We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, which are graphs that do not contain $H$ as a subgraph. To classify the complexity of Colouring on $H$-subgraph-free graphs for connected $H$, it remains to consider when $H$ is a tree of maximum degree $4$ with exactly one vertex of degree $4$, or a tree of maximum degree $3$ with at least two vertices of degree $3$. We let $H$ be a so-called subdivided ``H''-graph, which is either a subdivided $\mathbb{H}_0$: a tree of maximum degree $4$ that is a star, or a subdivided $\mathbb{H}_1$: a tree of maximum degree $3$ with exactly two vertices of degree $3$. We develop new decomposition theorems resulting in polynomial-time algorithms, and in combination with known results, fully classify all cases $\mathbb{H}_0$ and $\mathbb{H}_1$. To illustrate the wider applicability of our techniques, we also employ them to obtain similar new polynomial-time results for two other classic graph problems: Stable Cut and, in part, Feedback Vertex Set.

In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ℙ_k-free and ℙ_k-subgraph-free graphs. We consider the directed version of this research line, by addressing the question is it true that digraph homomorphism problems CSP(H) have a P versus NP-complete dichotomy when the input is restricted to ℙ→_k-free (resp. ℙ→_k-subgraph-free) digraphs? Our main contribution in this direction shows that if CSP(H) is NP-complete, then there is a positive integer N such that CSP(H) remains NP-hard even for ℙ→_N-subgraph-free digraphs. Moreover, CSP(H) becomes polynomial-time solvable for ℙ→_{N-1}-subgraph-free acyclic digraphs. We then verify the questions above for digraphs on three vertices and a family of smooth tournaments. We prove these results by establishing a connection between F-(subgraph)-free algorithmics and constraint satisfaction theory. On the way, we introduce restricted CSPs, i.e., problems of the form CSP(H) restricted to yes-instances of CSP(H') - these were called restricted homomorphism problems by Hell and Nešetřil. Another main result of this paper presents a P versus NP-complete dichotomy for these problems. Moreover, this complexity dichotomy is accompanied by an algebraic dichotomy in the spirit of the finite domain CSP dichotomy.

Abstract For a set of graphs $${\mathcal {H}}$$ H , a graph G is $${\mathcal {H}}$$ H -subgraph-free if G does not contain any graph from $${{{\mathcal {H}}}}$$ H as a subgraph. We propose general and easy-to-state conditions on graph problems that explain a large set of results for $${\mathcal {H}}$$ H -subgraph-free graphs. Namely, a graph problem must be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness must be preserved under edge subdivision of subcubic graphs. Our meta-classification says that if a graph problem $$\Pi $$ Π satisfies all three conditions, then for every finite set $${{{\mathcal {H}}}}$$ H , it is “efficiently solvable” on $${{{\mathcal {H}}}}$$ H -subgraph-free graphs if $${\mathcal {H}}$$ H contains a disjoint union of one or more paths and subdivided claws, and $$\Pi $$ Π is “computationally hard” otherwise. We apply our meta-classification on many well-known partitioning, covering and packing problems, network design problems and width parameter problems to obtain a dichotomy between polynomial-time solvability and -completeness. For distance-metric problems, we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). Apart from capturing a large number of explicitly and implicitly known results in the literature, we also prove a number of new results. Moreover, we perform an extensive comparison between the subgraph framework and the existing frameworks for the minor and topological minor relations, and pose several new open problems and research directions.

We study Steiner Forest on H -subgraph-free graphs, that is, graphs that do not contain some fixed graph H as a (not necessarily induced) subgraph. In contrast to the related Steiner Tree problem, Steiner Forest falls outside a recent framework that completely characterizes the complexity of many problems on H -subgraph-free graphs. Hence, the complexity of Steiner Forest on H -subgraph-free graphs remained open. Our main results are four polynomial-time algorithms for different excluded graphs H that are central to further understand its complexity. We also study the complexity of Steiner Forest for graphs with a small c -deletion set, that is, a small set X of vertices such that each connected component of G − X has size at most c . For this parameter, we give two algorithms that we later employ as subroutines (including a faster algorithm when c = 1 , that is, the vertex cover number) and exhibit a dichotomy theorem.

A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring , Star Colouring and Injective Colouring . We give almost complete complexity classifications for Acyclic Colouring , Star Colouring and Injective Colouring on H -free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours k is fixed, that is, not part of the input. From our study it follows that for fixed k , the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Abstract. We study the complexity of infinite-domain constraint satisfaction problems (CSPs): our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure [Formula: see text] can be transferred to a classification of the CSPs of first-order expansions of another structure [Formula: see text]. We exploit a product of structures (the algebraic product) that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSPs for first-order expansions of the [Formula: see text]-fold algebraic power of [Formula: see text]. This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of [Formula: see text] and explicit descriptions of the expressible relations in terms of syntactically restricted first-order formulas. By combining our classification result with general classification transfer techniques, we obtain surprisingly strong new classification results for highly relevant formalisms such as Allen’s Interval Algebra, the [Formula: see text]-dimensional Block Algebra, and the Cardinal Direction Calculus, even if higher-arity relations are allowed. Our results confirm the infinite-domain tractability conjecture for classes of structures that have been difficult to analyze with older methods. For the special case of structures with binary signatures, the results can be substantially strengthened and tightly connected to Ord-Horn formulas; this solves several longstanding open problems from the artificial intelligence (AI) literature.

Abstract. We consider proof complexity in light of the unusual binary encoding of certain combinatorial principles. We contrast this proof complexity with the normal unary encoding in several refutation systems, based on Resolution and Sherali–Adams. We first consider [Formula: see text], which is an extension of Resolution working on [Formula: see text]-DNFs (Disjunctive Normal Form formulas). We prove an exponential lower bound of [Formula: see text] for the size of refutations of the binary version of the [Formula: see text]-Clique Principle in [Formula: see text], where [Formula: see text] and [Formula: see text] is a doubly exponential function. Our result improves that of Lauria et al., who proved a similar lower bound for [Formula: see text], i.e., Resolution. For the [Formula: see text]-Clique and other principles we study, we show how lower bounds in Resolution for the unary version follow from lower bounds in [Formula: see text] for the binary version, so we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the (weak) Pigeonhole Principle [Formula: see text]. We prove that for any [Formula: see text], [Formula: see text] requires refutations of size [Formula: see text] in [Formula: see text] for [Formula: see text]. Our lower bound cannot be improved substantially with the same method since for [Formula: see text] we can prove there are [Formula: see text] size refutations of [Formula: see text] in [Formula: see text]. This is a consequence of the same upper bound for the unary weak Pigeonhole Principle of Buss and Pitassi. We contrast unary versus binary encoding in the Sherali–Adams (SA) refutation system where we prove lower bounds for both rank and size. For the unary encoding of the Pigeonhole Principle and the Ordering Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the (weak) Pigeonhole Principle [Formula: see text] requires exponentially sized (in [Formula: see text]) SA refutations, whereas the binary encoding of the Ordering Principle admits logarithmic rank, polynomially sized SA refutations. We continue by considering a natural refutation system we call “SA+Squares,” which is intermediate between SA and Lasserre (Sum-of-Squares). This has been studied under the name static-[Formula: see text] by Grigoriev et al. In this system, the unary encoding of the Linear Ordering Principle [Formula: see text] requires [Formula: see text] rank while the unary encoding of the Pigeonhole Principle becomes constant rank. Since Potechin has shown that the rank of [Formula: see text] in Lasserre is [Formula: see text], we uncover an almost quadratic separation between SA+Squares and Lasserre in terms of rank. Grigoriev et al. noted that the unary Pigeonhole Principle has rank 2 in SA+Squares and therefore polynomial size. Since we show the same applies to the binary [Formula: see text], we deduce an exponential separation for size between SA and SA+Squares.

For any finite set H = {H1, . . ., Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1, . . ., Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

We prove that QCSP(ℕ; x = y → y = z) is PSpace-complete, settling a question open for more than ten years. This completes the complexity classification for the QCSP over equality languages as a trichotomy between Logspace, NP-complete and PSpace-complete. We additionally settle the classification for bounded alternation QCSP(Γ), for Γ an equality language. Such problems are either in Logspace, NP-complete, co-NP-complete or rise in complexity in the Polynomial Hierarchy.

Abstract Paths $$P^1,\ldots ,P^k$$ P 1 , … , P k in a graph $$G=(V,E)$$ G = ( V , E ) are mutually induced if any two distinct $$P^i$$ P i and $$P^j$$ P j have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices $$(s_i,t_i)$$ ( s i , t i ) contains k mutually induced paths $$P^i$$ P i such that each $$P^i$$ P i starts from $$s_i$$ s i and ends at $$t_i$$ t i . This is a classical graph problem that is -complete even for $$k=2$$ k = 2 . We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.

Paths P1,…,Pk in a graph G=(V,E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices. For a fixed integer k, the k-Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi starts from si and ends at ti. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-Induced Disjoint Paths is NP-complete. We prove new complexity results for k-Induced Disjoint Paths if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where k is part of the input.

Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [ 1 ] and Zhuk [ 2 ], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture . This result becomes more significant now that the original Chen Conjecture (see [ 3 ]) is known to be false [ 4 ]. Switchability was introduced by Chen [ 1 ] as a generalization of the already-known collapsibility [ 5 ]. There, an algebra 𝔸 :=({ 0,1,2}; r ) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.

Stabbing Planes is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ is a finite language over three elements that contains all constants. In particular, such problems are in P, NP-complete, co-NP-complete, or PSpace-complete. Our classification refutes the hitherto widely believed Chen Conjecture. Additionally, we show that already on a 4-element domain there exists a constraint language Γ such that QCSP(Γ) is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class Θ P 2 . Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of such Γ has the polynomially generated powers property, then QCSP(Γ) is in NP. Otherwise, the polymorphism clone of Γ has the exponentially generated powers property and QCSP(Γ) is PSpace-complete. 1

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem \( \mathrm{QCSP}(\mathrm{H}) \) when \( \mathrm{H} \) is a reflexive tournament. It is well known that reflexive tournaments can be split into a sequence of strongly connected components \( \mathrm{H}_1,\ldots ,\mathrm{H}_n \) so that there exists an edge from every vertex of \( \mathrm{H}_i \) to every vertex of \( \mathrm{H}_j \) if and only if \( i\lt j \) . We prove that if \( \mathrm{H} \) has both its initial and final strongly connected component (possibly equal) of size 1, then \( \mathrm{QCSP}(\mathrm{H}) \) is in \( \mathsf {NL} \) and otherwise \( \mathrm{QCSP}(\mathrm{H}) \) is \( \mathsf {NP} \) -hard.

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as $L(1,1)$-Labelling). A classical complexity result on Colouring is a well-known dichotomy for $H$-free graphs (a graph is $H$-free if it does not contain $H$ as an induced subgraph). In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We perform such a study and give almost complete complexity classifications for Acyclic Colouring, Star Colouring and Injective Colouring on $H$-free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours $k$ is fixed, that is, not part of the input. From our study it follows that for fixed $k$ the three problems behave in the same way, but this is no longer true if $k$ is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.

A k-colouring c of a graph G is a mapping V(G) → {1,2,… k} such that c(u) ≠ c(v) whenever u and v are adjacent. The corresponding decision problem is Colouring. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1)-Labelling). A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as an induced subgraph). Moreover, this result has led to a large collection of results, which helped us to better understand the complexity of Colouring. In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We initiate such a systematic complexity study, and similar to the study of Colouring we use the class of H-free graphs as a testbed. We prove the following results: 1) We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs. 2) If the number of colours k is fixed, that is, not part of the input, we give full complexity classifications for each of the three problems for H-free graphs. From our study we conclude that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs.

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ is a finite language over 3 elements which contains all constants. In particular, such problems are either in P, NP-complete, co-NP-complete or PSpace-complete. Our classification refutes the hitherto widely-believed Chen Conjecture.

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut . This problem is known to be NP -hard on general graphs. We prove that it is polynomial-time solvable on claw-free graphs, answering a question of Ito et al. (TCS 2011). The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007–2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

Res(s) is an extension of Resolution working on s-DNFs. We prove tight n^{Omega(k)} lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [Massimo Lauria et al., 2017] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [Nathan Segerlind et al., 2004], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHP^m_n for m>n. Using the the same recursive approach we prove the new result that for any delta>0, Bin-PHP^m_n requires proofs of size 2^{n^{1-delta}} in Res(s) for s=o(log^{1/2}n). Our lower bound is almost optimal since for m >= 2^{sqrt{n log n}} there are quasipolynomial size proofs of Bin-PHP^m_n in Res(log n). Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in Pi_2-form and with no finite model.

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that Colouring is NP-complete for H-free graphs if H contains a cycle or claw, even for fixed k >= 3. We examine to what extent the situation may change if in addition the input graph has bounded diameter.

For $n\geq 3$, let $(H_n, E)$ denote the $n$th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on $n$ vertices. We show that for all structures $\Gamma$ with domain $H_n$ whose relations are first-order definable in $(H_n,E)$ the constraint satisfaction problem for $\Gamma$ either is in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.

We study formalisms for temporal and spatial reasoning in the modern context of Constraint Satisfaction Problems (CSPs). We show how questions on the complexity of their subclasses can be solved using existing results via the powerful use of primitive positive (pp) interpretations and pp-homotopy. We demonstrate the methodology by giving a full complexity classification of all constraint languages that are first-order definable in Allen's Interval Algebra and contain the basic relations (s) and (f). In the case of the Rectangle Algebra we answer in the affirmative the old open question as to whether ORD-Horn is a maximally tractable subset among the (disjunctive, binary) relations. We then generalise our results for the Rectangle Algebra to the r-dimensional Block Algebra.

We study the Constraint Satisfaction Problem CSP( A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete.

We apply the algebraic approach for Constraint Satisfaction Problems (CSPs) with counting quantifiers, developed by Bulatov and Hedayaty, for the first time to obtain classifications for computational complexity. We develop the consistency approach for expanding polymorphisms to deduce that, if H has an expanding majority polymorphism, then the corresponding CSP with counting quantifiers is tractable. We elaborate some applications of our result, in particular deriving a complexity classification for partially reflexive graphs endowed with all unary relations. For each such structure, either the corresponding CSP with counting quantifiers is in P, or it is NP-hard.

A discrete temporal constraint satisfaction problem is a constraint satisfaction problem (CSP) over the set of integers whose constraint language consists of relations that are first-order definable over the order of the integers. We prove that every discrete temporal CSP is in P or NP-complete, unless it can be formulated as a finite domain CSP, in which case the computational complexity is not known in general.

The complexity of the model checking problem for various fragments of first-order logic (FO) has attracted much attention over the last two decades, in particular for the fragment induced by $\exists$ and $\wedge$ and that induced by $\forall, \exists$, and $\wedge$, which are better known as the constraint satisfaction problem and the quantified constraint satisfaction problem, respectively. The former was conjectured to follow a dichotomy between P and NP-complete by Feder and Vardi [ SIAM J. Comput., 28 (1998), pp. 57--104]. For the latter, there are several partial trichotomy results between P, NP-complete, and Pspace-complete, and Chen [ Meditations on quantified constraint satisfaction, in Logic and Program Semantics, Springer, Heidelberg, 2012, pp. 35--49] ventured a conjecture regarding Pspace-completeness vs. membership in NP in the presence of constants. We give a comprehensive account of the whole field of the complexity of model checking similar syntactic fragments of FO. The above two fragments are in fact the only ones for which there is currently no known complexity classification. Indeed, we consider all other similar syntactic fragments of FO, induced by the presence or absence of quantifiers and connectives, and fully classify the complexities of the parameterization of the model-checking problem by a finite model $\mathcal{D}$, that is, the expression complexities for certain finite $\mathcal{D}$. Perhaps surprisingly, we show that for most of these fragments, “tractability” is witnessed by a generic solving algorithm which uses quantifier relativization. Our classification methodology relies on tailoring suitably the algebraic approach pioneered by Jeavons, Cohen, and Gyssens [ J. ACM, 44 (1997), pp. 527--548] for the constraint satisfaction problem and by Börner et al. [ Inform. and Comput., 207 (2009), pp. 923--944] for the quantified constraint satisfaction problem. Most fragments under consideration can be relatively easily classified, either directly or using Schaefer's dichotomy theorems for SAT and QSAT, with the notable exception of the positive equality-free fragment induced by $\exists,\forall, \wedge$, and $\vee$. This outstanding fragment can also be classified and enjoys a tetrachotomy: according to the model, the corresponding model checking problem is either tractable, NP-complete, co-NP-complete, or Pspace-complete.

A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide whether or not a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective $$H$$ -Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exactly two vertices with a self-loop as long as these two vertices are not adjacent. As a result, we can classify the computational complexity of Surjective $$H$$ -Colouring for every graph H on at most four vertices.

Let A be an idempotent algebra on a finite domain. We combine results of Chen, Zhuk and Carvalho et al. to argue that if A satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of Zhuk to prove a converse, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying the moral correctness of what we term the Chen Conjecture. We examine in closer detail the situation for domains of size three. Over any finite domain, the only type of PGP that can occur is switchability. Switchability was introduced by Chen as a generalisation of the already-known Collapsibility. For three-element domain algebras A that are Switchable, we prove that for every finite subset Delta of Inv(A), Pol(Delta) is Collapsible. The significance of this is that, for QCSP on finite structures (over three-element domain), all QCSP tractability explained by Switchability is already explained by Collapsibility. Finally, we present a three-element domain complexity classification vignette, using known as well as derived results.

We study the (non-uniform) quantified constraint satisfaction problem QCSP( H ) as H ranges over semicomplete digraphs. We obtain a complexity-theoretic trichotomy: QCSP( H ) is either in P, is NP-complete, or is Pspace-complete. The largest part of our work is the algebraic classification of precisely which semicomplete digraphs enjoy only essentially unary polymorphisms, which is combinatorially interesting in its own right.

We study the computational problem of checking whether a logical sentence is true in a finite abelian group. We prove that model checking first-order sentences on finite abelian groups is fixed-parameter tractable, when parameterized by the size of the sentence. We also prove that model checking monadic second-order sentences on finite abelian groups finitely presented by integer matrices is not fixed-parameter tractable (under standard assumptions in parameterized complexity).

Inspired by computational complexity results for the quantified constraint satisfaction problem, we study the clones of idem potent polymorphisms of certain digraph classes. Our first results are two algebraic dichotomy, even "gap", theorems. Building on and extending [Martin CP'11], we prove that partially reflexive paths bequeath a set of idem potent polymorphisms whose associated clone algebra has: either the polynomially generated powers property (PGP), or the exponentially generated powers property (EGP). Similarly, we build on [DaMM ICALP'14] to prove that semi complete digraphs have the same property. These gap theorems are further motivated by new evidence that PGP could be the algebraic explanation that a QCSP is in NP even for unbounded alternation. Along the way we also effect a study of a concrete form of PGP known as collapsibility, tying together the algebraic and structural threads from [Chen Sicomp'08], and show that collapsibility is equivalent to its Pi2-restriction. We also give a decision procedure for k-collapsibility from a singleton source of a finite structure (a form of collapsibility which covers all known examples of PGP for finite structures). Finally, we present a new QCSP trichotomy result, for partially reflexive paths with constants. Without constants it is known these QCSPs are either in NL or Pspace-complete [Martin CP'11], but we prove that with constants they attain the three complexities NL, NP-complete and Pspace-complete.

We initiate the study of constraint satisfaction problems (CSPs) in the presence of counting quantifiers $\exists^{\geq j}$ which assert the existence of at least $j$ elements such that the ensuing property holds. These are natural variants of CSPs in the mould of quantified CSPs (QCSPs). Namely, $\exists^{\geq 1}:=\exists$ and $\exists^{\geq n}:=\forall$ (for the domain of size $n$). We observe that a single counting quantifier $\exists^{\geq j}$ strictly between $\exists$ and $\forall$ already affords the maximal possible complexity of QCSPs (which have both $\exists$ and $\forall$), namely, being Pspace-complete for a suitably chosen template. Therefore, to better understand the complexity of this problem, we focus on restricted cases for which we derive the following results. First, for all subsets of counting quantifiers on clique and cycle templates, we give a full trichotomy---all such problems are in P, NP-complete, or Pspace-complete. Second, we consider the problem with exactly two quantifiers: $\exists^{\geq 1}:=\exists$ and $\exists^{\geq j}$ ($j \neq 1$). Such a CSP is already NP-hard on nonbipartite graph templates. We explore the situation of this generalized CSP on graph templates, giving various conditions for both tractability and hardness. For quantifiers $\exists^{\geq 1}$ and $\exists^{\geq 2}$, we give a dichotomy for all graphs, namely, the problem is NP-hard if the graph contains a triangle or has girth at least 5, and is in P otherwise. We strengthen this result in the following two ways. For bipartite graphs, the problem is in P for forests and graphs of girth 4, and is Pspace-hard otherwise. For complete multipartite graphs, the problem is in L, NP-complete, or Pspace-complete. Finally, using counting quantifiers we solve the complexity of a concrete QCSP whose complexity was previously open.

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B . This may be seen as a natural generalisation of the nonuniform quantified constraint satisfaction problem QCSP( B ). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterizes definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem either is in L, is NP-complete, is co-NP-complete, or is Pspace-complete.

We classify completely the complexity of evaluating positive equality-free sentences of first-order logic over a fixed, finite structure D. This problem may be seen as a natural generalisation of the quantified constraint satisfaction problem QCSP(D). We obtain a tetrachotomy for arbitrary finite structures: each problem is either in L, is NP-complete, is co-NP-complete or is P space-complete. Moreover, its complexity is characterised algebraically in terms of the presence or absence of specific surjective hyper-endomorphisms, and, logically, in terms of relativisation properties with respect to positive equality-free sentences. We prove that the meta-problem, to establish for a specific D into which of the four classes the related problem lies, is NP-hard.

The universal-algebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ω-categorical templates. Our first result is an exact characterization of those CSPs that can be formulated with (a finite or) an ω-categorical template. The universal-algebraic approach relies on the fact that in finite or ω-categorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. In this paper, we present results that can be used to study the computational complexity of CSPs with arbitrary infinite templates. Specifically, we prove that every CSP can be formulated with a template A such that a relation is primitive positive definable in A if and only if it is first-order definable on A and preserved by the infinitary polymorphisms of A. We present applications of our general results to the description and analysis of the computational complexity of CSPs. In particular, we present a polymorphism-based description of those CSPs that are first-order definable (and therefore can be solved in polynomial-time), and give general hardness criteria based on the absence of polymorphisms that depend on more than one argument.

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP(B). We introduce subjective hyper-endomorphisms and use them in proving a Galois connection that characterises definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem is either in Logspace, is NP-complete, is coNP-complete or is Pspace-complete.

We study two containment problems related to the quantified constraint satisfaction problem (QCSP). Firstly, we give a combinatorial condition on finite structures A and B that is necessary and sufficient to render QCSP(A) a subset of QCSP(B). The required condition is the existence of a positive integer r such that there is a surjective homomorphism from the power structure A^r to B. We note that this condition is already necessary to guarantee containment of the Pi_2 restriction of QCSP, that is Pi_2-CSP(A) a subset of Pi_2-CSP(B). Since we are able to give an effective bound on such an r, we provide a decision procedure for the model containment problem with non-deterministic double-exponential time complexity. Secondly, we prove that the entailment problem for quantified conjunctive-positive first-order logic is decidable. That is, given two sentences phi and psi of first-order logic with no instances of negation or disjunction, we give an algorithm that determines whether "phi implies psi" is true in all structures (models). Our result is in some sense tight, since we show that the entailment problem for positive first-order logic (i.e. quantified conjunctive-positive logic plus disjunction) is undecidable.

We propose a proof-theoretic approach for gaining evidence that certain parameterized problems are not fixed-parameter tractable. We consider proofs that witness that a given propositional CNF formula cannot be satisfied by a truth assignment that sets at most k variables to true, considering k as the parameter (we call such a formula a parameterized contradiction). One could separate the parameterized complexity classes FPT and W(M. Cesati, 2006) by showing that there is no fpt-bounded parameterized proof system, i.e., that there is no proof system that admits proofs of size f(k)nO(1)where f is a computable function and n represents the size of the propositional formula. By way of a first step, we introduce the system of parameterized tree-like resolution, and show that this system is not fpt-bounded. Indeed we give a general result on the size of shortest tree-like resolution proofs of parameterized contradictions that uniformly encode first-order principles over a universe of size n. We establish a dichotomy theorem that splits the exponential case of Riis's complexity-gap Theorem into two sub-cases, one that admits proofs of size f(k)nO(1)and one that does not. We also discuss how the set of parameterized contradictions may be embedded into the set of (ordinary) contradictions by the addition of new axioms. When embedded into general (DAG-like) resolution, we demonstrate that the pigeonhole principle has a proof of size 2kn2. This contrasts with the case of tree-like resolution where the embedded pigeonhole principle falls into the "non-FPT" category of our dichotomy.