. We analyze its relationship to a further type of reduction between classes of structures based on purely comparing for everynthe number of nonisomorphic structures of cardinality at mostnin both classes. Furthermore, in a more general setting we address the question of the existence of a maximal element in the partial ordering of the degrees.

Abstract The undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard” valid sentences? The polynomial time tractability of this problem would imply the fixed-parameter tractability of the parameterized problem that, given a natural number n in unary as input and a first-order sentence φ as parameter, asks whether φ has a proof of length ≤ n. As the underlying classical problem has been considered by Gödel we denote this problem by p-Gödel. We show that p-Gödel is not fixed-parameter tractable if DTIME(hO(1)) ≠ NTIME(hO(1)) for all time constructible and increasing functions h. Moreover we analyze the complexity of the construction problem associated with p-Gödel.

Recently it has been shown that the miniaturization mapping ℳ faithfully translates bexponential parameterized complexity into (unbounded) parameterized complexity. We determine the pre-images under ℳ of various (classes of) problems. For many parameterized problems whose underlying classical problem is in NP we show that the pre-images coincide with natural reparameterizations that take into account the amount of non-determinism needed to solve them.

We study the parameterized complexity of various path (and cycle) problems, the parameter being the length of the path. For example, we show that the problem of the existence of a maximal path of length k in a graph G is fixed-parameter tractable, while its counting version is #W[1]- complete. The corresponding problems for chordless (or induced) paths are W[2]-complete and #W[2]-complete respectively. With the tools developed in this paper we derive the NP-completeness of a related classical problem, thereby solving a problem due to Hedetniemi.

Abstract We observe that the W*-hierarchy, a variant (introduced by Downey, Fellows, and Taylor [7]) of the better known W-hierarchy, coincides with the W-hierarchy, though not level wise, but just as a whole hierarchy. More precisely, we prove that W[t] ⊆ W* [t] ⊆ W[2t − 2] for each t ≥ 2. It was known before that W[1] = W*[1] and W[2] = W*[2]. Our second main result is a new logical characterization of the W*-hierarchy in terms of “Fagin-definable problems.” As a by-product, we also obtain an improvement of our earlier characterization of the hierarchy in terms of model-checking problems. Furthermore, we obtain new complete problems for the classes W[3] and W*[3].

We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories.

We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories.

We give machine characterisations and logical descriptions of a number of parameterized complexity classes. The focus of our attention is the class W[P], which we characterise as the class of all parameterized problems decidable by a nondeterministic fixed-parameter tractable algorithm, whose use of nondeterminism is bounded in terms of the parameter. We give similar characterisations for AW[P], the "alternating version of W[P]", and various other parameterized complexity classes. We also give logical characterisations of the classes W[P] and AW[P] in terms of fragments of least fixed-point logic, thereby putting these two classes into a uniform framework that we have developed in earlier work. Furthermore, we investigate the relation between alternation and space in parameterized complexity theory. We prove that the compact Turing machine computation problem, shown to be hard for the class AW[SAT] in (K. A. Abrahamson et al., 1995) is complete for the class uniform-XNL.