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Tractability, hardness, and kernelization lower bound for and/or graph solution Uéverton dos Santos Souza, Fábio Protti * Institute of Computing—Fluminense Federal University, Niterói, RJ, Brazil
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Article history: Received 1 September 2014 Received in revised form 17 July 2017 Accepted 20 July 2017 Available online xxxx Keywords: And/or graphs W[P]-complete W[1]-hard FPT Tree-width Clique-width
a b s t r a c t And/or graphs are well-known data structures with several applications in many fields of computer science, such as Artificial Intelligence, Distributed Systems, Software Engineering, and Operations Research. An and/or graph is an acyclic digraph G containing a single source vertex s, where every vertex v ∈ V (G) has a label f (v ) ∈ {and,or}. In an and/or graph, (weighted) edges represent dependency relations between vertices: a vertex labeled and depends on all of its out-neighbors, while a vertex labeled or depends on only one of its out-neighbors. A solution subgraph H of an and/or graph G is a subdigraph of G containing its source vertex and such that if an and-vertex (resp. or-vertex) is included in H then all (resp. one) of its out-edges must also be included in H. In general, solution subgraphs represent feasible solutions of problems modeled by and/or graphs. The optimization problem associated with an and/or graph G consists of finding a minimum weight solution subgraph H of G, where the weight of a solution subgraph is the sum of the weights of its edges. Because of its wide applicability, in this work we develop a multivariate investigation of this optimization problem. In a previous paper (Souza et al., 2013) we have analyzed the complexity of such a problem under various aspects, including parameterized versions of it. However, the main open question has remained open: Is the problem of finding a solution subgraph of weight at most k (where k is the parameter) in FPT? In this paper we answer negatively to this question, proving the W[1]-hardness of the problem, and its W[P]-completeness when zero-weight edges are allowed. We also show that the problem is fixed-parameter tractable when parameterized by the tree-width, but it is W[SAT]-hard with respect to the clique-width and k as aggregated parameters. In addition, we show that when the out-edges of each or-vertex have all the same weight (a condition very common in practice), the problem becomes fixed-parameter tractable by the clique-width. Finally, using a framework developed by Bodlaender et al. (2009) and Fortnow and Santhanam (2011), based upon the notion of compositionality, we show that the tractable cases do not admit a polynomial kernel unless NP ⊆ coNP /poly, even restricted to instances without or-vertices with out-degree greater than two. © 2017 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we consider the complexity of a problem associated with relevant data structures, the so-called and/or graphs. An and/or graph is an acyclic digraph containing a single source vertex (that reaches all other vertices by directed paths), such that every vertex v ∈ V (G) has a label f (v ) ∈ { and,or}. In such digraphs, edges represent dependency
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Corresponding author. E-mail addresses: [email protected] (U.d.S. Souza), [email protected] (F. Protti).
http://dx.doi.org/10.1016/j.dam.2017.07.029 0166-218X/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Fig. 1. (a) An and/or graph; (b) An optimal solution subgraph of the graph in (a).
relations between vertices: a vertex labeled and depends on all of its out-neighbors (conjunctive dependency), while a vertex labeled or depends on only one of its out-neighbors (disjunctive dependency). In this paper, we denote by Ovi the set of out-neighbors of vertex vi . Fig. 1 shows examples of and/or graphs, where and-vertices have an arc around its out-edges. And/or graphs arise in several fields and applications, such as Artificial Intelligence [4,28,30], Automation and Robotics [10,15,21], Distributed Systems [3], Game Theory [22], Hypergraph Applications [20], Computational Logics [23], Operations Research [27,32], Scheduling [2], and Software Engineering [11,25], to name a few. The main optimization problem associated with and/or graphs is formally defined below. Min-and/or Instance: An and/or graph G = (V , E) containing a single source vertex s, and such that each edge e of G has an integer weight τ (e) > 0. Goal: Determine the minimum weight of a subdigraph H = (V ′ , E ′ ) of G (solution subgraph) satisfying the following properties:
• s ∈ V ′; • if a vertex v is in V ′ and f (v )=and then every out-edge of v belongs to E ′ ; • if a non-sink vertex v is in V ′ and f (v )=or then exactly one out-edge of v belongs to E ′ . Min-and/or0 is a generalization of Min-and/or where zero-weight edges are allowed. In 1974, Sahni [29] showed that Min-and/or is NP-hard; however, there are some cases for which it can be solved in polynomial time (for details, see [31]). We denote by Min-and/or(k) the parameterized version of Min-and/or which asks whether there is a solution subgraph of weight at most k. This approach is justified by the fact that many applications are concerned with satisfying a low cost limit. In a previous paper [31], we investigate some variants of Min-and/or(k): (a) Min-and/or(k, r), the parameterized version where every or-vertex of the input graph has at most r out-edges with the same weight; (b) Min-And/or0 (k), the parameterized version applied to and/or graphs where zero-weight edges are allowed. In [31] we prove that Minand/or(k, r) is in FPT, whereas Min-and/or0 (k) is W[2]-hard. However, the main question of classifying Min-and/or(k) has remained open up to now. In this paper we close this question by proving that Min-and/or(k) is W[1]-hard and in W[P]; we also show that Min-and/or0 (k) is in fact W[P]-complete. In addition, some fixed-parameter tractable cases are presented, and a proof that such tractable cases do not admit a polynomial kernel, unless NP ⊆ coNP /poly, is given. The latter condition, if true, would imply an unlikely collapse of the polynomial hierarchy to the third level (PH = Σp3 ). The remainder of this work is organized as follows. In Section 2 we show the W[P]-completeness of Min-and/or0 (k). In Section 3 we prove that Min-and/or(k) is W[1]-hard. In Section 4, we consider the tree-width and the clique-width as parameters. Finally, in Section 5, we present kernelization lower bounds to some cases. 2. W[P]-Completeness The class W[P] is defined as the class of parameterized problems fpt-reducible to Weighted Circuit Satisfiability [17]. Weighted Circuit Satisfiability Instance: A decision circuit C . Parameter: A positive integer k. Question: Does C have a satisfying assignment of weight k? Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Fig. 2. (a) An and/or graph G; (b) a decision circuit C constructed from G. The colors represent the injective function from non-sink vertices of G to gates of C .
Lemma 1. Min-and/or(k) and Min-and/or0 (k) are in W[P]. Proof. Let G be an instance of Min-and/or0 (k); first we apply the following reduction rules on G. 1. 2. 3. 4. 5.
For each edge e ∈ E(G), if τ (e) > k then remove it. If some vertex has become unreachable from s then remove it. Remove every vertex which was not a sink in the original instance and became a sink after some reduction. Remove each and-vertex such that some of its out-edges have been removed. Repeat Rules 2 and 4 while needed.
Rules 1–5 remove only vertices and edges which are guaranteed not to belong to any solution subgraph of cost at most k. Without loss of generality we assume that G does not have zero weight out-edges pointing to sinks. We construct a decision circuit C from G as follows: (i) for each edge e of G such that τ (e) > 0, create an and gate ge with fan-in τ (e); the inputs of ge are labeled by variables xe1 , xe2 , . . . , xeτ (e) ; (ii) for each non-sink and- (resp. or-) vertex v of G, create an and (resp. or) gate gv in C ; (iii) for each directed edge e = (vi , vj ) with τ (e) = 0, add an edge to gvi from gvj ; (iv) if there is an edge e from a vertex vi to a sink vertex vj , add an edge to gvi from ge ; (v) if there is an edge e from a vertex vi to a non-sink vertex vj in G with τ (e) > 0, create an and gate ge∗ , add edges to ge∗ from ge and gvj , and add an edge to gvi from ge∗ . Note that the same construction can be applied to Min-and/or(k). Fig. 2 illustrates an and/or graph G and a decision circuit C constructed from G. At this point, it is easy to see that if G has a solution subgraph S of size k then C has a Hamming assignment of weight k which sets to 1 all variables xek such that e ∈ E(S). Conversely, if C has a Hamming assignment of weight k then G has a solution subgraph S of size at most k which uses some zero weight edges and at most all edges e of G such that ge outputs 1. Finally, as G has no edge e with τ (e) > k, the construction of the circuit C can be done in O(k |E(G)|) time. Hence we have a fixed-parameter reduction from Min-and/or0 (k) to Weighted Circuit Satisfiability. □ For the next result we need to introduce the following W[P]-complete problem [17]. Weighted Monotone Circuit Satisfiability Instance: A monotone decision circuit C . Parameter: A positive integer k. Question: Does C have a satisfying assignment of weight k? Theorem 2. Min-and/or0 (k) is W[P]-complete. Proof. The membership to W[P] is shown in the previous lemma. The proof of W[P]-hardness uses a reduction from Weighted Monotone Circuit Satisfiability — WMCS. According to [33], a Boolean monotone circuit with n inputs and one output is defined as a finite directed acyclic graph, where each vertex corresponds to either a Boolean function ( And or Or) or one of the inputs, and there is exactly one node which is labeled as the output. Hence, given an instance C of WMCS, we construct an and/or graph G by: (a) reversing the direction of the edges of C ; (b) considering each vertex corresponding to an And (resp. Or) function as an and- (resp. or-) vertex; (c) setting the output vertex of C as the source vertex s, where f (s) = and; (d) assigning zero weight to all edges; (e) finally, for each input vertex v , adding a new vertex wv and a directed edge (v, wv ) with τ (v, wv ) = 1, setting f (v ) = and. Fig. 3 illustrates an and/or graph obtained from the Boolean monotone circuit C in Fig. 2(b). Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Fig. 3. An and/or graph obtained from the circuit C of Fig. 2(b).
By construction, it is easy to see that any solution subgraph S of G induces an assignment on the inputs of C such that its output is true. Note that the sinks of S indicate which variables should be ‘1’ in the assignment. On the other hand, a Hamming assignment of weight k of C suggests a bottom-up construction of a solution subgraph of size k in G. □
3. W[1]-hardness This section utilizes the useful Multicolored Clique problem, a well-known W[1]-complete problem [18]. Multicolored Clique Instance: A graph Q and a vertex-coloring ℓ : V (G) → {1, 2, . . . , c } for G. Parameter: A positive integer c. Question: Does Q have a clique including vertices of all c colors? Theorem 3. Min-and/or(k) is W[1]-hard and in W[P]. Proof. The W[P]-membership is shown in Lemma 1. The W[1]-hardness proof uses a reduction from Multicolored Clique. Let (Q , c) be an instance of Multicolored Clique, and let V1 , V2 , . . . , Vc be the color classes of Q . Without loss of generality we consider that each vertex in Vi has at least one neighbor in Vj (i ̸ = j). We construct an instance (G, k) of Min-and/or(k) as follows (see Fig. 4): 1. 2. 3. 4. 5.
create a source vertex s in G with f (s) = and; for each color ci of Q , create a vertex wi with f (wi ) = or and add an edge from s to wi ; for each color class Vi of Q , create copies Vi1 , Vi2 , Vi3 , Vi4 in G; add edges from wi to each vertex in Vi1 in G; let v 1 , v 2 , v 3 , v 4 be the copies of a vertex v ∈ V (Q ); add edges (v 1 , v 2 ), (v 2 , v 3 ), (v 3 , v 4 ) to G; assign f (v 1 ) = and and f (v 2 ) = f (v 3 ) = f (v 4 ) = or; j 6. for each vertex v 1 ∈ Vi1 (1 ≤ i ≤ c), create c − 1 new or-out-neighbors av 1 (1 ≤ j ≤ c and i ̸ = j), and add an edge j
from av 1 to a vertex u3 ∈ Vj3 if and only if v and u are neighbors in Q ; 7. assign weight 1 to all edges of G; 8. finally, set k = 2c 2 + 3c.
If Q contains a multicolored clique C such that |C | = c then it is possible to construct a solution subgraph S of G with weight k as follows: (a) s and all of its out-edges belong to S; (b) for each or-vertex wi ∈ V (S), include in S an edge (wi , v 1 ) of G such that v ∈ C ; (c) for each v 1 ∈ V (S), add all of its out-edges to S; (d) for each v 2 ∈ V (S) add its out-edges to S; (e) j j for each v 3 ∈ V (S) add its out-edges to S; (f) finally, for each av 1 in S, choose an out-edge (av 1 , u3 ) to be added to S such that 2 u ∈ C . Note that S has weight k = 2c + 3c. Conversely, if G has a solution subgraph S with weight k = 2c 2 + 3c then it is possible to obtain a multicolored clique C of Q as follows: a vertex v of Q belongs to C if and only if v 3 belongs to V (S). Since any solution subgraph of G has weight at j least k, if S has weight k = 2c 2 + 3c then S has only c vertices of type v 3 , and for each pair v 3 , au1 (u ̸ = v ) in S there is an edge between them, implying that C is a multicolored clique of size c of Q . □
Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Fig. 4. An and/or graph G obtained from Q . Edges of G are directed according to a breadth-first search from s.
4. Bounded tree- and clique-width Instances of Min-and/or and Min-and/or0 are directed acyclic graphs, and as both problems are solvable in polynomial time when the underlying undirected graph is a tree [31], it is still interesting to ask about the complexity of both problems when the underlying undirected graph has bounded tree-width or clique-width. 4.1. Tree-width Let G be a graph, T a tree, and let V = (Vt )t ∈T be a family of vertex sets Vt ⊆ V (G) indexed by the vertices t of T . The pair (T , V ) is called a tree-decomposition of G if it satisfies the following three conditions [16]:
⋃
(T1) V (G) = t ∈T Vt ; (T2) for every edge e ∈ G there exists t ∈ T such that both ends of e lie in Vt ; (T3) If Vti and Vtj both contain a vertex v , then v ∈ Vtk for all nodes tk in the path between ti and tj . The width of (T , V ) is the number max{|Vt | − 1 : t ∈ T }, and the tree-width t w (G) of G is the minimum width of any tree-decomposition of G. Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Let Min-and/or0 (t w ) be the parameterized version when the tree-width of the underlying undirected graph is considered as a single parameter. Theorem 4. Min-and/or0 (t w ) is fixed-parameter tractable. Proof. A solution subgraph of an and/or graph is MSO2 -expressible [12,13] by the following formula ϕ (A, X ): there is a set of edges A and a set of vertices X such that
• • • • •
s ∈ X; every edge of A has its two endpoints in X ; every vertex of X except s is the head of an edge in A; every vertex u ∈ X ∩ And has all its out-edges in A; every vertex u ∈ X ∩ Or has a unique out-edge in A.
Let now Ψ (A) ⇔ ∃X ϕ (A, X ). Min-and/or0 consists of finding a subset A of minimum weight τ (A) satisfying Ψ (A), and is fixed-parameter tractable when parameterized by the tree-width via Courcelle’s meta-theorem for graphs of bounded tree-width and properties expressed by MSO formulas using edge set quantifications (MSO2 properties) [12,13]. □
4.2. Clique-width The clique-width of a graph G is based on the number of labels needed to construct G by means of the following four operations [9]:
• • • •
Creation of a new vertex v with label i. Disjoint union of two labeled graphs G and H. Joining by an edge every vertex labeled i to every vertex labeled j, where i ̸ = j Renaming label i to label j.
The clique-width c w (G) of a graph G is defined in [14] as the minimum number of labels which are necessary to generate G by using these operations. Let Min-and/or0 (k, c w ) and Min-and/or(k, c w ) be parameterized versions where k and the clique-width c w are aggregated parameters. In contrast to the previous result on the tree-width, the next theorem shows that Min-and/or0 and Min-and/or are unlikely to be FPT-time solvable when parameterized by k and c w . W[SAT] is defined as the class of parameterized problems reducible to Weighted Satisfiability [17] and Weighted Monotone Satisfiability is W[SAT]-complete [1]. Weighted Monotone Satisfiability Instance: A monotone Boolean formula X . Parameter: A positive integer k. Question: Does X have a satisfying assignment of weight k? Theorem 5. Min-and/or0 (k, c w ) is W[SAT]-hard and in W[P]. Proof. From an instance (X , k) of Weighted Monotone Satisfiability we construct an instance (G, k) of Min-and/or0 (k, c w ) as follows. First construct the tree circuit C corresponding to X ; we can assume that all leaves of the tree circuit C are at the same level. Next, we obtain G as described in the proof of Theorem 2. The only vertices of G with in-degree greater than one are those representing inputs of C , and G has a solution subgraph of size k if and only if X has a satisfying assignment of weight k. Let Vi be the set of vertices of G at a distance i from s, and let j be the distance from s to the sinks. Without loss of generality Vj−2 contains only or-vertices. For the pair Vj−2 , Vj−1 add edges from Vj−2 to Vj−1 in order to make the underlying undirected graph induced by Vj−2 , Vj−1 a complete bipartite graph; assign weight k + 1 for these new edges. Now, note that G has clique-with at most four, since complete bipartite graphs have clique-width two, trees have clique-width at most three [24], and using four labels we can construct trees reserving an exclusive label to the leaves. □ Corollary 6. Min-and/or(k, c w ) is W[1]-hard and in W[P]. Proof. Simply change the construction of Theorem 3 similarly as described in the previous theorem. □ Although Theorem 5 and Corollary 6 show the parameterized intractability of both problems with respect to the cliquewidth, the next result provides a sufficient condition to solve them in FPT-time when parameterized by c w . Theorem 7. Min-and/or0 (c w ) is fixed-parameter tractable if for each or-vertex its out-edges have the same weight. Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Proof. Let G = (V , E) be an and/or graph where for each or-vertex ∑ v of G its out-edges have the same weight. To each vertex v ∈ V assign a weight τ (v ) such that τ (v ) = 0 if v is a sink, τ (v ) = e∈Ov τ (e) if v is an and-vertex, and τ (v ) = τ (e) (e ∈ Ov ) if v is an or-vertex. subgraph S of G satisfies ∑ for any or-vertex of G its out-edges have the same weight, any solution ∑ Note that since 0 τ ( v ). At this point, we consider the objective function of Min-and/or (c w ) as the sum of weights τ (e) = v∈V (S) e∈E(S) of vertices in the solution subgraph. Thus, a solution subgraph of G can be MSO1 -expressible [12,13] by a formula Φ (X ) (only vertex and vertex set variables are used), as follows. Let And and Or be the subsets of V containing the and- and the or-vertices, respectively.
Φ (X ) = ‘‘there is a set of vertices X such that
• • • •
s ∈ X; every vertex of X except s has an in-neighbor in X ; every vertex u ∈ X ∩ And has all its out-neighbors in X ; every vertex u ∈ X ∩ Or has some out-neighbor in X ’’.
By Courcelle’s meta-theorem for graphs of bounded clique-width and properties expressed by MSO1 formulas [12,13], we can determine in FPT time the minimum τ (X ) for X satisfying Φ (X ), and the corresponding set X , which induces an optimal solution subgraph of G. □ 5. Infeasibility of polynomial kernels The study of problems which admit kernelization algorithms that reduce instances to a size which is polynomially bounded in the parameter has become an interesting field of investigation. Such problems are said to have a polynomial kernelization algorithm, or a polynomial kernel. Results on infeasibility of polynomial kernels for specific problems have appeared recently. Bodlaender et al. [6] and Fortnow and Santhanam [19] have developed a framework based on the notion of compositionality, to show that a problem does not admit a polynomial kernel unless NP ⊆ coNP/poly. Definition 1. [5] An or-composition algorithm for a parameterized problem Π ⊆ Σ ∗ × N is an algorithm that gets as input a sequence ((x1 , k), . . . , (xr , k)), with each (xi , ki ) ∈ Σ ∗ × N, and outputs a pair (x′ , k′ ), such that
∑
1. the algorithm uses time polynomial in 1≤i≤r |xi | + k; 2. k′ is bounded by a polynomial in k; 3. (x′ , k′ ) ∈ Π if and only if there exists an index i, 1 ≤ i ≤ r, such that (xi , k) ∈ Π . A problem is said to be or-compositional if it admits an or-composition algorithm. Corollary 8 ([6,19]). Let Π be a parameterized problem with Π c its corresponding classical decision variant. If Π is orcompositional and Π c is NP-complete, then Π has no kernel of polynomial size, unless NP ⊆ coNP/poly. Algorithms that compose multiple instances of a problem into one instance have been developed for various problems [6–8,26]. Definition 2. Let H be a class of and/or graphs such that every non-sink or-vertex v of a graph G ∈ H has out-degree at most two, and every edge has weight equal to one. As shown in [31], Min-and/or remains NP-hard even restricted to and/or graphs in H . Lemma 9. Min-and/or(k) restricted to graphs in H can be solved in O(2k × n) time. Proof. This proof uses the method of bounded search trees (see [17]). Consider the following algorithm for G ∈ H , where each vertex v ∈ V (G) has a label ℓ(v ) initially equals zero; Pand = {s}, Por = ∅ if f (s) = and; Pand = ∅, Por = {s} if f (s) = or.
Sol -Sub( G ,k ,Pand ,Por ) While Pand ̸= ∅ do . Remove a vertex v from Pand . k := k − |Ov | . For each non -sink w ∈ Ov do . . If ℓ(w) = 0 then . . . If f (w) = and then . . . . Pand := Pand ∪ {w} . . . Else . . . . Por := Por ∪ {w} . . ℓ(w) := 1 Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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If ( k < 0 ) or ( k = 0 and Por ̸= ∅ ) then . return false If Por = ∅ then . return true Else . remove a vertex v from Por . ctrl := false . For each w ∈ Ov do % Note that |Ov |≤ 2 . . If ( ℓ(w) = 0 ) and ( w is not a sink) then . . . If f (w) = and then . . . . ctrl := (ctrl or Sol -Sub( G ,k − 1 ,Pand ∪ {w} ,Por )) . . . Else . . . . ctrl := (ctrl or Sol -Sub( G ,k − 1 ,Pand ,Por ∪ {w} )) . . Else . . . ctrl := (ctrl or Sol -Sub( G ,k − 1 ,Pand ,Por )) return ctrl It is easy to see that the previous recursive algorithm exhaustively checks the existence of a solution subgraph containing at most k edges. As it implicitly constructs a binary search tree with height at most k (size O(2k )), where for each node is performed a O(n) routine, the algorithm runs in O(2k × n) time. □ At this point, we will show that Min-and/or(k) restricted to graphs in H does not have a polynomial kernel unless NP ⊆ coNP/poly. Theorem 10. Min-and/or(k) restricted to graphs in H is or-compositional. Proof. Suppose the input sequence Q = ( (G1 , k), (G2 , k), . . . , (Gt , k) ), where Gi is an and/or graph in H , 1 ≤ i ≤ t, and k is a fixed parameter. Let (G′ , k′ ) be an instance obtained from Q as follows: (a) If t ≥ 2k then solve each instance (Gi , k), 1 ≤ i ≤ t, in O(2k × n) time and return an instance (G′ , k) such that G′ has a solution subgraph of size k if and only if some Gi has a solution subgraph of size k. Note that this algorithm uses time polynomial in (t , n). (b) If t < 2k , assume without loss of generality that t is a power of two. Then, • initially, set G′ as the disjoint union of all the graphs in Q ; • add a new vertex s′ to G′ ; • construct a full binary tree T with source s′ and leaves si , for each graph Gi , and assign weight 1 to each edge of T and label or to each internal vertex of T ; • set k′ = k + log2 t; • output the pair (G′ , k′ ). Since the source vertex of G′ is s′ , clearly G′ has a solution subgraph of weight at most k′ if and only if there is a graph Gi in Q with a solution subgraph of weight at most k. As k′ = k + log2 t and t < 2k , we have k′ ≤ 2k. □ Corollary 11. Unless NP ⊆ coNP/poly, Min-and/or(k, t w ) and Min-and/or(k, c w ) restricted to graphs in H do not admit polynomial kernels. Proof. Follows from the above, or-composition which preserves tree-width and increases by at most one the cliquewidth. □ 6. Conclusion In this paper a multivariate investigation of the problem of finding a solution subgraph of an and/or graph with weight at most k is made. We answer negatively to the following open question: ‘‘Is the problem of finding a solution subgraph of weight at most k (where k is the parameter) in FPT?’’. We show that Min-and/or(k) is W[1]-hard and in W[P]; Min-and/or0 (k) is W[P]-complete; Min-and/or0 (t w ) is fixedparameter tractable; Min-and/or0 (k, c w ) is W[SAT]-hard and in W[P]; and Min-and/or0 (c w ) is FPT time solvable when each or-vertex has all of its out-edges with the same weight. In addition, it is shown that the tractable cases do not admit a polynomial kernel unless NP ⊆ coNP/poly, even restricted to instances without or-vertices with out-degree greater than two. Table 1 summarizes the results obtained in this work. Our results suggest the following questions: (a) for which class of the W-hierarchy Min-and/or(k) is complete?; (b) Is Min-and/or0 (k, c w ) W[P]-complete? Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Table 1 Complexity of and/or graph solution problems.
k tw cw k, t w k, c w k, c w a a
Min-and/or
Min-and/or0
W[1]-hard, in W[P] FPT NP-complete when c w ≥ 4 or-compositional W[1]-hard, in W[P] FPT, or-compositional
W[P]-complete FPT NP-complete when c w ≥ 4 or-compositional W[SAT]-hard, in W[P] FPT, or-compositional
In this version, for any or-vertex its out-edges have the same weight.
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Tractability, hardness, and kernelization lower bound for and/or graph solution Uéverton dos Santos Souza, Fábio Protti * Institute of Computing—Fluminense Federal University, Niterói, RJ, Brazil
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Article history: Received 1 September 2014 Received in revised form 17 July 2017 Accepted 20 July 2017 Available online xxxx Keywords: And/or graphs W[P]-complete W[1]-hard FPT Tree-width Clique-width
a b s t r a c t And/or graphs are well-known data structures with several applications in many fields of computer science, such as Artificial Intelligence, Distributed Systems, Software Engineering, and Operations Research. An and/or graph is an acyclic digraph G containing a single source vertex s, where every vertex v ∈ V (G) has a label f (v ) ∈ {and,or}. In an and/or graph, (weighted) edges represent dependency relations between vertices: a vertex labeled and depends on all of its out-neighbors, while a vertex labeled or depends on only one of its out-neighbors. A solution subgraph H of an and/or graph G is a subdigraph of G containing its source vertex and such that if an and-vertex (resp. or-vertex) is included in H then all (resp. one) of its out-edges must also be included in H. In general, solution subgraphs represent feasible solutions of problems modeled by and/or graphs. The optimization problem associated with an and/or graph G consists of finding a minimum weight solution subgraph H of G, where the weight of a solution subgraph is the sum of the weights of its edges. Because of its wide applicability, in this work we develop a multivariate investigation of this optimization problem. In a previous paper (Souza et al., 2013) we have analyzed the complexity of such a problem under various aspects, including parameterized versions of it. However, the main open question has remained open: Is the problem of finding a solution subgraph of weight at most k (where k is the parameter) in FPT? In this paper we answer negatively to this question, proving the W[1]-hardness of the problem, and its W[P]-completeness when zero-weight edges are allowed. We also show that the problem is fixed-parameter tractable when parameterized by the tree-width, but it is W[SAT]-hard with respect to the clique-width and k as aggregated parameters. In addition, we show that when the out-edges of each or-vertex have all the same weight (a condition very common in practice), the problem becomes fixed-parameter tractable by the clique-width. Finally, using a framework developed by Bodlaender et al. (2009) and Fortnow and Santhanam (2011), based upon the notion of compositionality, we show that the tractable cases do not admit a polynomial kernel unless NP ⊆ coNP /poly, even restricted to instances without or-vertices with out-degree greater than two. © 2017 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we consider the complexity of a problem associated with relevant data structures, the so-called and/or graphs. An and/or graph is an acyclic digraph containing a single source vertex (that reaches all other vertices by directed paths), such that every vertex v ∈ V (G) has a label f (v ) ∈ { and,or}. In such digraphs, edges represent dependency
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Corresponding author. E-mail addresses: [email protected] (U.d.S. Souza), [email protected] (F. Protti).
http://dx.doi.org/10.1016/j.dam.2017.07.029 0166-218X/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Fig. 1. (a) An and/or graph; (b) An optimal solution subgraph of the graph in (a).
relations between vertices: a vertex labeled and depends on all of its out-neighbors (conjunctive dependency), while a vertex labeled or depends on only one of its out-neighbors (disjunctive dependency). In this paper, we denote by Ovi the set of out-neighbors of vertex vi . Fig. 1 shows examples of and/or graphs, where and-vertices have an arc around its out-edges. And/or graphs arise in several fields and applications, such as Artificial Intelligence [4,28,30], Automation and Robotics [10,15,21], Distributed Systems [3], Game Theory [22], Hypergraph Applications [20], Computational Logics [23], Operations Research [27,32], Scheduling [2], and Software Engineering [11,25], to name a few. The main optimization problem associated with and/or graphs is formally defined below. Min-and/or Instance: An and/or graph G = (V , E) containing a single source vertex s, and such that each edge e of G has an integer weight τ (e) > 0. Goal: Determine the minimum weight of a subdigraph H = (V ′ , E ′ ) of G (solution subgraph) satisfying the following properties:
• s ∈ V ′; • if a vertex v is in V ′ and f (v )=and then every out-edge of v belongs to E ′ ; • if a non-sink vertex v is in V ′ and f (v )=or then exactly one out-edge of v belongs to E ′ . Min-and/or0 is a generalization of Min-and/or where zero-weight edges are allowed. In 1974, Sahni [29] showed that Min-and/or is NP-hard; however, there are some cases for which it can be solved in polynomial time (for details, see [31]). We denote by Min-and/or(k) the parameterized version of Min-and/or which asks whether there is a solution subgraph of weight at most k. This approach is justified by the fact that many applications are concerned with satisfying a low cost limit. In a previous paper [31], we investigate some variants of Min-and/or(k): (a) Min-and/or(k, r), the parameterized version where every or-vertex of the input graph has at most r out-edges with the same weight; (b) Min-And/or0 (k), the parameterized version applied to and/or graphs where zero-weight edges are allowed. In [31] we prove that Minand/or(k, r) is in FPT, whereas Min-and/or0 (k) is W[2]-hard. However, the main question of classifying Min-and/or(k) has remained open up to now. In this paper we close this question by proving that Min-and/or(k) is W[1]-hard and in W[P]; we also show that Min-and/or0 (k) is in fact W[P]-complete. In addition, some fixed-parameter tractable cases are presented, and a proof that such tractable cases do not admit a polynomial kernel, unless NP ⊆ coNP /poly, is given. The latter condition, if true, would imply an unlikely collapse of the polynomial hierarchy to the third level (PH = Σp3 ). The remainder of this work is organized as follows. In Section 2 we show the W[P]-completeness of Min-and/or0 (k). In Section 3 we prove that Min-and/or(k) is W[1]-hard. In Section 4, we consider the tree-width and the clique-width as parameters. Finally, in Section 5, we present kernelization lower bounds to some cases. 2. W[P]-Completeness The class W[P] is defined as the class of parameterized problems fpt-reducible to Weighted Circuit Satisfiability [17]. Weighted Circuit Satisfiability Instance: A decision circuit C . Parameter: A positive integer k. Question: Does C have a satisfying assignment of weight k? Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Fig. 2. (a) An and/or graph G; (b) a decision circuit C constructed from G. The colors represent the injective function from non-sink vertices of G to gates of C .
Lemma 1. Min-and/or(k) and Min-and/or0 (k) are in W[P]. Proof. Let G be an instance of Min-and/or0 (k); first we apply the following reduction rules on G. 1. 2. 3. 4. 5.
For each edge e ∈ E(G), if τ (e) > k then remove it. If some vertex has become unreachable from s then remove it. Remove every vertex which was not a sink in the original instance and became a sink after some reduction. Remove each and-vertex such that some of its out-edges have been removed. Repeat Rules 2 and 4 while needed.
Rules 1–5 remove only vertices and edges which are guaranteed not to belong to any solution subgraph of cost at most k. Without loss of generality we assume that G does not have zero weight out-edges pointing to sinks. We construct a decision circuit C from G as follows: (i) for each edge e of G such that τ (e) > 0, create an and gate ge with fan-in τ (e); the inputs of ge are labeled by variables xe1 , xe2 , . . . , xeτ (e) ; (ii) for each non-sink and- (resp. or-) vertex v of G, create an and (resp. or) gate gv in C ; (iii) for each directed edge e = (vi , vj ) with τ (e) = 0, add an edge to gvi from gvj ; (iv) if there is an edge e from a vertex vi to a sink vertex vj , add an edge to gvi from ge ; (v) if there is an edge e from a vertex vi to a non-sink vertex vj in G with τ (e) > 0, create an and gate ge∗ , add edges to ge∗ from ge and gvj , and add an edge to gvi from ge∗ . Note that the same construction can be applied to Min-and/or(k). Fig. 2 illustrates an and/or graph G and a decision circuit C constructed from G. At this point, it is easy to see that if G has a solution subgraph S of size k then C has a Hamming assignment of weight k which sets to 1 all variables xek such that e ∈ E(S). Conversely, if C has a Hamming assignment of weight k then G has a solution subgraph S of size at most k which uses some zero weight edges and at most all edges e of G such that ge outputs 1. Finally, as G has no edge e with τ (e) > k, the construction of the circuit C can be done in O(k |E(G)|) time. Hence we have a fixed-parameter reduction from Min-and/or0 (k) to Weighted Circuit Satisfiability. □ For the next result we need to introduce the following W[P]-complete problem [17]. Weighted Monotone Circuit Satisfiability Instance: A monotone decision circuit C . Parameter: A positive integer k. Question: Does C have a satisfying assignment of weight k? Theorem 2. Min-and/or0 (k) is W[P]-complete. Proof. The membership to W[P] is shown in the previous lemma. The proof of W[P]-hardness uses a reduction from Weighted Monotone Circuit Satisfiability — WMCS. According to [33], a Boolean monotone circuit with n inputs and one output is defined as a finite directed acyclic graph, where each vertex corresponds to either a Boolean function ( And or Or) or one of the inputs, and there is exactly one node which is labeled as the output. Hence, given an instance C of WMCS, we construct an and/or graph G by: (a) reversing the direction of the edges of C ; (b) considering each vertex corresponding to an And (resp. Or) function as an and- (resp. or-) vertex; (c) setting the output vertex of C as the source vertex s, where f (s) = and; (d) assigning zero weight to all edges; (e) finally, for each input vertex v , adding a new vertex wv and a directed edge (v, wv ) with τ (v, wv ) = 1, setting f (v ) = and. Fig. 3 illustrates an and/or graph obtained from the Boolean monotone circuit C in Fig. 2(b). Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Fig. 3. An and/or graph obtained from the circuit C of Fig. 2(b).
By construction, it is easy to see that any solution subgraph S of G induces an assignment on the inputs of C such that its output is true. Note that the sinks of S indicate which variables should be ‘1’ in the assignment. On the other hand, a Hamming assignment of weight k of C suggests a bottom-up construction of a solution subgraph of size k in G. □
3. W[1]-hardness This section utilizes the useful Multicolored Clique problem, a well-known W[1]-complete problem [18]. Multicolored Clique Instance: A graph Q and a vertex-coloring ℓ : V (G) → {1, 2, . . . , c } for G. Parameter: A positive integer c. Question: Does Q have a clique including vertices of all c colors? Theorem 3. Min-and/or(k) is W[1]-hard and in W[P]. Proof. The W[P]-membership is shown in Lemma 1. The W[1]-hardness proof uses a reduction from Multicolored Clique. Let (Q , c) be an instance of Multicolored Clique, and let V1 , V2 , . . . , Vc be the color classes of Q . Without loss of generality we consider that each vertex in Vi has at least one neighbor in Vj (i ̸ = j). We construct an instance (G, k) of Min-and/or(k) as follows (see Fig. 4): 1. 2. 3. 4. 5.
create a source vertex s in G with f (s) = and; for each color ci of Q , create a vertex wi with f (wi ) = or and add an edge from s to wi ; for each color class Vi of Q , create copies Vi1 , Vi2 , Vi3 , Vi4 in G; add edges from wi to each vertex in Vi1 in G; let v 1 , v 2 , v 3 , v 4 be the copies of a vertex v ∈ V (Q ); add edges (v 1 , v 2 ), (v 2 , v 3 ), (v 3 , v 4 ) to G; assign f (v 1 ) = and and f (v 2 ) = f (v 3 ) = f (v 4 ) = or; j 6. for each vertex v 1 ∈ Vi1 (1 ≤ i ≤ c), create c − 1 new or-out-neighbors av 1 (1 ≤ j ≤ c and i ̸ = j), and add an edge j
from av 1 to a vertex u3 ∈ Vj3 if and only if v and u are neighbors in Q ; 7. assign weight 1 to all edges of G; 8. finally, set k = 2c 2 + 3c.
If Q contains a multicolored clique C such that |C | = c then it is possible to construct a solution subgraph S of G with weight k as follows: (a) s and all of its out-edges belong to S; (b) for each or-vertex wi ∈ V (S), include in S an edge (wi , v 1 ) of G such that v ∈ C ; (c) for each v 1 ∈ V (S), add all of its out-edges to S; (d) for each v 2 ∈ V (S) add its out-edges to S; (e) j j for each v 3 ∈ V (S) add its out-edges to S; (f) finally, for each av 1 in S, choose an out-edge (av 1 , u3 ) to be added to S such that 2 u ∈ C . Note that S has weight k = 2c + 3c. Conversely, if G has a solution subgraph S with weight k = 2c 2 + 3c then it is possible to obtain a multicolored clique C of Q as follows: a vertex v of Q belongs to C if and only if v 3 belongs to V (S). Since any solution subgraph of G has weight at j least k, if S has weight k = 2c 2 + 3c then S has only c vertices of type v 3 , and for each pair v 3 , au1 (u ̸ = v ) in S there is an edge between them, implying that C is a multicolored clique of size c of Q . □
Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Fig. 4. An and/or graph G obtained from Q . Edges of G are directed according to a breadth-first search from s.
4. Bounded tree- and clique-width Instances of Min-and/or and Min-and/or0 are directed acyclic graphs, and as both problems are solvable in polynomial time when the underlying undirected graph is a tree [31], it is still interesting to ask about the complexity of both problems when the underlying undirected graph has bounded tree-width or clique-width. 4.1. Tree-width Let G be a graph, T a tree, and let V = (Vt )t ∈T be a family of vertex sets Vt ⊆ V (G) indexed by the vertices t of T . The pair (T , V ) is called a tree-decomposition of G if it satisfies the following three conditions [16]:
⋃
(T1) V (G) = t ∈T Vt ; (T2) for every edge e ∈ G there exists t ∈ T such that both ends of e lie in Vt ; (T3) If Vti and Vtj both contain a vertex v , then v ∈ Vtk for all nodes tk in the path between ti and tj . The width of (T , V ) is the number max{|Vt | − 1 : t ∈ T }, and the tree-width t w (G) of G is the minimum width of any tree-decomposition of G. Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Let Min-and/or0 (t w ) be the parameterized version when the tree-width of the underlying undirected graph is considered as a single parameter. Theorem 4. Min-and/or0 (t w ) is fixed-parameter tractable. Proof. A solution subgraph of an and/or graph is MSO2 -expressible [12,13] by the following formula ϕ (A, X ): there is a set of edges A and a set of vertices X such that
• • • • •
s ∈ X; every edge of A has its two endpoints in X ; every vertex of X except s is the head of an edge in A; every vertex u ∈ X ∩ And has all its out-edges in A; every vertex u ∈ X ∩ Or has a unique out-edge in A.
Let now Ψ (A) ⇔ ∃X ϕ (A, X ). Min-and/or0 consists of finding a subset A of minimum weight τ (A) satisfying Ψ (A), and is fixed-parameter tractable when parameterized by the tree-width via Courcelle’s meta-theorem for graphs of bounded tree-width and properties expressed by MSO formulas using edge set quantifications (MSO2 properties) [12,13]. □
4.2. Clique-width The clique-width of a graph G is based on the number of labels needed to construct G by means of the following four operations [9]:
• • • •
Creation of a new vertex v with label i. Disjoint union of two labeled graphs G and H. Joining by an edge every vertex labeled i to every vertex labeled j, where i ̸ = j Renaming label i to label j.
The clique-width c w (G) of a graph G is defined in [14] as the minimum number of labels which are necessary to generate G by using these operations. Let Min-and/or0 (k, c w ) and Min-and/or(k, c w ) be parameterized versions where k and the clique-width c w are aggregated parameters. In contrast to the previous result on the tree-width, the next theorem shows that Min-and/or0 and Min-and/or are unlikely to be FPT-time solvable when parameterized by k and c w . W[SAT] is defined as the class of parameterized problems reducible to Weighted Satisfiability [17] and Weighted Monotone Satisfiability is W[SAT]-complete [1]. Weighted Monotone Satisfiability Instance: A monotone Boolean formula X . Parameter: A positive integer k. Question: Does X have a satisfying assignment of weight k? Theorem 5. Min-and/or0 (k, c w ) is W[SAT]-hard and in W[P]. Proof. From an instance (X , k) of Weighted Monotone Satisfiability we construct an instance (G, k) of Min-and/or0 (k, c w ) as follows. First construct the tree circuit C corresponding to X ; we can assume that all leaves of the tree circuit C are at the same level. Next, we obtain G as described in the proof of Theorem 2. The only vertices of G with in-degree greater than one are those representing inputs of C , and G has a solution subgraph of size k if and only if X has a satisfying assignment of weight k. Let Vi be the set of vertices of G at a distance i from s, and let j be the distance from s to the sinks. Without loss of generality Vj−2 contains only or-vertices. For the pair Vj−2 , Vj−1 add edges from Vj−2 to Vj−1 in order to make the underlying undirected graph induced by Vj−2 , Vj−1 a complete bipartite graph; assign weight k + 1 for these new edges. Now, note that G has clique-with at most four, since complete bipartite graphs have clique-width two, trees have clique-width at most three [24], and using four labels we can construct trees reserving an exclusive label to the leaves. □ Corollary 6. Min-and/or(k, c w ) is W[1]-hard and in W[P]. Proof. Simply change the construction of Theorem 3 similarly as described in the previous theorem. □ Although Theorem 5 and Corollary 6 show the parameterized intractability of both problems with respect to the cliquewidth, the next result provides a sufficient condition to solve them in FPT-time when parameterized by c w . Theorem 7. Min-and/or0 (c w ) is fixed-parameter tractable if for each or-vertex its out-edges have the same weight. Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Proof. Let G = (V , E) be an and/or graph where for each or-vertex ∑ v of G its out-edges have the same weight. To each vertex v ∈ V assign a weight τ (v ) such that τ (v ) = 0 if v is a sink, τ (v ) = e∈Ov τ (e) if v is an and-vertex, and τ (v ) = τ (e) (e ∈ Ov ) if v is an or-vertex. subgraph S of G satisfies ∑ for any or-vertex of G its out-edges have the same weight, any solution ∑ Note that since 0 τ ( v ). At this point, we consider the objective function of Min-and/or (c w ) as the sum of weights τ (e) = v∈V (S) e∈E(S) of vertices in the solution subgraph. Thus, a solution subgraph of G can be MSO1 -expressible [12,13] by a formula Φ (X ) (only vertex and vertex set variables are used), as follows. Let And and Or be the subsets of V containing the and- and the or-vertices, respectively.
Φ (X ) = ‘‘there is a set of vertices X such that
• • • •
s ∈ X; every vertex of X except s has an in-neighbor in X ; every vertex u ∈ X ∩ And has all its out-neighbors in X ; every vertex u ∈ X ∩ Or has some out-neighbor in X ’’.
By Courcelle’s meta-theorem for graphs of bounded clique-width and properties expressed by MSO1 formulas [12,13], we can determine in FPT time the minimum τ (X ) for X satisfying Φ (X ), and the corresponding set X , which induces an optimal solution subgraph of G. □ 5. Infeasibility of polynomial kernels The study of problems which admit kernelization algorithms that reduce instances to a size which is polynomially bounded in the parameter has become an interesting field of investigation. Such problems are said to have a polynomial kernelization algorithm, or a polynomial kernel. Results on infeasibility of polynomial kernels for specific problems have appeared recently. Bodlaender et al. [6] and Fortnow and Santhanam [19] have developed a framework based on the notion of compositionality, to show that a problem does not admit a polynomial kernel unless NP ⊆ coNP/poly. Definition 1. [5] An or-composition algorithm for a parameterized problem Π ⊆ Σ ∗ × N is an algorithm that gets as input a sequence ((x1 , k), . . . , (xr , k)), with each (xi , ki ) ∈ Σ ∗ × N, and outputs a pair (x′ , k′ ), such that
∑
1. the algorithm uses time polynomial in 1≤i≤r |xi | + k; 2. k′ is bounded by a polynomial in k; 3. (x′ , k′ ) ∈ Π if and only if there exists an index i, 1 ≤ i ≤ r, such that (xi , k) ∈ Π . A problem is said to be or-compositional if it admits an or-composition algorithm. Corollary 8 ([6,19]). Let Π be a parameterized problem with Π c its corresponding classical decision variant. If Π is orcompositional and Π c is NP-complete, then Π has no kernel of polynomial size, unless NP ⊆ coNP/poly. Algorithms that compose multiple instances of a problem into one instance have been developed for various problems [6–8,26]. Definition 2. Let H be a class of and/or graphs such that every non-sink or-vertex v of a graph G ∈ H has out-degree at most two, and every edge has weight equal to one. As shown in [31], Min-and/or remains NP-hard even restricted to and/or graphs in H . Lemma 9. Min-and/or(k) restricted to graphs in H can be solved in O(2k × n) time. Proof. This proof uses the method of bounded search trees (see [17]). Consider the following algorithm for G ∈ H , where each vertex v ∈ V (G) has a label ℓ(v ) initially equals zero; Pand = {s}, Por = ∅ if f (s) = and; Pand = ∅, Por = {s} if f (s) = or.
Sol -Sub( G ,k ,Pand ,Por ) While Pand ̸= ∅ do . Remove a vertex v from Pand . k := k − |Ov | . For each non -sink w ∈ Ov do . . If ℓ(w) = 0 then . . . If f (w) = and then . . . . Pand := Pand ∪ {w} . . . Else . . . . Por := Por ∪ {w} . . ℓ(w) := 1 Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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If ( k < 0 ) or ( k = 0 and Por ̸= ∅ ) then . return false If Por = ∅ then . return true Else . remove a vertex v from Por . ctrl := false . For each w ∈ Ov do % Note that |Ov |≤ 2 . . If ( ℓ(w) = 0 ) and ( w is not a sink) then . . . If f (w) = and then . . . . ctrl := (ctrl or Sol -Sub( G ,k − 1 ,Pand ∪ {w} ,Por )) . . . Else . . . . ctrl := (ctrl or Sol -Sub( G ,k − 1 ,Pand ,Por ∪ {w} )) . . Else . . . ctrl := (ctrl or Sol -Sub( G ,k − 1 ,Pand ,Por )) return ctrl It is easy to see that the previous recursive algorithm exhaustively checks the existence of a solution subgraph containing at most k edges. As it implicitly constructs a binary search tree with height at most k (size O(2k )), where for each node is performed a O(n) routine, the algorithm runs in O(2k × n) time. □ At this point, we will show that Min-and/or(k) restricted to graphs in H does not have a polynomial kernel unless NP ⊆ coNP/poly. Theorem 10. Min-and/or(k) restricted to graphs in H is or-compositional. Proof. Suppose the input sequence Q = ( (G1 , k), (G2 , k), . . . , (Gt , k) ), where Gi is an and/or graph in H , 1 ≤ i ≤ t, and k is a fixed parameter. Let (G′ , k′ ) be an instance obtained from Q as follows: (a) If t ≥ 2k then solve each instance (Gi , k), 1 ≤ i ≤ t, in O(2k × n) time and return an instance (G′ , k) such that G′ has a solution subgraph of size k if and only if some Gi has a solution subgraph of size k. Note that this algorithm uses time polynomial in (t , n). (b) If t < 2k , assume without loss of generality that t is a power of two. Then, • initially, set G′ as the disjoint union of all the graphs in Q ; • add a new vertex s′ to G′ ; • construct a full binary tree T with source s′ and leaves si , for each graph Gi , and assign weight 1 to each edge of T and label or to each internal vertex of T ; • set k′ = k + log2 t; • output the pair (G′ , k′ ). Since the source vertex of G′ is s′ , clearly G′ has a solution subgraph of weight at most k′ if and only if there is a graph Gi in Q with a solution subgraph of weight at most k. As k′ = k + log2 t and t < 2k , we have k′ ≤ 2k. □ Corollary 11. Unless NP ⊆ coNP/poly, Min-and/or(k, t w ) and Min-and/or(k, c w ) restricted to graphs in H do not admit polynomial kernels. Proof. Follows from the above, or-composition which preserves tree-width and increases by at most one the cliquewidth. □ 6. Conclusion In this paper a multivariate investigation of the problem of finding a solution subgraph of an and/or graph with weight at most k is made. We answer negatively to the following open question: ‘‘Is the problem of finding a solution subgraph of weight at most k (where k is the parameter) in FPT?’’. We show that Min-and/or(k) is W[1]-hard and in W[P]; Min-and/or0 (k) is W[P]-complete; Min-and/or0 (t w ) is fixedparameter tractable; Min-and/or0 (k, c w ) is W[SAT]-hard and in W[P]; and Min-and/or0 (c w ) is FPT time solvable when each or-vertex has all of its out-edges with the same weight. In addition, it is shown that the tractable cases do not admit a polynomial kernel unless NP ⊆ coNP/poly, even restricted to instances without or-vertices with out-degree greater than two. Table 1 summarizes the results obtained in this work. Our results suggest the following questions: (a) for which class of the W-hierarchy Min-and/or(k) is complete?; (b) Is Min-and/or0 (k, c w ) W[P]-complete? Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.
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Table 1 Complexity of and/or graph solution problems.
k tw cw k, t w k, c w k, c w a a
Min-and/or
Min-and/or0
W[1]-hard, in W[P] FPT NP-complete when c w ≥ 4 or-compositional W[1]-hard, in W[P] FPT, or-compositional
W[P]-complete FPT NP-complete when c w ≥ 4 or-compositional W[SAT]-hard, in W[P] FPT, or-compositional
In this version, for any or-vertex its out-edges have the same weight.
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Please cite this article in press as: U.d.S. Souza, F. Protti, Tractability, hardness, and kernelization lower bound for and/or graph solution, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.029.