Counting Small Induced Subgraphs Satisfying Monotone Properties
CCounting Small Induced SubgraphsSatisfying Monotone Properties
Marc Roth
Merton College, University of Oxford, United [email protected]
Johannes Schmitt
Mathematical Institute, University of Bonn, [email protected]
Philip Wellnitz
Max Planck Institute for Informatics,Saarland Informatics Campus (SIC), Saarbrücken, [email protected]
Abstract
Given a graph property Φ, the problem
IndSub (Φ) asks, on input a graph G and a positive integer k , to computethe number IndSub (Φ , k → G ) of induced subgraphs of size k in G that satisfy Φ. The search for explicit criteriaon Φ ensuring that IndSub (Φ) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is partof the major line of research on counting small patterns in graphs. However, apart from an implicit result dueto Curticapean, Dell and Marx [STOC 17] proving that a full classification into “easy” and “hard” properties ispossible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and Dörfleret al. [MFCS 19], not much is known.In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties:We show that for any non-trivial monotone property Φ, the problem
IndSub (Φ) cannot be solved in time f ( k ) · | V ( G ) | o ( k/ log / ( k )) for any function f , unless the Exponential Time Hypothesis fails. By this, we establishthat any significant improvement over the brute-force approach is unlikely; in the language of parameterizedcomplexity, we also obtain a W [ ]-completeness result.To prove our result, we use that for fixed Φ and k , we can express the function G IndSub (Φ , k → G )as a finite linear-combination of homomorphism counts from graphs H i to G . The coefficient vectors of thesehomomorphism counts in the linear combination are called the homomorphism vectors associated to Φ; by theComplexity Monotonicity framework of Curticapean, Dell and Marx [STOC 17], the positions of non-zero entriesof these vectors are known to determine the complexity of IndSub (Φ). Our main technical result lifts the notionof f -polynomials from simplicial complexes to graph properties and relates the derivatives of the f -polynomialof Φ to its homomorphism vector. We then apply results from the theory of Hermite-Birkhoff interpolation to the f -polynomial to establish sufficient conditions on Φ which ensure that certain entries in the homomorphism vectordo not vanish—which in turn implies hardness. For monotone graph properties, non-triviality then turns out tobe a sufficient condition. Using the same method, we also prove a conjecture by Jerrum and Meeks [TOCT 15,Combinatorica 19]: IndSub (Φ) is W [ ]-complete if Φ is a non-trivial graph property only depending on thenumber of edges of the graph. Keywords and phrases
Counting complexity, fine-grained complexity, graph homomorphisms, induced subgraphs,parameterized complexity a r X i v : . [ c s . CC ] A p r Counting Small Induced Subgraphs Satisfying Monotone Properties
Detection, enumeration and counting of patterns in graphs are among the most well-studied computationalproblems in theoretical computer science with a plethora of applications in diverse disciplines, includingbiology [59, 33], statistical physics [62, 41, 42], neural and social networks [50] and database theory [34], toname but a few. At the same time, those problems subsume in their unrestricted forms some of the mostinfamous NP-hard problems such as Hamiltonicity, the clique problem, or, more generally, the subgraphisomorphism problem [17, 65]. In the modern-day era of “big data”, where even quadratic-time algorithmsmay count as inefficient, it is hence crucial to find relaxations of hard computational problems that allowfor tractable instances.A very successful approach for a more fine-grained understanding of hard computational problems is amultivariate analysis of the complexity of the problem: Instead of establishing upper and (conditional)lower bounds only depending on the input size, we aim to find additional parameters that, in the bestcase, are small in real-world instances and allow for efficient algorithms if assumed to be bounded. Incase of detection and counting of patterns in graphs, it turns out that the size of the pattern is oftensignificantly smaller than the size of the graph: Consider as an example the evaluation of database queries.While a classical analysis of this problem requires considering instances where the size of the query is aslarge as the database, a multivariate analysis allows us to impose the restriction of the query being muchsmaller than the database, which is the case for real-world instances. More concretely, suppose we aregiven a query ϕ of size k and a database B of size n , and we wish to evaluate the query ϕ on B . Assumefurther, that we are given two algorithms for the problem: One has a running time of O ( n k ), and theother one has a running time of O (2 k · n ). While, classically, both algorithms are inefficient in the sensethat their running times are not bounded by a polynomial in the input size n + k , the second algorithm issignificantly better than the first one for real-world instances and can even be considered efficient.In this work, we focus on counting of small patterns in large graphs. The field of counting complexitywas founded by Valiant’s seminal result on the complexity of computing the permanent [66, 67], whereit was shown that computing the number of perfect matchings in a graph is finding a perfect matching in a graph can be done in polynomial-time [28]. Hence, a perfectmatching is a pattern that allows for efficient detection but is unlikely to admit efficient counting. Initiatedby Valiant, computational counting evolved into a well-studied subfield of theoretical computer science.In particular, it turns out that counting problems are closely related to computing partition functionsin statistical physics [62, 41, 42, 32, 15, 3]. Indeed, one of the first algorithmic result in the field ofcomputational counting is the famous FKT-Algorithm by the statistical physicists Fisher, Kasteleynand Temperley [62, 41, 42] that computes the partition function of the so-called dimer model on planarstructures, which is essentially equivalent to computing the number of perfect matchings in a planargraph. The FKT-Algorithm is the foundation of the framework of holographic algorithms, which, amongothers, have been used to identify the tractable cases of a variety of complexity classifications for countingconstraint satisfaction problems [68, 7, 5, 6, 35, 8, 9, 2]. Unfortunately, the intractable cases of thoseclassifications indicate that, except for rare examples, counting is incredibly hard (from a complexity theorypoint of view). In particular, many efficiently solvable combinatorial decision problems turn out to beintractable in their counting versions, such as counting of satisfying assignments of monotone 2-CNFs [67],counting of independent sets in bipartite graphs [52] or counting of s - t -paths [67], to name but a few. Forthis reason, we follow the multivariate approach as outlined previously and restrict ourselves in this workon counting of small patterns in large graphs. Among others, problems of this kind find applications inneural and social networks [50], computational biology [1, 57], and database theory [27, 11, 12, 24]. . Roth, J. Schmitt, and P. Wellnitz 2 Formally, we follow the approach of Jerrum and Meeks [37] and study the family of problems
IndSub (Φ):Given a graph property Φ, the problem
IndSub (Φ) asks, on input a graph G and a positive integer k ,to compute the number of induced subgraphs of size k in G that satisfy Φ. As observed by Jerrum andMeeks, the generality of the definition allows to express counting of almost arbitrary patterns of size k ina graph, subsuming counting of k -cliques and k -independent sets as very special cases.Assuming that Φ is computable, we note that the problem IndSub (Φ) can be solved by brute-force intime O ( f ( k ) · | V ( G ) | k ) for some function f only depending on Φ. The corresponding algorithm enumeratesall subsets of k vertices of G and counts how many of those subsets satisfy Φ. As we consider k to besignificantly smaller than | V ( G ) | , we are interested in the dependence of the exponent on k . More precisely,the goal is to find the best possible g ( k ) such that IndSub (Φ) can be solved in time O ( f ( k ) · | V ( G ) | g ( k ) ) (1)for some function f such that f and g only depend on Φ. Readers familiar with parameterized complexitytheory will identify the case of g ( k ) ∈ O (1) as fixed-parameter tractability (FPT) results. We first providesome background and elaborate on the existing results on IndSub (Φ) before we present the contributionsof this paper.
So far, the problem
IndSub (Φ) has been investigated using primarily the framework of parameterizedcomplexity theory. As indicated before,
IndSub (Φ) is in FPT if the function g in Equation (1) isbounded by a constant (independent of k ), and the problem is W [ ]-complete, if it is at least as hard asthe parameterized clique problem; here W [ ] should be considered a parameterized counting equivalentof NP and we provide the formal details in Section 2. In particular, the so-called Exponential TimeHypothesis (ETH) implies that W [ ]-complete problems are not in FPT; again, this is made formal inSection 2.The problem IndSub (Φ) was first studied by Jerrum and Meeks [37]. They introduced the problemand proved that
IndSub (Φ) is W [ ]-complete if Φ is the property of being connected. Implicitly, theirproof also rules out the function g of Equation (1) being in o ( k ), unless ETH fails, which establishes atight conditional lower bound. In a subsequent line of research [38, 48, 39], Jerrum and Meeks proved IndSub (Φ) to be W [ ]-complete if at least one of the following is true: (1) Φ has low edge-densities ; this is true for instance for all sparse properties such as planarity and madeformal in Section 4.1. (2)
Φ holds for a graph H if and only if the number of edges of H is even/odd. (3) Φ is closed under the addition of edges, and the minimal elements have large treewidth.Unfortunately, none of the previous results establishes a conditional lower bound that comes close tothe upper bound given by the brute-force algorithm. This is particularly true due to the application ofRamsey’s Theorem in the proofs of many of the prior results: Ramsey’s Theorem states that there is afunction R ( k ) = 2 Θ( k ) such that every graph with at least R ( k ) vertices contains either a k -independentset or a k -clique [53, 61, 29]. Relying on this result for a reduction from finding or counting k -independentsets or k -cliques, the best implicit conditional lower bounds achieved only rule out an algorithm runningin time f ( k ) · | V ( G ) | o (log k ) for any function f .Moreover, the previous results only apply to a very specific set of properties. In particular, Jerrumand Meeks posed the following open problem concerning a generalization of the second result (2); we saythat Φ is k -trivial, if it is either true or false for all graphs with k vertices. Note that
IndSub (Φ) is identical to p - UnlabelledInducedSubgraphWithProperty (Φ) as defined in [37].
Counting Small Induced Subgraphs Satisfying Monotone Properties
Conjecture 1.1 ([38, 39]) . Let Φ be a graph property that only depends on the number of edges of agraph. If for infinitely many k the property Φ is not k -trivial, then IndSub (Φ) is W [ ] -complete. Note that the condition of Φ not being k -trivial for infinitely many k is necessary for hardness, as otherwise,the problem becomes trivial if k exceeds a constant depending only on Φ.The first major breakthrough towards a complete understanding of the complexity of IndSub (Φ) isthe following implicit classification due to Curticapean, Dell and Marx [20]:
Theorem 1.2 ([20]) . Let Φ denote a graph property. Then the problem IndSub (Φ) is either
FPT or W [ ] -complete. While the previous classification provides a very strong result for the structural complexity of
IndSub (Φ),it leaves open the question of the precise bound on the function g . Furthermore, it is implicit in the sensethat it does not reveal the complexity of IndSub (Φ) if a concrete property Φ is given. Nevertheless, thetechnique introduced by Curticapean, Dell and Marx, which is now called
Complexity Monotonicity , turnedout to be the right approach for the treatment of
IndSub (Φ). In particular, the subsequent resultson
IndSub (Φ), including the classification in this work, have been obtained by strong refinements of
Complexity Monotonicity ; we provide a brief introduction when we discuss the techniques used in this paper.More concretely, a superset of the authors established the following classifications for edge-monotoneproperties in recent years [56, 26]: The problem
IndSub (Φ) is W [ ]-complete and, assuming ETH,cannot be solved in time f ( k ) · | V ( G ) | o ( k ) for any function f , if at least one of the following is true: Φ is non-trivial, closed under the removal of edges and false on odd cycles.Φ is non-trivial on bipartite graphs and closed under the removal of edges.While the second result completely answers the case of edge-monotone properties on bipartite graphs, ageneral classification of edge-monotone properties is still unknown.
We begin with monotone properties, that is, properties that are closed under taking subgraphs. Weclassify those properties completely and explicitly; the following theorem establishes hardness and analmost tight conditional lower bound.
Main Theorem 1.
Let Φ denote a monotone graph property. Suppose that for infinitely many k the property Φ is not k -trivial. Then IndSub (Φ) is W [ ] -complete and cannot be solved in time f ( k ) · | V ( G ) | o ( k/ √ log k ) for any function f , unless ETH fails. In fact, we obtain a tight bound, that is, we can drop the factor of 1 / √ log k in the exponent, assumingthe conjecture that “you cannot beat treewidth” [47]. The latter is an important open problem inparameterized and fine-grained complexity theory asking for a tight conditional lower bound for theproblem of finding a homomorphism from a small graph H to a large graph G : The best known algorithmfor that problem runs in time poly ( | V ( H ) | ) · | V ( G ) | O ( tw ( H )) , where tw ( H ) is the treewidth of H (see forinstance [25, 47, 20]), and the question is whether this running time is essentially optimal; we discuss thedetails later in the paper. We provide simplified statements; the formal and more general results can be found in [56, 26]. We will only rely on treewidth in a black-box manner in this paper and thus refer the reader for instance to [22,Chapter 7] for a detailed treatment. . Roth, J. Schmitt, and P. Wellnitz 4
As a concrete example of a property that is classified by Main Theorem 1, but that was not classifiedbefore, consider the (monotone) property of being 3-colorable: Recall that a 3-coloring of a graph is afunction mapping each vertex to one of three colors such that no two adjacent vertices are mapped to thesame color. Clearly any subgraph of a graph G admits a 3-coloring if G does.Note that in Main Theorem 1, the assumption of Φ not being k -trivial for infinitely many k is necessary,as otherwise the problem IndSub (Φ) becomes trivial for all k that are greater than a constant onlydepending on Φ.Note further, that W [ ]-completeness in Main Theorem 1 is not surprising, as the decision version of IndSub (Φ) was implicitly shown to be W [ ]-complete by Khot and Raman [43]; W [ ] is the decisionversion of W [ ] and should be considered a parameterized equivalent of NP . However, their reduction isnot parsimonious. Also, their proof uses Ramsey’s Theorem and thus only yields an implicit conditionallower bound of f ( k ) · | V ( G ) | o (log k ) , whereas our lower bound is almost tight.Our second result establishes an almost tight lower bound for sparse properties, that is, properties Φthat admit a constant s such that ever graph H for which Φ holds has at most s · | V ( H ) | many edges.Furthermore, the bound can be made tight if the set K (Φ) of positive integers k for which Φ is not k -trivialis additionally dense . By this we mean that there is a constant ‘ such that for every positive integer n ,there exists n ≤ k ≤ ‘n such that Φ is not k -trivial. Note that density rules out artificial properties suchas Φ( H ) = 1 if and only if H is an independent set and has precisely 2 ↑ n vertices for some positiveinteger n , with 2 ↑ n the n -fold exponential tower with base 2. In particular, for every property Φ that is k -trivial only for finitely many k , the set K (Φ) is dense. Main Theorem 2.
Let Φ denote a sparse graph property such that Φ is not k -trivial for infinitelymany k . Then, IndSub (Φ) is W [ ] -complete and cannot be solved in time f ( k ) · | V ( G ) | o ( k/ log k ) forany function f , unless ETH fails.If K (Φ) is additionally dense, then IndSub (Φ) cannot be solved in time f ( k ) · | V ( G ) | o ( k ) for anyfunction f , unless ETH fails. Our third result solves the open problem posed by Jerrum and Meeks by proving that (a strengthenedversion of) Conjecture 1.1 is true.
Main Theorem 3.
Let Φ denote a computable graph property that only depends on the number of edgesof a graph. If Φ is not k -trivial for infinitely many k , then IndSub (Φ) is W [ ] -complete and cannot besolved in time f ( k ) · | V ( G ) | o ( k/ log k ) for any function f , unless ETH fails.If K (Φ) is additionally dense, then IndSub (Φ) cannot be solved in time f ( k ) · | V ( G ) | o ( k/ √ log k ) forany function f , unless ETH fails. Note that, similar to Main Theorem 1, the conditional lower bounds in the previous two theorems becometight, if “you cannot beat treewidth” [47]; in particular, the condition of being dense can be removed inthat case.Finally, we consider properties that are hereditary , that is, closed under taking induced subgraphs. Weobtain a criterion on such graph properties that, if satisfied, yields a tight conditional lower bound for thecomplexity of
IndSub (Φ). While the statement of the criterion is deferred to the technical discussion,we can see that every hereditary property that is defined by a single forbidden induced subgraph satisfiesthe criterion.
Main Theorem 4.
Let H be a graph with at least vertices and let Φ denote the property of being H -free, that is, a graph satisfies Φ if and only if it does not contain H as an induced subgraph. Then, IndSub (Φ) is W [ ] -complete and cannot be solved in time f ( k ) · | V ( G ) | o ( k ) for any function f , unlessETH fails. Counting Small Induced Subgraphs Satisfying Monotone Properties
Note that the case of H being the graph with one vertex, which is excluded above, yields the property Φwhich is false on all graphs G with at least one vertex, for which IndSub (Φ) is the constant zero-functionand thus trivially solvable. Hence, Main Theorem 4 establishes indeed a complete classification for allproperties Φ=“ H -free”. We rely on the framework of Complexity Monotonicity of computing linear combinations of homomorphismcounts [20]. More precisely, it is known that for every computable graph property Φ and positive integer k ,there exists a unique computable function a from graphs to rational numbers such that for all graphs G IndSub (Φ , k → G ) = X H a ( H ) · Hom ( H → G ) , (2)where IndSub (Φ , k → G ) denotes the number of induced subgraphs of size k in G that satisfy Φ, and Hom ( H → G ) denotes the number of graph homomorphisms from H to G . It is known that the function a has finite support, that is, there is only a finite number of graphs H for which a ( H ) = 0.Intuitively, Complexity Monotonicity states that computing a linear combination of homomorphismcounts is precisely as hard as computing its hardest term [20]. Furthermore, the complexity of computingthe number of homomorphisms from a small graph H to a large graph G is (almost) precisely understoodby the dichotomy result of Dalmau and Jonsson [23] and the conditional lower bound under ETH dueto Marx [47]: Roughly speaking, it is possible to compute Hom ( H → G ) efficiently if and only if H has small treewidth. As a consequence, the complexity of computing IndSub (Φ , k → G ) is preciselydetermined by the support of the function a . Unfortunately, determining the latter turned out to be anincredibly hard task: It was shown in [56] and [26] that the function a subsumes a variety of algebraic andeven topological invariants. As a concrete example, a subset of the authors showed that for edge-monotoneproperties Φ, the coefficient a ( K k ) of the complete graph in Equation (2) is, up to a factor of k !, equal tothe reduced Euler characteristic of what is called the simplicial graph complex of Φ and k [56]. By this, aconnection to Karp’s Evasiveness Conjecture was established. In particular, it is known that Φ is evasiveon k -vertex graphs if the reduced Euler characteristic is non-zero [40]. As a consequence, the coefficient a ( K k ) can reveal a property to be evasive on k -vertex graphs if shown to be non-zero.The previous example illustrated that identifying the support of the function a in Equation (2) is ahard task, but using the framework of Complexity Monotonicity requires us to solve this task. In thiswork, we present a solution for properties whose f -vectors (see below) have low Hamming weight: Givena property Φ and a positive integer k , we define a (cid:0) k (cid:1) + 1 dimensional vector f Φ ,k by setting f Φ ,ki to bethe number of edge-subsets of size i of the complete graph with k vertices such that the induced graphsatisfies Φ, that is, f Φ ,ki := { A ⊆ E ( K k ) | A = i ∧ Φ( K k [ A ]) = 1 } for all i = 0 , . . . , (cid:0) k (cid:1) . By this, we lift the notion of f -vectors from abstract simplicial complexes to graphproperties; readers familiar with the latter will observe that the f -vector of an edge-monotone property Φequals the f -vector of its associated graph complex (see for instance [4]). Similarly, we introduce thenotions of h -vectors h Φ ,k and f -polynomials f Φ ,k of graph properties, defined as follows; we set d = (cid:0) k (cid:1) . h Φ ,k‘ := ‘ X i =0 ( − ‘ − i · (cid:18) d − i‘ − i (cid:19) · f Φ ,ki , where ‘ ∈ { , . . . , d } ; and f Φ ,k ( x ) := d X i =0 f Φ ,ki · x d − i . Intuitively, the Evasiveness Conjecture states that every decision tree algorithm verifying a non-trivial edge-monotonegraph property has to query every edge of the input graph in the worst case [54, 49]. . Roth, J. Schmitt, and P. Wellnitz 6
Our main combinatorial insight relates the function a in Equation (2) to the h -vector of Φ. For the formalstatement, we let H (Φ , k, i ) denote the set of all graphs H with k vertices and i edges that satisfy Φ. Wethen show that for all i = 0 , . . . , d , we have k ! X H ∈H (Φ ,k,i ) a ( H ) = h Φ ,ki . In particular, the previous equation shows that there is a graph H with i edges that survives with a non-zero coefficient a ( H ) in Equation (2) whenever the i -th entry of the h -vector h Φ ,k is non-zero. As graphswith many edges have high treewidth, we can thus establish hardness of computing IndSub (Φ , k → G )by proving that there is a non-zero entry with a high index in h Φ ,k . To this end, we relate h Φ ,k and f Φ ,k by observing that their entries are evaluations of the derivatives of the f -polynomial f Φ ,k ( x ). Moreconcretely, our goal is to show that a large amount of high-indexed zero entries of h Φ ,k yields that the onlypolynomial of degree at most d that satisfies the constrains given by the evaluations of the derivativesis the zero polynomial. However, the latter can only be true if Φ is trivially false on k -vertex graphs.Using Hermite-Birkhoff interpolation and Pólya’s Theorem we are able to achieve this goal whenever theHamming weight of f Φ ,k is small. Our meta-theorem thus classifies the complexity of IndSub (Φ) interms of the Hamming weight of the f -vectors of Φ. Main Theorem 5.
Let Φ denote a computable graph property and suppose that Φ is not k -trivial forinfinitely many k . Let β : K (Φ) → Z ≥ denote the function that maps k to (cid:0) k (cid:1) − hw ( f Φ ,k ) . If β ( k ) ∈ ω ( k ) then the problem IndSub (Φ) is W [ ] -complete and cannot be solved in time g ( k ) · n o (( β ( k ) /k ) / (log( β ( k ) /k ))) for any function g , unless ETH fails. For the refined conditional lower bounds in case of monotone properties and properties for which theset K (Φ) is dense (see Main Theorems 1 to 3), we furthermore rely on a consequence of the Kostochka-Thomason-Theorem [44, 63] that establishes a lower bound on the size of the smallest clique-minors ofgraphs with many edges.In contrast to the previous families of properties, we do not rely on the general meta-theorem (MainTheorem 5) for our treatment of hereditary properties. Instead, we carefully construct a reduction fromcounting k -independent sets in bipartite graphs. Given a hereditary graph property Φ defined by the(possibly infinite) set Γ(Φ) of forbidden induced subgraphs, we say that Φ is critical if there is a graph H ∈ Γ(Φ) and an edge e of H such that the graph obtained from H by deleting e and then cloning theformer endpoints of e satisfies Φ; the formal definition is provided in Section 6. The reduction fromcounting k -independent sets in bipartite graphs then yields the following result: Main Theorem 6.
Let Φ denote a computable and critical hereditary graph property. Then IndSub (Φ) is W [ ] -complete and cannot be solved in time g ( k ) · n o ( k ) for any function g , unless ETH fails. We then establish that every hereditary property with precisely one non-trivial forbidden subgraph H iscritical, which yields Main Theorem 4. We begin with providing all necessary technical background in Section 2. In particular, we introduce themost important notions in parameterized and fine-grained complexity theory, as well as the principle ofHermite-Birkhoff interpolation.
Counting Small Induced Subgraphs Satisfying Monotone Properties
Section 3 presents and proves our main combinatorial result which relates the f -vectors and h -vectors of acomputable graph property Φ on k -vertex graphs to the coefficients in the associated linear combinationof homomorphisms as given by Equation (2).We derive the meta-theorem for the complexity classification of IndSub (Φ) afterwards in Section 4and illustrate its applicability by establishing new conditional lower bounds for properties that aremonotone, that have low edge-densities, and that depend only on the number of edges of a graph. Thosebounds are refined to match the statements of Main Theorems 1 and 2 in Section 5 by combining ourmain combinatorial result with results from extremal graph theory that relate the number of edges of agraph to the size of its largest clique-minor.Finally, we present our treatment of hereditary properties in Section 6.
Acknowledgements
We thank Dániel Marx for pointing out a proof of Lemma 6.2 and Jacob Focke for helpful comments.
Given a finite set S , we write S and | S | for the cardinality of S . Further, given a non-negative integer r ,we set [ r ] := { , . . . , r } ; in particular, we have [0] = ∅ . The hamming weight of a vector f ∈ Q n , denotedby hw ( f ), is defined to be the number of non-zero entries of f .Graphs in this work are simple and do not contain self-loops. Given a graph G , we write V ( G ) for thevertices and E ( G ) for the edges of G . Furthermore, we define G to be the set of all (isomorphism classesof) graphs. The complement G of a graph G has vertices V ( G ) and edges E ( G ) \ {{ v, v } | v ∈ V ( G ) } .Given a subset ˆ E of edges of a graph G , we write G [ ˆ E ] for the graph with vertices V ( G ) andedges ˆ E . Given a subset ˆ V of vertices of a graph G , we write G [ ˆ V ] for the graph with vertices ˆ V andedges E ( G ) ∩ ˆ V . In particular, we say that G [ ˆ V ] is an induced subgraph of G . Given graphs H and G ,we define IndSub ( H → G ) to be the set of all induced subgraphs of G that are isomorphic to H .Given graphs H and G , a homomorphism from H to G is a function ϕ : V ( H ) → V ( G ) such that { ϕ ( u ) , ϕ ( v ) } ∈ E ( G ) whenever { u, v } ∈ E ( H ). We write Hom ( H → G ) for the set of all homomorph-isms from H to G . In particular, we write Hom ( H → ? ) for the function that maps a graph G to Hom ( H → G ). A bijective homomorphism from a graph H to itself is an automorphism and we write Aut ( H ) to denote the set of all automorphisms of H .For a graph H , we define the average degree of H as d ( H ) := 1 | V ( H ) | · X v ∈ V ( H ) deg ( v ) . Further, we rely on the treewidth of a graph, which is a graph parameter tw : G → N . However, we onlywork with the treewidth in a black-box manner, and thus we omit the definition and refer the interestedreader to the literature, (see for instance [22, Chapter 7]). In particular, we use the following well-knownresult from extremal graph theory, which relates the treewidth of a graph H to its average degree. Lemma 2.1 (Folklore, see for instance [10, Corollary 1]) . For any graph H with average degree at least d ,we have tw ( H ) ≥ d . Finally, we also rely on the following celebrated result from extremal graph theory:
Theorem 2.2 (Turán’s Theorem, see for instance [46, Section 2.1]) . A graph H with more than (cid:0) − r (cid:1) · | V ( H ) | edges contains the clique K r +1 as a subgraph. . Roth, J. Schmitt, and P. Wellnitz 8Graph Properties A graph property Φ is a function from graphs to { , } such that Φ( H ) = Φ( G ) whenever H and G are isomorphic. We say that a graph H satisfies Φ if Φ( H ) = 1. Given a positive integer k and agraph property Φ, we write Φ k for the set of all (isomorphism classes of) graphs with k vertices thatsatisfy Φ. Furthermore, given a graph G , a positive integer k , and a graph property Φ, we write IndSub (Φ , k → G ) for the set of all induced subgraphs with k vertices of G that satisfy Φ. In particular,we write IndSub (Φ , k → ? ) for the function that maps a graph G to IndSub (Φ , k → G ).Given a graph property Φ, we define ¬ Φ( H ) = 1 : ⇔ Φ( H ) = 0 as the negation of Φ. Furthermore, wedefine Φ( H ) = 1 : ⇔ Φ( H ) = 1 as the inverse of Φ. We observe the following identities:
Fact 2.3.
For every graph property Φ , graph G and positive integer k , we have IndSub ( ¬ Φ , k → G ) = (cid:18) | V ( G ) | k (cid:19) − IndSub (Φ , k → G ) , and IndSub (Φ , k → G ) = IndSub (Φ , k → G ) . Proof.
The first identity is immediate. For the second one, we observe that
IndSub (Φ , k → G ) = X H ∈ Φ k IndSub ( H → G ) = X H ∈ Φ k IndSub ( H → G )= X H ∈ Φ k IndSub ( H → G ) = IndSub (Φ , k → G ) , where we use the equality IndSub ( H → G ) = IndSub ( H → G ) from [46, Section 5.2.3]. Fine-Grained and Parameterized Complexity Theory
Given a computable graph property Φ, the problem
IndSub (Φ) asks, given a graph G with n verticesand a positive integer k , to compute IndSub (Φ , k → G ), that is, the number of induced subgraphs ofsize k in G that satisfy Φ. Note that the problem can be solved by brute-force in time f ( k ) · O ( n k ) byiterating over all subsets of k vertices in G and testing which of the subsets induce a graph that satisfies Φ;the latter part takes time f ( k ) for some f depending on Φ.As elaborated in the introduction, our goal is to understand the complexity of IndSub (Φ) forinstances with small k and large n . More precisely, we wish to identify the best possible exponentof n in the running time. To this end, we rely on the frameworks of fine-grained and parameterizedcomplexity theory. Regarding the former, we prove conditional lower bounds based on the ExponentialTime Hypothesis due to Impagliazzo and Paturi [36]:
Conjecture 2.4 (Exponential Time Hypothesis (ETH)) . The problem - SAT cannot be solved in time exp ( o ( m )) , where m is the number of clauses of the input formula. Assuming ETH, we are able to prove that the exponent ( k ) of the brute-force algorithm for IndSub (Φ)cannot be improved significantly for non-trivial monotone properties by establishing that no algorithmwith a running time of f ( k ) · | V ( G ) | o ( k/ √ log k ) for any function f exists. We omit using the word “complement” for graph properties to avoid confusion on whether we mean ¬ Φ or Φ.
Counting Small Induced Subgraphs Satisfying Monotone Properties
In the language of parameterized complexity theory, our reductions also yield W [ ]-completeness results,where W [ ] should be considered the parameterized counting equivalent of NP; we provide a roughintroduction in what follows and refer the interested reader to references like [22] and [30] for a detailedtreatment.A parameterized counting problem is a pair of a function P : Σ ∗ → N and a computable parameterization κ : Σ ∗ → N . Examples include the problems VertexCover and
Clique which ask, given a graph G and a positive integer k , to compute the number P ( G, k ) of vertex covers or cliques, respectively, of size k . Both problems are parameterized by the solution size, that is κ ( G, k ) := k . Similarly, the problem IndSub (Φ) can be viewed as a parameterized counting problem when parameterized by κ ( G, k ) := k ;we implicitly assume this parameterization of IndSub (Φ) in the remainder of this paper.A parameterized counting problem is called fixed-parameter tractable (FPT) if there is a computablefunction f such that the problem can be solved in time f ( κ ( x )) · | x | O (1) , where | x | is the input size.Given two parameterized counting problems ( P, κ ) and ( ˆ
P , ˆ κ ), a parameterized Turing-reduction from( P, κ ) to ( ˆ
P , ˆ κ ) is an algorithm A that is given oracle access to ˆ P and, on input x , computes P ( x ) in time f ( κ ( x )) · | x | O (1) for some computable function f ; furthermore, the parameter κ ( y ) of every oracle queryposed by A must be bounded by g ( κ ( x )) for some computable function g .While VertexCover is known to be fixed-parameter tractable [30],
Clique is not fixed-parametertractable, unless ETH fails [13, 14]. Moreover,
Clique is the canonical complete problem for theparameterized complexity class W [ ], see [30]; in particular, we use the following definition of W [ ]-completeness in this work. Definition 2.5.
A parameterized counting problem is W [ ] - complete if it is interreducible with Clique with respect to parameterized Turing-reductions.
Note that that the absence of an FPT algorithm for
Clique under ETH and the definition of parameter-ized Turing-reductions yield that W [ ]-complete problems are not fixed-parameter tractable, unless ETHfails, legitimizing the notion of W [ ]-completeness as evidence for (fixed-parameter) intractability. Jerrumand Meeks [37] have shown that IndSub (Φ) reduces to
Clique for every computable property Φwith respect to parameterized Turing-reductions. Thus we will only treat the “hardness part” of the W [ ]-completeness results in this paper.The fine-grained and parameterized complexity of the homomorphism counting problem are thefoundation of the lower bounds established in this work: Given a class of graphs H , the problem Hom ( H )asks, on input a graph H ∈ H and an arbitrary graph G , to compute Hom ( H → G ). The parameter isgiven by | V ( H ) | . The following classification shows that, roughly speaking, the complexity of Hom ( H )is determined by the treewidth of the graphs in H . Theorem 2.6 ([23, 47]) . Let H denote a recursively enumerable class of graphs. If the treewidth of H is bounded by a constant, then Hom ( H ) is solvable in polynomial time. Otherwise, the problem is W [ ] -complete and cannot be solved in time f ( | V ( H ) | ) · | V ( G ) | o (cid:0) tw ( H )log tw ( H ) (cid:1) for any function f , unless ETH fails. Note that the classification of
Hom ( H ) into polynomial-time and W [ ]-complete cases is explicitlystated and proved in the work of Dalmau and Jonson [23]. However, the conditional lower bound followsonly implicitly by a result of Marx [47]. We provide a proof for completeness. . Roth, J. Schmitt, and P. Wellnitz 10 Proof.
As discussed, we focus on the lower bound, which follows implicitly from a result of Marx [47]on the complexity of
Partitioned Subgraph Isomorphism : The problem
PartitionedSub ( H ) asks, givena graph H ∈ H , an arbitrary graph G and a (not necessarily proper) vertex coloring c : V ( G ) → V ( H ),to decide whether there is an injective homomorphism ϕ from H to G such that c ( ϕ ( v )) = v for eachvertex v of H .The result of Marx [47, Corollary 6.2] states that for every H of unbounded treewidth, the problem PartitionedSub ( H ) cannot be solved in time f ( | V ( H ) | ) · | V ( G ) | o ( tw ( H ) / log tw ( H )) (3)for any function f , unless ETH fails. Now suppose we are given a graph H ∈ H , an arbitrary graph G anda coloring c : V ( G ) → V ( H ). We wish to decide whether there is an injective homomorphism ϕ from H to G such that c ( ϕ ( v )) = v holds for each vertex v of H . Note first, that we can drop the requirementof ϕ being injective, as every homomorphism that preserves the coloring is injective. Note further, thatwithout loss of generality, we can assume that c is a homomorphism from G to H : Every edge { u, v } of G such that { c ( u ) , c ( v ) } / ∈ E ( H ) is irrelevant for finding a homomorphism ϕ from H to G that preserves thecoloring c . Hence, we can delete all of those edges from G . Thus, the problem PartitionedSub ( H ) isequivalent to the problem cp - Hom ( H ) which asks, given a graph H ∈ H , an arbitrary graph G , and ahomomorphism c ∈ Hom ( G → H ), to decide whether there is a ϕ ∈ Hom ( H → G ) such that c ( ϕ ( v )) = v for each v ∈ V ( H ). Finally, it is known that (the counting version of) cp - Hom ( H ) tightly reduces to Hom ( H ) via the principle of inclusion and exclusion [55, Lemma 2.52] or polynomial interpolation [24,Section 3.2]. Thus the conditional lower bound in Equation (3) holds for Hom ( H ) as well.The question whether the lower bound from Theorem 2.6 can be strengthened to f ( | V ( H ) | ) · | V ( G ) | o ( tw ( H )) is known as “Can you beat treewidth?” and constitutes a major open problem in parameterized com-plexity theory and an obstruction for tight conditional lower bounds on the complexity of a variety of(parameterized) problems, see for instance [45, 18, 21, 20].As described in the introduction, the complexity of computing a finite linear combination of homo-morphism counts is precisely determined by the complexity of computing the non-vanishing terms. Theformal statement is provided subsequently. Theorem 2.7 (Complexity Monotonicity [12, 20]) . Let a : G → Q denote a function of finite supportand let F denote a graph such that a ( F ) = 0 . There are a computable function g and a deterministicalgorithm A with oracle access to the function G X H ∈G a ( H ) · Hom ( H → G ) , and which, given a graph G with n vertices, computes Hom ( F → G ) in time g ( a ) · n c , where c is aconstant independent of a . Furthermore, each queried graph has at most g ( a ) · n vertices. As observed by Curticapean, Dell and Marx [20], counting induced subgraphs of size k that satisfy Φ isequivalent to computing a finite linear combination of homomorphism counts. Thus, the previous resultsyield an implicit dichotomy for IndSub (Φ).
Theorem 2.8 ([20]) . Let Φ denote a computable graph property and let k denote a positive integer.There is a unique and computable function a : G → Q of finite support such that IndSub (Φ , k → ? ) = X H ∈G a ( H ) · Hom ( H → ? ) . Furthermore, the problem
IndSub (Φ) is either fixed-parameter tractable or W [ ] -complete. Note that the result on
IndSub (Φ) in the previous theorem does not concern the fine-grained complexityof the problem. To reveal the latter, it is necessary to understand the support of the function a ; we tacklethis task in detail in Section 3. f -Vectors and h -Vectors It was observed in [56] that there is a close connection between the structure of the simplicial graphcomplex of edge-monotone properties Φ and the complexity of
IndSub (Φ). In this work, we generalizetwo important topological invariants of simplicial complexes to arbitrary graph properties: The f -vectorand the h -vector. Definition 2.9.
Let Φ denote a graph property, let k denote a positive integer and set d = (cid:0) k (cid:1) . The f -vector f Φ ,k = ( f Φ ,ki ) di =0 of Φ and k is defined by f Φ ,ki := { A ⊆ E ( K k ) | A = i ∧ Φ( K k [ A ]) = 1 } , where i ∈ { , . . . , d } , that is, f Φ ,ki is the number of edge-subsets of size i of K k such that the induced graph satisfies Φ .The h -vector h Φ ,k = ( h Φ ,k‘ ) d‘ =0 is defined by h Φ ,k‘ := ‘ X i =0 ( − ‘ − i · (cid:18) d − i‘ − i (cid:19) · f Φ ,ki , where ‘ ∈ { , . . . , d } . As mentioned before, note that those notions of f and h -vectors correspond to the eponymous notions forsimplicial (graph) complexes. We omit the definition of the latter as we are only concerned with thegeneralized notions and refer the interested reader e.g. to [4].It turns out that the non-vanishing of suitable entries h Φ ,k‘ of the h -vector implies hardness for IndSub (Φ). The result in [56] can be considered as a very restricted special case as it shows that thenon-vanishing of the reduced Euler characteristic of the complex (which is equal to the entry h Φ ,kd ) implieshardness. On the other hand, for many graph properties it is easy to deduce information about the f -vector (for instance that f Φ ,k‘ = 0 for sufficiently large ‘ with respect to k ). We observe that the f and h -vectors of a graph property are related by the so-called the f -polynomial which is again a generalizationof the epynomous notion for simplicial complexes: Definition 2.10.
Let Φ denote a graph property, let k denote a positive integer and set d = (cid:0) k (cid:1) . The f -polynomial of Φ and k is a univariate polynomial of degree at most d defined as follows: f Φ ,k ( x ) := d X i =0 f Φ ,ki · x d − i . As we see in the proof of Lemma 3.5, the entries of the f and h -vectors are given up to combinatorialfactors by derivatives of the f -polynomial at 0 and −
1. Intuitively, we apply Hermite-Birkhoff interpolationon f Φ ,k and its derivatives to prove that specific entries of h Φ ,k cannot vanish in case a sufficient numberof entries of f Φ ,k do, unless Φ is trivially false on k -vertex graphs. In some parts of the literature, the f -vector comes with an index shift of − . Roth, J. Schmitt, and P. Wellnitz 12Hermite-Birkhoff Interpolation and Pólya’s Theorem While a univariate polynomial of degree d is uniquely determined by d + 1 evaluations in pairwise differentpoints, the problem of Hermite-Birkhoff interpolation asks under which conditions we can uniquely recoverthe polynomial if we instead impose conditions on the derivatives of the polynomial at m distinct points.Following the notation of Schoenberg [58], the problem is formally expressed as follows. Given a matrix E = ( ε ij ) ∈ { , } m × d +1 where i ∈ { , . . . , m } and j ∈ { , . . . d } , as well as reals x < · · · < x m , the goalis to find a polynomial f of degree at most d such that for all i and j with ε ij = 1 we have f ( j ) ( x i ) = 0Here, f ( j ) denotes the j -th derivative of f . In particular, we are interested under which conditions on thematrix E , the zero polynomial is the unique solution. In this case, E is called poised . It turns out thatthe case m = 2 is sufficient for our purposes; fortunately, this case was fully solved by Pólya: Theorem 2.11 (Pólya’s Theorem [51, 58]) . Let E be defined as above with m = 2 . Suppose that P i,j ε ij = d + 1 and for every j ∈ { , . . . , d } set M j := j X i =0 ε ,i + ε ,i . Then, E is poised if and only if M j ≥ j + 1 holds true for all j ∈ { , . . . , d − } . In this section we discuss and prove our main technical result:
Theorem 3.1.
Let Φ denote a computable graph property, let k denote a positive integer, and let w denote the Hamming weight of the f -vector f Φ ,k . Suppose that Φ is not trivially false on k -vertex graphs.Then there is a unique and computable function a : G → Q of finite support such that IndSub (Φ , k → ? ) = X H ∈G a ( H ) · Hom ( H → ? ) , satisfying that there is a graph K on k vertices and at least (cid:0) k (cid:1) − w + 1 edges such that a ( K ) = 0 . First, recall from Theorem 2.8 that for any computable graph property Φ and positive integer k , there isa unique computable function a : G → Q (with finite support) satisfying IndSub (Φ , k → ? ) = X H ∈G a ( H ) · Hom ( H → ? ) . (4)Now, for the remainder of the section, fix a (computable) graph property Φ and a positive integer k (andthus the function a ). This allows us to simplify the notation for the f and h -vectors, as well as for the f -polynomial: We write f := f Φ ,k , h := h Φ ,k , and f := f Φ ,k . Furthermore, we set d := (cid:0) k (cid:1) and we write H i for the set of all graphs on k vertices and with i edges.Next, we define the vector ˜ h i as˜ h i := X K ∈H i a ( K ) , where i ∈ { , . . . , d } , that is, the i -th entry of ˜ h is the sum of the coefficients of graphs with k vertices and i edges in Equation (4).Now we establish the aforementioned connection between the coefficients of Equation (4) and the h -vectorof the property Φ. Lemma 3.2.
We have k ! · ˜ h = h . Note that as a consequence, the h -vector of a simplicial graph complex is determined by the coefficients ofits associated linear combination of homomorphisms. Proof.
Given two graphs H and H on k vertices each, we write { H ⊇ H } for the number of possibilitiesof adding edges to H such that (a graph isomorphic to) H is obtained. We start with the following claimwhich was implicitly shown in [56]; we include a proof for completeness. Claim 3.3.
Let K denote a graph with k vertices and define a as in Equation (4). We have a ( K ) = X H ∈ Φ k Aut ( H ) − · ( − E ( K ) − E ( H ) · { K ⊇ H } . Proof.
Fix a graph K with k vertices. Using the standard transformations from strong embeddingsto embeddings and from embeddings to homomorphisms (see for instance Lovász [46]), we obtain thefollowing: IndSub (Φ , k → ? )= X H ∈ Φ k Aut ( H ) − X H ∈G ( − E ( H ) − E ( H ) · { H ⊇ H } X ρ ≥∅ µ ( ∅ , ρ ) · Hom ( H /ρ → ? ) , where µ denotes the Möbius function and the rightmost sum ranges over the partition lattice of the setof vertices of H . Furthermore, H /ρ is the quotient graph obtained by identifying vertices of H alongthe partition ρ . In particular, H / ∅ = H . We omit the details, which can be found in [46, 56], as weonly need that H /ρ has strictly less than k vertices for all ρ > ∅ and that µ ( ∅ , ∅ ) = 1. This allows us torewrite the previous equation as follows: IndSub (Φ , k → ? )= X H ∈ Φ k Aut ( H ) − X H ∈G ( − E ( H ) − E ( H ) { H ⊇ H } · Hom ( H → ? ) + R (Φ k ) , where the remainder R (Φ k ) does not depend on any numbers Hom ( F → ? ) for graphs F with k vertices.In particular, reordering and grouping the coefficients of Hom ( K → ? ) yields the claim.Next, we investigate the term P K ∈H ‘ { K ⊇ H } . Claim 3.4.
Let ‘ ∈ { , . . . , d } denote an integer and let H denote a graph with k vertices and at most ‘ edges. Then, we have X K ∈H ‘ { K ⊇ H } = (cid:18) d − E ( H ) ‘ − E ( H ) (cid:19) . Proof.
Any extension from the graph H to a graph with ‘ edges has to add ‘ − E ( H ) edges to H ; thereare exactly d − E ( H ) possible choices for these ‘ − E ( H ) edges. Hence the claim follows from basiccombinatorics. This step is done explicitly in [56]. . Roth, J. Schmitt, and P. Wellnitz 14
Now, fix an ‘ ∈ { , . . . , d } ; we proceed to show that k ! · ˜ h ‘ = h ‘ , which proves the lemma. To that end,from the definition of ˜ h , we obtain k ! · ˜ h ‘ = k ! · X K ∈H ‘ a ( K )= k ! · X K ∈H ‘ X H ∈ Φ k Aut ( H ) − · ( − ‘ − E ( H ) · { K ⊇ H } = X H ∈ Φ k k ! · Aut ( H ) − · ( − ‘ − E ( H ) X K ∈H ‘ { K ⊇ H } , where the second equality holds due to Claim 3.3. Now observe that { K ⊇ H } = 0 if H has more edgesthan K . Thus we see that k ! · ˜ h ‘ = X H ∈ Φ k E ( H ) ≤ ‘ k ! · Aut ( H ) − · ( − ‘ − E ( H ) X K ∈H ‘ { K ⊇ H } = X H ∈ Φ k E ( H ) ≤ ‘ k ! · Aut ( H ) − · ( − ‘ − E ( H ) · (cid:18) d − E ( H ) ‘ − E ( H ) (cid:19) , where the last equality holds due to Claim 3.4. Next we use the fact that k ! is the order of the symmetricgroup Sym k : For any graph H in the above sum, choose a set A of edges of the labeled complete graph K k on k vertices such that the corresponding subgraph K k [ A ] is isomorphic to H . The group Sym k acts onthe vertices and thus on the edges of K k . By the definition of a graph automorphism, the stabilizer of theset A has exactly Aut ( H ) elements.Now observe that the orbit of A under Sym k is the collection of all sets A such that K k [ A ] ∼ = H .Therefore, by the Orbit Stabilizer Theorem, we have k ! · Aut ( H ) − = { A ⊆ E ( K k ) | K k [ A ] ∼ = H } . Hence we can conclude that k ! · ˜ h ‘ = X H ∈ Φ k E ( H ) ≤ ‘ { A ⊆ E ( K k ) | K k [ A ] ∼ = H } · ( − ‘ − E ( H ) · (cid:18) d − E ( H ) ‘ − E ( H ) (cid:19) = ‘ X i =0 X H ∈ Φ k E ( H )= i { A ⊆ E ( K k ) | K k [ A ] ∼ = H } · ( − ‘ − i · (cid:18) d − i‘ − i (cid:19) = ‘ X i =0 { A ⊆ E ( K k ) | A = i ∧ Φ( K k [ A ]) = 1 } · ( − ‘ − i · (cid:18) d − i‘ − i (cid:19) = ‘ X i =0 f i · ( − ‘ − i · (cid:18) d − i‘ − i (cid:19) = h ‘ , completing the proof.In the next step, we use Pólya’s Theorem to prove that the Hamming weight of the f -vector determines anindex β of the h -vector such that at least one entry of h with index at least β is non-zero. By Lemma 3.2the same then follows for ˜ h . Lemma 3.5.
Let w denote the Hamming weight of f and set β = d − w . If Φ is not trivially false on k -vertex graphs then at least one of the values h d , . . . , h β +1 is non-zero. Proof.
Recall the definition of the f -polynomial f ( x ) = P di =0 f i · x d − i and observe that f ( j ) ( x ) = d − j X i =0 f i · ( d − i ) j · x d − j − i . By j j = j !, we immediately obtain f ( j ) (0) = f d − j · j !. Therefore, by assumption, we have f ( j ) (0) = 0 for β + 1 many indices j .Furthermore, we see that f ( j ) ( −
1) = d − j X i =0 f i · ( d − i ) j · ( − d − j − i = j ! · d − j X i =0 f i · (cid:18) d − ij (cid:19) · ( − d − j − i = j ! · d − j X i =0 f i · (cid:18) d − i ( d − j ) − i (cid:19) · ( − d − j − i = j ! · h d − j . Now assume for the sake of contradiction that each of the values h d , . . . , h β +1 is zero. Consequently, f ( j ) ( −
1) = 0 for j = 0 , . . . , w −
1. Interpreting those evaluations of the derivatives of the f -polynomial asan instance of Hermite-Birkhoff interpolation, the corresponding matrix E looks as follows: . . . w − w . . . d (cid:18) (cid:19) . . . . . . ε ε ε . . . ε w − ε w . . . ε d In particular, at least β + 1 = d + 1 − w of the values ε j are 1; As β + 1 and w sum up to d + 1, we caneasily verify that the conditions of Pólya’s Theorem (Theorem 2.11) are satisfied: Let us modify E byarbitrarily choosing precisely β + 1 of the ε ,j that are 1 and set the others to 0, and call the resultingmatrix ˆ E . We then have both M j ≥ j + 1 (for all j ∈ { , . . . , d − } ) and the first and second row of ˆ E sum up to precisely d + 1. Hence the matrix ˆ E is poised, that is, the only polynomial of degree at most d that satisfies the corresponding instance of Hermite-Birkhoff interpolation is the zero polynomial. As weobtained ˆ E from E just by ignoring some vanishing conditions, the same conclusion is true for E andthus f = 0 is the unique solution. This, however, contradicts the fact that the property Φ is not triviallyfalse on k -vertex graphs, completing the proof.Combining Lemmas 3.2 and 3.5 yields our main technical result, which we restate here for convenience. Theorem 3.1.
Let Φ denote a computable graph property, let k denote a positive integer, and let w denote the Hamming weight of the f -vector f Φ ,k . Suppose that Φ is not trivially false on k -vertex graphs.Then there is a unique and computable function a : G → Q of finite support such that IndSub (Φ , k → ? ) = X H ∈G a ( H ) · Hom ( H → ? ) , satisfying that there is a graph K on k vertices and at least (cid:0) k (cid:1) − w + 1 edges such that a ( K ) = 0 . Recall that an entry 1 in the matrix E represents an evaluation f ( j ) ( −
1) = 0 in the first row and an evaluation f ( j ) (0) = 0 in the second row. . Roth, J. Schmitt, and P. Wellnitz 16 Proof.
Set d = (cid:0) k (cid:1) and β = d − w . By Equation (4) the function a exists and is computable and hasa finite support. Now, Lemma 3.5 implies that at least one of the values h Φ ,kd , . . . , h Φ ,kβ +1 is non-zeroand thus, by Lemma 3.2, at least one of the values ˜ h d , . . . , ˜ h β +1 is non-zero as well. Next, observe that˜ h i = P K ∈H i a ( K ) for all i ∈ { , . . . , d } , where H i is the set of all graphs on k vertices and i edges. Inparticular, ˜ h i = 0 implies that a ( K ) = 0 for at least one K ∈ H i , yielding the claim. (Φ) by the Hamming Weight of the f -Vectors In this section, we derive a general hardness result for
IndSub (Φ) based on the Hamming weight ofthe f -vector. In a sense, we “black-box” Theorem 3.1; using the resulting classification, we establish firsthardness results and almost tight conditional lower bounds for a variety of families of graph properties.However, note that taking a closer look at the number of edges of the graphs with non-vanishingcoefficients (as provided by Theorem 3.1) often yields improved, sometimes even matching conditionallower bounds; we defer the treatment of the refined analysis to Section 5.In what follows, we write K (Φ) for the set of all k such that Φ k is non-empty. Theorem 4.1.
Let Φ denote a computable graph property and suppose that the set K (Φ) is infinite.Let β : K (Φ) → Z ≥ denote the function that maps k to (cid:0) k (cid:1) − hw ( f Φ ,k ) . If β ( k ) ∈ ω ( k ) then the problem IndSub (Φ) is W [ ] -complete and cannot be solved in time g ( k ) · | V ( G ) | o (( β ( k ) /k ) / log( β ( k ) /k )) for any function g , unless ETH fails. The same statement holds for IndSub (Φ) and
IndSub ( ¬ Φ) . Note that the condition of K (Φ) being infinite is necessary for hardness: Otherwise there is a constant c such that we can output 0 whenever k ≥ c and solve the problem by brute-force if k < c , yielding analgorithm with a polynomial running time. Note further that the (log( β ( k ) /k )) − -factor in the exponentis related to the question of whether it is possible to “beat treewidth” [47]. In particular, if the factor of(log tw ( H )) − in Theorem 2.6 can be dropped, then all further results in this section can be strengthenedto yield tight conditional lower bounds under ETH. Proof.
By Theorem 3.1, for each k ∈ K (Φ) we obtain a graph H k with k vertices and at least β ( k ) edgessuch that a ( H k ) = 0, where a is the function in Equation (4). The average degree of H k satisfies d ( H k ) = 1 k · X v ∈ V ( H k ) deg ( v ) = 2 | E ( H k ) | k ≥ β ( k ) k , where the second equality is due to the Handshaking Lemma. By Lemma 2.1, we thus obtain that tw ( H k ) ≥ β ( k ) k , which is unbounded as β ( k ) ∈ ω ( k ) by assumption.Now let H denote the set of all graphs H k for k ∈ K (Φ). By Theorem 2.6, we obtain that Hom ( H )is W [ ]-complete and cannot be solved in time g ( k ) · | V ( G ) | o (( β ( k ) /k ) / log( β ( k ) /k )) for any function g , unless ETH fails. Further, by Complexity Monotonicity (Theorem 2.7), the same istrue for IndSub (Φ) as well. Finally, we use Fact 2.3 to obtain the same result for
IndSub (Φ) and
IndSub ( ¬ Φ); completing the proof. See Theorem 2.6 and its discussion.
As a first application of Theorem 4.1, we consider properties Φ that satisfy hw ( f Φ ,k ) ∈ o ( k ) . We say that such a property Φ has a low edge-densities . Properties with low edge-density subsume,for example, exclusion of a set of fixed minors such as planarity. They have been studied by Jerrumand Meeks [38], where they show that
IndSub (Φ) is W [ ]-complete for these properties. However,their proof uses Ramsey’s Theorem and thus only establishes an implicit conditional lower bound of g ( k ) · | V ( G ) | o (log k ) . In contrast, we achieve the following, almost tight lower bound: Theorem 4.2.
Let Φ denote a computable graph property with low edge-densities. Suppose that theset K (Φ) is infinite. Then IndSub (Φ) is W [ ] -complete and cannot be solved in time g ( k ) · | V ( G ) | o ( k/ log k ) for any function g , unless ETH fails. The same is true for IndSub (Φ) and
IndSub ( ¬ Φ) . Proof.
If Φ has low edge-densities, then we have β ( k ) = (cid:0) k (cid:1) − hw ( f Φ ,k ) ∈ Θ( k ). Thus o (cid:18) β ( k ) /k log( β ( k ) /k ) (cid:19) = o ( k/ log k ) . The claim hence follows by Theorem 4.1.The previous result applies, in particular, to sparse properties. In Section 5 we show, that a refinedanalysis based on Turán’s Theorem as well as Theorem 3.1 establishes a tight conditional lower bound forsparse properties Φ that additionally satisfy a density condition on K (Φ); the combination of those tworesults then implies Main Theorem 2. Jerrum and Meeks [38, 39] asked whether
IndSub (Φ) is W [ ]-complete whenever Φ is non-trivialinfinitely often and only depends on the number of edges of a graph, that is, ∀ H , H : | E ( H ) | = | E ( H ) | ⇒ Φ( H ) = Φ( H ) . We answer this question affirmatively, even for properties that can depend both on the number of edgesand vertices of the graph, and additionally provide an almost tight conditional lower bound:
Theorem 4.3.
Let Φ denote a computable graph property that only depends on the number of edgesand the number of vertices of a graph. If Φ k is non-trivial only for finitely many k then IndSub (Φ) isfixed-parameter tractable. Otherwise,
IndSub (Φ) is W [ ] -complete and cannot be solved in time g ( k ) · | V ( G ) | o ( k/ log k ) for any function g , unless ETH fails. Note that Theorem 4.3 is also true for
IndSub (Φ) and
IndSub ( ¬ Φ), as ¬ Φ and Φ depend only onthe number of edges and vertices of a graph if and only if Φ does. . Roth, J. Schmitt, and P. Wellnitz 18
Proof.
First, assume that Φ k is non-trivial only for finitely many k . Then, there is a constant c suchthat for every k > c , the property Φ k is either trivially true or trivially false. Hence, given as input agraph G and an integer k , we check whether k ≤ c . If this is the case, we solve the problem by brute-force.Otherwise, we check whether Φ k is trivially false or trivially true. If Φ k is false, we output 0; otherwisewe output (cid:0) nk (cid:1) . It is immediate that this algorithm yields fixed-parameter tractability.Now assume that Φ k is non-trivial for infinitely many k . Since for Φ k we fix the number of vertices tobe k , by assumption Φ k only depends on the number of edges of a graph. Thus, we have hw ( f ¬ Φ ,k ) = (cid:18) k (cid:19) − hw ( f Φ ,k ) . (5)Hence, setˆΦ k := ( Φ k if hw ( f Φ ,k ) ≤ (cid:0) k (cid:1) ¬ Φ k if hw ( f Φ ,k ) > (cid:0) k (cid:1) . We observe that, by assumption, K ( ˆΦ) is infinite, and by Fact 2.3 the problems IndSub (Φ) and
IndSub ( ˆΦ) are equivalent. By definition and by Equation (5), we see that hw ( f ˆΦ ,k ) ≤ (cid:0) k (cid:1) andtherefore β ( k ) = (cid:0) k (cid:1) − hw ( f ˆΦ ,k ) ∈ Θ( k ). Thus, we have o (cid:18) β ( k ) /k log( β ( k ) /k ) (cid:19) = o ( k/ log k ) . The claim now follows by Theorem 4.1.
Recall that a property Φ is called monotone if it is closed under taking subgraphs. The decision version of
IndSub (Φ), that is, deciding whether there is an induced subgraph of size k that satisfies Φ is known tobe W [ ]-complete if Φ is monotone. Further, Φ is non-trivial and K (Φ) is infinite (this follows implicitly bya result of Khot and Raman [43]). However, as the reduction of Khot and Raman is not parsimonious, thereduction does not yield W [ ]-completeness of the counting version. More importantly, the proof of Khotand Raman uses Ramsey’s Theorem and thus only implies a conditional lower bound of g ( k ) · | V ( G ) | o (log k ) .Using our main result, we achieve a much stronger and almost tight lower bound under ETH. Theorem 4.4.
Let Φ denote a computable graph property that is monotone and non-trivial. Supposethat K (Φ) is infinite. Then IndSub (Φ) is W [ ] -complete and cannot be solved in time g ( k ) · | V ( G ) | o ( k/ log k ) for any function g , unless ETH fails. The same is true for IndSub (Φ) and
IndSub ( ¬ Φ) . Proof.
As Φ is non-trivial, there is a graph F such that Φ( H ) is false for every H that contains F as a(not necessarily induced) subgraph. Set r = | V ( F ) | and fix k ∈ K (Φ). By Turáns Theorem (Theorem 2.2)we have that every graph H on k vertices with more than (cid:0) − r (cid:1) · k edges contains the clique K r +1 andthus F as a subgraph. Consequently, Φ is false on every graph with k vertices and more than (cid:0) − r (cid:1) · k edges. Therefore, we have β ( k ) = (cid:18) k (cid:19) − hw ( f Φ ,k ) ≥ (cid:18) k (cid:19) − (cid:18) − r (cid:19) · k k r − k ∈ Ω( k ) . This step is the reason why we only get fixed-parameter tractability and not necessarily polynomial-time tractability.
Thus β ( k ) ∈ Θ( k ) and we conclude that o (cid:18) β ( k ) /k log( β ( k ) /k ) (cid:19) = o ( k/ log k ) . The claim hence follows by Theorem 4.1.
Recall that the lower bounds of the previous section become tight if it is impossible to “beat treewidth”,that is, if the (log k ) − factor in the exponent of Theorem 2.6 can be dropped. In this section, we showthat the lower bounds of the previous section can also be refined—and in case of sparse properties evenbe made tight—without the latter assumption. This requires relying on two results from extremal graphtheory on forbidden cliques and clique-minors. The first one is Turán’s Theorem, which we have seenalready. The second one is a consequence of the Kostochka-Thomason-Theorem: Theorem 5.1 ([44, 63]) . There is a constant c > such that every graph H with an average degree ofat least ct √ log t contains the clique K t as a minor. Note that Theorem 5.1 is often stated in terms of the number of edges of a graph, instead of its averagedegree. However, due to the Handshaking-Lemma, both statements are equivalent.Roughly speaking, we combine the Kostochka-Thomason-Theorem with Theorem 3.1 to find graphswith large clique-minors in the linear combination of homomorphisms associated with a graph property Φas given by Equation (4). This then allows us to derive hardness by a reduction from the problem of finding cliques, instead of relying on Theorem 2.6.In what follows, we say that a subset K of the natural numbers is dense if there is a constant ‘ ≥ n ∈ N > there is a k ∈ K that satisfies n ≤ k ≤ ‘n . Now recall that K (Φ) is the setof all k such that Φ k is not empty. The lower bounds for IndSub (Φ) in the current section requirethe set K (Φ) to be dense. Roughly speaking, this is to exclude artificial properties (such as for instanceΦ( H ) = 1 if and only if H is an independent set and | V ( H ) | = 2 ↑ m for some m ∈ N , where 2 ↑ m isthe m -fold exponential tower of 2). While the latter property satisfies the conditions of Theorem 4.1, wecannot construct a tight reduction to IndSub (Φ) as the only non-trivial oracle queries satisfy k = 2 ↑ m for some m ∈ N . Fortunately, monotone properties exclude such artificial properties: Lemma 5.2.
Let Φ denote a non-trivial monotone graph property such that K (Φ) is infinite. Then K (Φ) is the set of all positive integers and thus dense. Proof.
Fix a n ∈ N > . As K (Φ) is infinite, there is a k ≥ n in K (Φ). Thus there is a graph H ∈ Φ k . Nowdelete k − n arbitrary vertices of H and call the resulting graph H . As Φ is monotone, we have that H ∈ Φ n and hence n ∈ K (Φ) as well.The following technical lemma is the basis for the lower bounds in this section; recall that theproblem Hom ( H ) asks, given a graph H ∈ H and an arbitrary graph G , to compute the number ofhomomorphisms from H to G . Lemma 5.3.
Let r ≥ denote a constant and let H denote a decidable class of graphs. Further,let K ( H ) denote the set of all positive integers k such that there is a graph H ∈ H with k vertices and atleast k / r − k/ edges. If K ( H ) is dense, then Hom ( H ) cannot be solved in time f ( | V ( H ) | ) · | V ( G ) | o ( | V ( H ) | / √ log | V ( H ) | ) for any function f , unless ETH fails. . Roth, J. Schmitt, and P. Wellnitz 20 Proof.
We construct a tight reduction from the problem
Clique , which asks, given a graph G and aparameter ˆ k ∈ N > , to decide whether there is a clique of size ˆ k in G . It is known that Clique cannot besolved in time ˆ f (ˆ k ) · | V ( G ) | o (ˆ k ) for any function ˆ f , unless ETH fails [13, 14].Now assume there is an algorithm A that solves IndSub (Φ) in time f ( | V ( H ) | ) · | V ( G ) | o ( | V ( H ) | / √ log | V ( H ) | ) for some function f . We use A to solve Clique in time ˆ f (ˆ k ) · | V ( G ) | o (ˆ k ) for some function ˆ f .As K ( H ) is dense, there is a constant ‘ ≥ k ∈ N > there is a k ∈ K ( H ) such thatˆ k ≤ k ≤ ‘ ˆ k . Given a graph G with n vertices and ˆ k ∈ N > , we construct the graph G as follows: We firstsearch for a k ∈ K ( H ) that satisfies ˆ k ≤ k ≤ ‘ ˆ k . Note that finding k is computable as K ( H ) is decidable.Then, we add k − ˆ k “fresh” vertices to G and add edges between all pairs of new vertices and between allpairs of a new vertex and an old vertex.Next, let c denote the constant from Theorem 5.1 and set h ( k ) := cr · p log( k/cr ) , where r := max { /c, r + 1 } . Now, we construct a graph ˆ G from G as follows: The vertices of ˆ G are the h ( k )-cliques of G , and two vertices C and C of ˆ G are made adjacent if all edges between vertices in C and vertices in C are present in G . Note that ˆ G has O ( n h ( k ) ) vertices and can be constructed intime g (ˆ k ) · O ( n h ( k ) ) for some computable function g . Claim 5.4.
The graph G contains a clique of size ˆ k if and only if ˆ G contains a clique of size k/h ( k ). Proof.
Let C denote a ˆ k -clique in G . Then we obtain a k -clique in G by adding the fresh k − ˆ k verticesto C . Next, partition C in blocks of size h ( k ). Each block will be a vertex of ˆ G and all correspondingvertices are pairwise adjacent by the construction of ˆ G . As there are k/h ( k ) many blocks, we found thedesired clique in ˆ G .For the other direction, let ˆ C denote a k/h ( k )-clique in ˆ G . By the definition of ˆ G , each vertex of ˆ C corresponds to a clique of size h ( k ) in G . Furthermore, two different of those cliques cannot share acommon vertex as the corresponding vertices in ˆ G are adjacent—recall that we do not allow self-loops.Consequently, the union of the k/h ( k ) many cliques constitutes a clique of size k in G . Finally, at most k − ˆ k of the vertices of this clique can be fresh vertices, and thus G contains a clique of size (at least) ˆ k .Next, we search for a graph H ∈ H with k vertices and at least k / r − k/ g ( k ) for some computable function g as H is decidable. Now we see that d ( H ) = 1 k · X v ∈ V ( H ) deg ( v ) = 2 | E ( H ) | k ≥ kr − . Formally, we have to round h ( k ) and later k/h ( k ). For the sake of readability, we assume that all logs and fractionsyield integers, but we point out that this might require to find a d k/h ( k ) e -clique at the end of the proof, while theoracle can only determine the existence of a b k/h ( k ) c -clique. However, using the latter, we can easily decide whetherthere is a d k/h ( k ) e -clique by checking for each vertex whether its neighbourhood contains a b k/h ( k ) c -clique. Set t = kh ( k ) . For k large enough, we have ct p log t = c · kcr · p log( k/cr ) · q log( k/cr ) − log p log( k/cr ) ≤ c · kcr · p log( k/cr ) · p log( k/cr ) = kr ≤ kr − ≤ d ( H )Theorem 5.1 thus implies that K t is a minor of H . Furthermore, it is known that, whenever a graph F is a minor of a graph H , there is a tight reduction from counting homomorphisms from F to countinghomomorphisms from H — see for instance [55, Chapter 2.5] and Section 3 in the full version of [24].Consequently, we can use the algorithm A to compute the number of homomorphisms from K t to ˆ G . Byassumption on A , this takes time at most f ( | V ( H ) | ) · | V ( ˆ G ) | o ( | V ( H ) | / √ log | V ( H ) | ) = f ( k ) · ( n h ( k ) ) o ( k/ √ log k ) = f ( k ) · n o ( k ) , where the latter holds as h ( k ) ∈ Θ( √ log k )—recall that r and c are constants. However, it is easy to seethat ˆ G contains a t -clique if and only if the number of homomorphisms from K t to ˆ G is at least 1. ByClaim 5.4, this is equivalent to G having a clique of size ˆ k . As k ∈ O (ˆ k ) (recall that ‘ is a constant), theoverall running time is bounded by g (ˆ k ) · O ( n h ( k ) ) + g ( k ) + f ( k ) · n o ( k ) ≤ ˆ f (ˆ k ) · n o (ˆ k ) for ˆ f (ˆ k ) := g (ˆ k ) + g ( ‘ ˆ k ) + f ( ‘ ˆ k ). This yields the desired contradiction and concludes the proof.We are now able to establish the refined lower bounds for IndSub (Φ).
Theorem 5.5.
Let Φ denote a computable graph property that is monotone and non-trivial. Supposethat K (Φ) is infinite. Then IndSub (Φ) cannot be solved in time g ( k ) · | V ( G ) | o ( k/ √ log k ) for any function g , unless ETH fails. The same is true for IndSub (Φ) and
IndSub ( ¬ Φ) . Proof.
We begin similarly as in the proof of Theorem 4.4: As Φ is non-trivial, there is a graph F suchthat Φ( H ) is false for every H that contains F as a (not necessarily induced) subgraph. Set r = | V ( F ) | and fix k ∈ K (Φ). By Turáns Theorem (Theorem 2.2) we have that every graph H on k vertices withmore than (cid:0) − r (cid:1) · k edges contains the clique K r +1 and thus F as a subgraph. Consequently, Φ is falseon every graph with k vertices and more than (cid:0) − r (cid:1) · k edges. We now use Theorem 3.1 and obtain acomputable and unique function a of finite support such that IndSub (Φ , k → ∗ ) = X H ∈G a ( H ) · Hom ( H → ∗ ) , satisfying that there is a graph H k on k vertices and at least (cid:18) k (cid:19) − hw ( f Φ ,k ) + 1 ≥ (cid:18) k (cid:19) − (cid:18) − r (cid:19) · k ≥ k r − k If k is not large enough for the inequalities to hold, then k and thus ˆ k are bounded by a constant, and we can computethe number of ˆ k -cliques in G by brute-force. Full version available at https://arxiv.org/abs/1902.04960 . . Roth, J. Schmitt, and P. Wellnitz 22 edges such that a ( H k ) = 0. Consequently, Complexity Monotonicity (Theorem 2.7) yields a tight reductionfrom the problem Hom ( H ) where H := { H k | k ∈ K (Φ) } . By the previous observation, the graph H k has k vertices and at least k / r − k/ k such that H k ∈ H isdense. Thus we can use Lemma 5.3, which concludes the proof—note that the results for IndSub (Φ)and
IndSub ( ¬ Φ) follow by Fact 2.3.We continue with the refined lower bound for properties that only depend on the number of edges of agraph; in this case, we have to assume density.
Theorem 5.6.
Let Φ denote a computable graph property that only depends on the number of edges ofa graph. If the set of k for which Φ k is non-trivial is dense, then IndSub (Φ) cannot be solved in time g ( k ) · | V ( G ) | o ( k/ √ log k ) for any function g , unless ETH fails. Proof.
We use the same set-up as in the proof of Theorem 4.3. In particular, we obtain ˆΦ such that
IndSub (Φ) and
IndSub ( ˆΦ) are equivalent, K ( ˆΦ) is dense, and hw ( f ˆΦ ,k ) ≤ (cid:0) k (cid:1) for all k ∈ H ( ˆΦ). Weuse Theorem 3.1 and obtain a computable and unique function a of finite support such that IndSub (Φ , k → ∗ ) = X H ∈G a ( H ) · Hom ( H → ∗ ) , satisfying that there is a graph H k on k vertices and at least (cid:18) k (cid:19) − hw ( f Φ ,k ) + 1 ≥ k − k a ( H k ) = 0. The application of Complexity Monotonicity (2.7) and Lemma 5.3 is nowsimilar to the previous proof; the only difference is, that we can choose r = 2.As a final result in this section, we establish a tight conditional lower bound for sparse properties; recallthat a property Φ is called sparse if there is a constant s only depending on Φ such that Φ is false on everygraph with k vertices and more than sk edges. Instead of relying in the Kostochka-Thomason-Theorem,it suffices to use Turán’s Theorem, however. Theorem 5.7.
Let Φ denote a computable sparse graph property such that K (Φ) is dense. Then IndSub (Φ) cannot be solved in time g ( k ) · | V ( G ) | o ( k ) for any function g , unless ETH fails. The same is true for IndSub (Φ) and
IndSub ( ¬ Φ) . Proof.
Let s denote the constant given by the definition of sparsity, and fix k ∈ K (Φ). We use Theorem 3.1and obtain a computable and unique function a of finite support such that IndSub (Φ , k → ∗ ) = X H ∈G a ( H ) · Hom ( H → ∗ ) , satisfying that there is a graph H k on k vertices and at least (cid:0) k (cid:1) − hw ( f Φ ,k ) + 1 edges such that a ( H k ) = 0.Now choose r := d k s +1)+1 e , and observe that (cid:18) k (cid:19) − hw ( f Φ ,k ) + 1 > (cid:18) k (cid:19) − sk > (cid:18) k (cid:19) − ( s + 1) k ≥ (cid:18) − r (cid:19) · | V ( H k ) | . By Turáns Theorem (Theorem 2.2), H k hence contains K r +1 as a subgraph and, in particular, K r asa minor. Furthermore, Complexity Monotonicity (Theorem 2.7) shows that we can, given a graph G compute Hom ( H k → G ) in linear time if we are given oracle access to IndSub (Φ , k → ? ). As we haveseen in the proof of Lemma 5.3, it is known that, whenever a graph F is a minor of a graph H , there is atight reduction from counting homomorphisms from F to counting homomorphisms from H [24, 55]. Inparticular, this implies that we can compute Hom ( K r → G ) in linear time if we are given oracle accessto IndSub (Φ , k → ? ). Note further that Hom ( K r → G ) is at least 1 if and only if G contains a cliqueof size r .We continue similarly as in the proof of Lemma 5.3: Assume that there is an algorithm A that solves IndSub (Φ) in time g ( k ) · | V ( G ) | o ( k ) for some function g . We show that A can be used to solve theproblem of finding a clique of size ˆ k in a graph G in time ˆ f (ˆ k ) · | V ( G ) | o (ˆ k ) for some function ˆ f . Given ˆ k and G , search for the smallest k ∈ K (Φ) such thatˆ k ≤ d k s + 1) + 1 e , and note that finding such a k is computable in time only depending on ˆ k as Φ is computable. Notefurther that k ∈ O (ˆ k ) as s is a constant and K (Φ) is dense.We construct the graph G from G by adding d k s +1)+1 e − ˆ k “fresh” vertices and adding edges betweenevery pair of new vertices and every pair containing one new and one old vertex. It is easy to see that G hasa clique of size ˆ k if and only if G has a clique of size d k s +1)+1 e . By the analysis and assumptions above,we can decide whether the latter is true by using A in time g ( k ) · | V ( G ) | o ( k ) . As | V ( G ) | ∈ O ( | V ( G ) | ) and k ∈ O (ˆ k ), the overall time to decide whether G has a clique of size ˆ k is hence bounded by ˆ f (ˆ k ) · | V ( G ) | o (ˆ k ) for some function ˆ f , which is impossible, unless ETH fails [13, 14]. The results for IndSub (Φ) and
IndSub ( ¬ Φ) follow by Fact 2.3.
While we established hardness for a variety of graph properties with f -vectors of small hamming weightin the previous sections, we observe that our meta-theorem (Theorem 4.1) does not apply for properties Φfor which f Φ ,k , f ¬ Φ ,k and f Φ ,k have large hamming weight. A well-studied class of properties containingexamples of such Φ is the family of hereditary graph properties: In contrast to monotone properties,which are closed under taking subgraphs, a property Φ is called hereditary if it is closed under taking induced subgraphs. It is a well-known fact that every hereditary property Φ is characterized by a (possiblyinfinite) set Γ(Φ) of forbidden induced subgraphs, that isΦ( G ) = 1 ⇔ ∀ H ∈ Γ(Φ) :
IndSub ( H → G ) = 0 . Given any graph H , the property Φ of being H -free , that is, not containing H as an induced subgraph, ishereditary with Γ(Φ) = { H } .In this section, we settle the hardness-question for many hereditary graph properties as well. Notethat Φ is hereditary if and only if its inverse Φ is. In particular, H ∈ Γ(Φ) ⇔ H ∈ Γ(Φ).
Theorem 6.1.
Let H denote graph that is not the trivial graph with a single vertex and let Φ denotethe property Φ( G ) = 1 : ⇔ G is H -free. Then IndSub (Φ) is W [ ] -complete and cannot be solved intime g ( k ) · | V ( G ) | o ( k ) for any function g , unless ETH fails. The same is true for the problem IndSub ( ¬ Φ) . . Roth, J. Schmitt, and P. Wellnitz 24 uvH uuuvvH , u,v vvvH , u,v Figure 1
Different explosions of the edge { u, v } of a graph H . We start with some terminology used in this section. Given a graph H , a pair ( u, v ) ∈ V ( H ) , and twonon-negative integers x and y , we construct the exploded graph H ( u, v, x, y ) by adding x − u and y − v , including all incident edges; if x or y are zero, then we delete u or v , respectively.Given an edge e = { u, v } of H and two non-negative integers x and y , we define the e - exploded graph as H x,yu,v := ( V ( H ) , E ( H ) \ { u, v } )( u, v, x, y ). Consult Figure 1 for a visualization. An edge e = { u, v } of agraph H is called critical , if IndSub ( H → H x,yu,v ) = 0 for every pair x, y ∈ N ≥ .Now let Φ denote a hereditary graph property and let Γ(Φ) denote the associated set of forbiddeninduced subgraphs. We say that Φ has a critical edge if there is a graph H ∈ Γ(Φ) and an edge { u, v } ∈ E ( H ) such that for all positive integers x and y , the graph H x,yu,v satisfies Φ, that is, for everyˆ H ∈ Γ(Φ), we have
IndSub ( ˆ H → H x,yu,v ) = 0 . Finally, we say that a hereditary property Φ is critical if either Φ or its inverse Φ has a critical edge. Wewill see later in this section that every critical property Φ will induce hardness of
IndSub (Φ).First, we establish that for every graph H with at least two vertices, the property “ H -free” is critical.To this end, we rely on neighbour-sharing vertices: Given two vertices u and v of a graph H , we say that u and v are false twins if they have the same set of adjacent vertices. Note that, in particular, false twinscannot be adjacent as we consider graphs without self-loops. Furthermore, for a graph H , we define thepartition P ( H ) by adding two vertices to the same block if and only if they are false twins. Finally, wedefine the graph H ↓ by identifying all vertices in H with the block of P ( H ) they belong to, that is, thevertices of H ↓ are the blocks of P ( H ) and two blocks B and B are adjacent if there are vertices v ∈ B and v ∈ B such that { v, v } ∈ E ( H ). Lemma 6.2.
Let H denote a graph with at least vertices and let Φ denote a hereditary graph propertysuch that Γ(Φ) = { H } . Then Φ is critical. Proof.
We show that either Φ or Φ has a critical edge. As Γ(Φ) = { H } , we need to prove that at leastone of H and H has a critical edge. Let us start with the following claim. Claim 6.3.
Let F denote a graph with an edge { u, v } such that { u } and { v } are singleton sets in thepartition P ( F ). Then { u, v } is a critical edge of F . Proof.
Suppose there are integers x, y ∈ N ≥ such that there is an induced subgraph F of F x,yu,v that isisomorphic to F . Then there is a natural bijection between the blocks of F and the blocks of F x,yu,v sendingthe class of each vertex not equal to u, v to themselves, sending the class { u } to the class of its x clonesand similar for v . But note that the graph F x,yu,v ↓ has one fewer edge than F ↓ (since the previous edge { u, v } was removed). However, the induced subgraph F of F x,yu,v satisfies that F ↓ is a subgraph of F x,yu,v ↓ and thus also has at least one fewer edges than F ↓ , a contradiction to F being isomorphic to F . Using the previous claim, it suffices to show that there are two vertices u and v such that { u, v } is anedge and { u } and { v } are singletons in either one of H and H .We first show that for every vertex z ∈ V ( H ) = V ( H ), the set { z } is either a singleton in P ( H ) orin P ( H ). To this end, assume that z has a false twin z in H and a false twin z in H . Consequently, { z, z } / ∈ E ( H ) and { z, z } / ∈ E ( H ), and thus { z, z } ∈ E ( H ) and { z, z } ∈ E ( H ). Now, as z and z arefalse twins in H and { z, z } ∈ E ( H ), we see that { z , z } ∈ E ( H ). However, as z and z are false twinsin H and { z, z } ∈ E ( H ), we see that { z , z } ∈ E ( H ) as well, which leads to the desired contradiction.Now assume without loss of generality that H has at least one edge; otherwise we consider H . Ifthere are false twins u and v in H , then, by the previous argument, { u } and { v } are singletons in P ( H )and u and v are adjacent in H . By Claim 6.3, we obtain a critical edge of H . If there are no false twinsin H , we can choose an arbitrary edge of H which is then again critical by Claim 6.3. This concludes theproof. We have shown that every hereditary property defined by a single (non-trivial) forbidden induced subgraphis critical. Let us now provide some examples of critical hereditary properties that are defined by multipleforbidden induced subgraphs: Φ( H ) = 1 : ⇔ H is perfect. A graph H is perfect if for every induced subgraph of H , the size of thelargest clique equals the chromatic number. By the Strong Perfect Graph Theorem [16], we havethat Γ(Φ) is the set of all odd cycles of length at least 5 and their complements. Now observe thatevery edge of the cycle of length 5 is critical for Φ as the exploded graph is bipartite and thus perfect. Φ( H ) = 1 : ⇔ H is chordal. A graph is chordal if it does not contain an induced cycle of length 4 ormore. Consequently, we can choose an arbitrary edge of the cycle of length 4 as a critical edge for Φ,as the resulting exploded graphs do not contain any cycle. Φ( H ) = 1 : ⇔ H is a split graph. A split graph is a graph whose vertices can be partitioned into aclique and an independent set. It is known [31] that Γ(Φ) contains the cycles of length 4 and 5, andthe complement of the cycle of length 4. The latter is the graph containing two disjoint edges, and itis easy to see that any of those two edges is critical for Φ.We now establish hardness of IndSub (Φ) for critical hereditary properties. We start with thefollowing lemma, which constructs a (tight) parameterized Turing-reduction from counting cliques ofsize k in bipartite graphs. Lemma 6.4.
Let Φ denote a computable and critical hereditary graph property. There is an algorithm A with oracle access to IndSub (Φ) that expects as input a bipartite graph G and a positive integer k , andcomputes the number of independent sets of size k in G in time O ( | G | ) . Furthermore, the number of callsto the oracle is bounded by O (1) and every queried pair ( ˆ G, ˆ k ) satisfies | V ( ˆ G ) | ∈ O ( | V ( G ) | ) and ˆ k ∈ O ( k ) . Proof.
Assume without loss of generality that Φ has a critical edge; otherwise we use Fact 2.3 and proceedwith Φ. Hence choose H ∈ Γ(Φ) and e = { u, v } ∈ E ( H ) such that IndSub ( ˆ H → H x,yu,v ) = 0 for everyˆ H ∈ Γ(Φ) and for all non-negative integers x, y .Now let G = ( U ˙ ∪ V, E ) and k denote the given input. If U or V are empty, then we can trivially computethe number of independent sets of size k . Hence, we have U = { u , . . . , u n } and V = { v , . . . , v n } forsome integers n , n >
0. Now, we proceed as follows: In the first step, we construct the graph H n ,n u,v . Inthe next step, we identify u and its n − U and v and its n − For your amusement: if you color all vertices red that are singletons in H and color all vertices blue that are singletonsonly in H , then the fact that the Ramsey number R (2) is 3 shows that for H having at least 3 vertices, we find acritical edge in H or H . . Roth, J. Schmitt, and P. Wellnitz 26 uvH VU G ˆ G Figure 2
The construction of ˆ G from Lemma 6.4. the vertices of V . Finally, we add the edges E of G . We call the resulting graph ˆ G and we observe that ˆ G can clearly be constructed in time O ( | G | ); note that | H | is a constant as Φ is fixed. Consult Figure 2 fora visualization of the construction.Note that the construction induces a partition of the vertices of ˆ G into three sets: V ( ˆ G ) = R ˙ ∪ U ˙ ∪ V, where R = V ( H ) \ { u, v } ; we set r := | R | . Now define IndSub (Φ , k + r → ˆ G )[ R ] := { F ∈ IndSub (Φ , k + r → ˆ G ) | R ⊆ V ( F ) } , that is, IndSub (Φ , k + r → ˆ G )[ R ] is the set of all induced subgraphs F of size k + r in ˆ G that satisfy Φand that contain all vertices in R . Next, we show that the cardinality of IndSub (Φ , k + r → ˆ G )[ R ] revealsthe number of independent sets of size k in G . Claim 6.5.
Let IS k denote the set of independent sets of size k in G . We have IndSub (Φ , k + r → ˆ G )[ R ] = IS k . Proof.
Let b denote the function that maps a graph F ∈ IndSub (Φ , k + r → ˆ G )[ R ] to a k -vertex subsetof G given by b ( F ) := V ( F ) ∩ ( U ˙ ∪ V ) . We show that im ( b ) = IS k .“ ⊆ ”: Fix a graph F ∈ IndSub (Φ , k + r → ˆ G )[ R ] and write b ( F ) = U ˙ ∪ V where U ⊆ U and V ⊆ V .As | V ( F ) | = k + r , as well as V ( F ) = R ˙ ∪ U ˙ ∪ V , and | R | = r , we see that | b ( F ) | = k . Now assumethat b ( F ) is not an independent set, that is, there is an edge ( u, v ) ∈ U × V in F . Observing that theinduced subgraph F [ R ∪ { u, v } ] of F is isomorphic to H , and that H is a forbidden induced subgraph ofthe property Φ, yields the desired contradiction.“ ⊇ ”: Let U ˙ ∪ V denote an independent set of size k of G with | U | = k and | V | = k ; note that k or k might be zero. Let F denote the induced subgraph of ˆ G with vertices U ˙ ∪ V ˙ ∪ R . Then F has k + r vertices and is isomorphic to H k ,k u,v . Suppose F does not satisfy Φ. Then F has an inducedsubgraph isomorphic to a graph in Γ(Φ). However, this is impossible as IndSub ( ˆ H → H x,yu,v ) = ∅ for allˆ H ∈ Γ(Φ) and non-negative integers x, y .This shows that b is a surjective function from IndSub (Φ , k + r → ˆ G )[ R ] to IS k . Furthermore, injectivityis immediate by the definition of b as V ( F ) \ ( U ˙ ∪ V ) = R for every F ∈ IndSub (Φ , k + r → ˆ G )[ R ], whichproves the claim. It hence remains to show how our algorithm A can compute the cardinality of IndSub (Φ , k + r → ˆ G )[ R ]. Claim 6.6.
Write R = { z , . . . , z r } . We see that IndSub (Φ , k + r → ˆ G )[ R ] = X J ⊆ [ r ] ( − | J | · IndSub (Φ , k + r → ˆ G \ J ) , where ˆ G \ J is the graph obtained from ˆ G by deleting all vertices z i with i ∈ J . Proof.
Using the principle of inclusion and exclusion, we obtain that
IndSub (Φ , k + r → ˆ G )[ R ]= IndSub (Φ , k + r → ˆ G ) − { F ∈ IndSub (Φ , k + r → ˆ G ) | ∃ i ∈ [ r ] : z i / ∈ V ( F ) } = IndSub (Φ , k + r → ˆ G ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r [ i =1 { F ∈ IndSub (Φ , k + r → ˆ G ) | z i / ∈ V ( F ) } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = IndSub (Φ , k + r → ˆ G ) − X ∅6 = J ⊆ [ r ] ( − | J | +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) \ i ∈ J { F ∈ IndSub (Φ , k + r → ˆ G ) | z i / ∈ V ( F ) } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = IndSub (Φ , k + r → ˆ G ) − X ∅6 = J ⊆ [ r ] ( − | J | +1 { F ∈ IndSub (Φ , k + r → ˆ G ) | ∀ i ∈ J : z i / ∈ V ( F ) } = IndSub (Φ , k + r → ˆ G ) − X ∅6 = J ⊆ [ r ] ( − | J | +1 IndSub (Φ , k + r → ˆ G \ J )= X J ⊆ [ r ] ( − | J | · IndSub (Φ , k + r → ˆ G \ J ) . Consequently, the algorithm A requires linear time in | G | and 2 r ∈ O (1) oracle calls, each query of theform ( ˆ G \ J, k + r ), to compute the number of independent sets of size k in G as shown in the previousclaims. In particular, | V ( ˆ G \ J ) | ∈ O ( | V ( G ) | ) for each J ⊆ [ r ] and k + r ∈ O ( k ); this completes theproof. Theorem 6.7.
Let Φ denote a computable and critical hereditary graph property. Then IndSub (Φ) is W [ ] -complete and cannot be solved in time g ( k ) · | V ( G ) | o ( k ) for any function g , unless ETH fails. The same is true for the problem IndSub ( ¬ Φ) . Proof.
It is known that counting independent sets of size k in bipartite graphs is W [ ]-hard [19] andcannot be solved in time g ( k ) · | V ( G ) | o ( k ) for any function g , unless ETH fails [26]. The theorem thusfollows by Lemma 6.4.As a particular consequence we establish a complete classification for the properties of being H -free,including for instance claw-free graphs and co-graphs [60]. . Roth, J. Schmitt, and P. Wellnitz 28 Theorem 6.1.
Let H denote graph that is not the trivial graph with a single vertex and let Φ denotethe property Φ( G ) = 1 : ⇔ G is H -free. Then IndSub (Φ) is W [ ] -complete and cannot be solved intime g ( k ) · | V ( G ) | o ( k ) for any function g , unless ETH fails. The same is true for the problem IndSub ( ¬ Φ) . Proof.
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