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Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number
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Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number✩ Michael Fellows a , Fábio Protti b , Frances Rosamond a , Maise Dantas da Silva b , Uéverton S. Souza b, * a b
University of Bergen, Bergen, Norway Fluminense Federal University, Niterói, Brazil
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Article history: Received 30 December 2015 Received in revised form 22 June 2017 Accepted 3 July 2017 Available online xxxx Keywords: Flood-It Flood-filling Vertex cover FPT Kernel ETH
a b s t r a c t Flood-It is a combinatorial problem on a colored graph whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a pivot vertex p. A flooding move consists of changing the color of the monochromatic component (maximal monochromatic connected subgraph) containing p. This problem generalizes a combinatorial game named alike which is played on m × n grids. It is known that FloodIt is NP-hard even for 3 × n grids and for instances with bounded number of colors, diameter, treewidth, or pathwidth. In [Fellows, Souza, Protti, Dantas da Silva, Tractability and hardness of flood-filling games on trees, Theoretical Computer Science, 576, 102-116, 2015] it is shown that Flood-It is W[1]-hard when played on trees with bounded number of colors, and the number of leaves is a single parameter. Contrasting with such results, in this work we show that Flood-It is fixed-parameter tractable when parameterized by either the vertex cover number or the neighborhood diversity. Additionally, we prove that Flood-It does not admit a polynomial kernel when the vertex cover number is a single parameter, unless coNP ⊆ NP /poly. Finally, lower bounds based on the (Strong) Exponential Time Hypothesis as well as an upper bound for the required time to solve Flood-It are also provided. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Let G be a vertex-colored graph and let p ∈ V (G) be a pivot vertex of G. A flooding move m = (p, c) in G consists of changing to c the color of p and of all vertices in the monochromatic component (maximal monochromatic connected subgraph) containing p in G. The problem of determining the minimum number of flooding moves to make the graph monochromatic is called Flood-It. Fig. 1 shows a sequence of moves to flood a graph. As shown in [13], Flood-It played on trees is analogous to a restricted case of the Shortest Common Supersequence Problem (SCS) [18], where no string has the same symbol in consecutive positions. Consequently, Flood-It inherits many applications from this special case of SCS, such as: microarray production [27], DNA sequence assembly [2], and multiple sequence alignment [28]. In particular, when each symbol occurs at most once in any path from the pivot to a leaf of the tree, each path is analogous to a phylogenetic sequence (see [14]). ✩ This project was partially supported by the FAPERJ-Brazil (grant no. E-26/010.001578/2016) and CNPq-Brazil (grant no. 459051/2014-8). Corresponding author. E-mail addresses: [email protected] (M. Fellows), [email protected] (F. Protti), [email protected] (F. Rosamond), [email protected] (M.D. da Silva), [email protected] (U.S. Souza).
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http://dx.doi.org/10.1016/j.dam.2017.07.004 0166-218X/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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Fig. 1. An optimal sequence of moves to flood a 6-colored Petersen graph.
As described in [13], Flood-It on trees can also be applied to scheduling. Each color corresponds to an operation in a sequential process of manufacturing an object. In the input tree T , paths from the pivot to the leaves correspond to the manufacturing sequences for a number of different objects that share the same production line. A flooding of T then corresponds to a schedule of operations for the production line that allows all of the different objects to be manufactured. Beyond these applications when the underlying graph is a tree, some disease spreading models described in [1] work in a similar way as the Flood-It game. The computational problem Flood-It is generalization of a combinatorial game named alike, which is originally played on a colored board consisting of an m × n grid, where each tile of the board has an initial color from a fixed color set. Many complexity issues on Flood-It have recently been investigated. In [9], Clifford, Jalsenius, Montanaro, and Sach show that Flood-It is NP-hard on n × n grids colored with at least three colors. Meeks and Scott [22] prove that Flood-It remains NP-hard on 3 × n grids colored with at least four colors. Clifford, Jalsenius, Montanaro, and Sach present in [9] a polynomialtime algorithm for Flood-It on 2 × n grids. Regarding the complexity of Flood-It played on general graphs, Fleischer and Woeginger [15] proved that Flood-It (denoted by Honey-Bee-Solitaire) remains NP-hard even when the game is restricted to trees or split graphs, but it is polynomial-time solvable on co-comparability graphs. In [13,29], Fellows, Souza, Protti, and Dantas da Silva show that Flood-It played on trees is analogous to an important subcase of SCS, as indicated earlier in this section. In [30], Souza, Protti and Dantas da Silva describe polynomial-time algorithms to play Flood-It on Cn2 or Pn2 (the second power of a cycle or a path on n vertices) and 2 × n circular grids, and Fellows, Souza, Protti, and Dantas da Silva [13] develop a multivariate investigation of the complexity of Flood-It when played on trees, analyzing the complexity consequences of parameterizing flood-filling problems in various ways. Besides that, a variant of Flood-It, where in each move the player can choose a new pivot vertex (so-called Free-Flood-It) was also studied in [9,13,20,23,24,30]. Flood-It remains NP-hard even assuming constant values for: number of colors [9]; diameter [29]; or treewidth [15]. In [13,29], Fellows, Souza, Protti, and Dantas da Silva show parameterized complexity results on Flood-It on trees; for instance, Flood-It on trees is W[1]-hard when parameterized by the aggregate parameter (number of leaves, number of colors). Therefore, finding interesting parameters for which Flood-It is fixed-parameter tractable seems to be a challenge. The main goal of this paper is to analyze the parameterized complexity of Flood-It when parameterized by the vertex cover number. Results. We describe an FPT-algorithm for Flood-It with either the vertex cover number or the neighborhood diversity as a single parameter, and we present a polynomial kernelization algorithm when the neighborhood diversity and the number of colors of the input graph form an aggregate parameter. In addition, we show the following results: Flood-It does not admit polynomial kernel when the vertex cover number is a single parameter, unless coNP ⊆ NP /poly; no 2o(k+ic ) nO(1) time algorithm for Flood-It is possible unless the Exponential Time Hypothesis (ETH) fails; and no (2 −ε )ic nO(1) time algorithm for Flood-It exists unless the Strong Exponential Time Hypothesis (SETH) fails, where k is the cardinality of a minimum vertex cover and ic is the number of colors of a maximum independent set. On the other hand, an O(2O(k log(ic k)) nO(1) ) algorithm for Flood-It is provided. Definitions and notation. - Two vertices a and b are m-connected when there is a monochromatic path between them. - A subgraph H is said to be flooded when H becomes monochromatic. Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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- A vertex v is flooded by a move m if the color of v is played in m and v becomes m-connected to p after playing m. We say that a move m floods a vertex v by a vertex w if v and w are neighbors, m changes the color of w to flood v , and w was the first neighbor of v to be flooded in G. - If v is a neighbor of p then v is flooded by p, and if v is flooded by w we say that w is the link to v . - A move m = (p, c) is played on subgraph H if p ∈ V (H). A maximal monochromatic connected subgraph H ′ of a subgraph H is abbreviated an mcs of H. - An island is a vertex v colored with a color c such that no neighbor of v is colored with c. - A good move for a color ca is a move that floods all non-flooded vertices having color ca . A move that is not good for some color is a bad move. - The graph G on which Flood-It is played is called the board of the game. 2. The vertex cover number and the neighborhood diversity as parameters In this section we present a parameterized complexity analysis of Flood-It with respect to the vertex cover number and with respect to the neighborhood diversity. Theorem 1. Flood-It is in FPT when parameterized by the vertex cover number (k). Proof. The proof of Theorem 1 is based in the following facts:
• Enumerating all the vertex covers of size at most k can be done in O(2k n) time [12]. • Given a vertex cover C of a graph G and a flooding F of G, in order to flood the vertices in C it is always possible to play a subset of moves in F .
• In any optimal flooding of G, after first flooding a vertex cover of G, each color is played at most one more time in order to complete the flooding. Since a minimum vertex cover C of G can be found in O(2k n) time, the main idea of this proof is to construct in f (k)nO(1) time a subgraph G′ of G, consisting of: (i) the pivot vertex p; (ii) a minimum vertex cover C of G; (iii) possible and nonredundant links to flood C in G. We need the following terminology: two distinct colors c1 and c2 are called twin colors if for every vertex v colored with ci , i ∈ {1, 2}, there is a vertex w colored with cj , j ∈ {1, 2} \ {i}, such that v and w have the same neighborhood. The construction of G′ is presented below. (a) Set G′ = G. (b) For each independent set R in V (G′ ) \ (C ∪{p}) such that all vertices in R have the same color and the same neighborhood, do: select one vertex v ∈ R and remove all the vertices in R \ {v}. (c) Let S be the set of colors not in C ∪ {p}; for each subset Q of S consisting of mutually twin colors do: select one color ci ∈ Q and remove all vertices colored with a color in Q \ {ci }. k
Graph G′ has O(2k+2 ) vertices: by (b) G′ has at most 2k + k + 1 vertices for each color; by (c) each color class that is k not the color of a vertex in C ∪ {p} represents a subset of P(C ) (the power set of C ), i.e., G′ has at most 22 + k + 1 colors. Thus, we can construct an optimal flooding S of G in FPT time as follows: (1) obtain a minimum vertex cover C in FPT time; (2) construct the graph G′ in polynomial time; (3) exhaustively analyze all the possible sequences of moves to flood C in G′ ; (4) for each possible flooding Fi′ of C in G′ , create a flooding Fi of G applying Fi′ , and next play each color at most one more time in order to flood G; and (5) finally set S as the flooding Fi with minimum number of moves. To show that such algorithm finds the optimal flooding, it is enough to verify that the removals made by Rules (b) and (c) are always safe. In fact, vertices with the same color and the same neighborhood work as a single vertex, thus Rule (b) can be applied without loss of generality. In addition, bad moves using colors that are not in C ∪ {p} are played to create links to flood some vertices in C ; hence the set of such bad moves can be played using no twin colors, because one can always play an already played color i instead of one of its twin colors. This means that Rule (c) can also be applied without loss of generality. □ To prove Theorem 1 we choose to present the simplest proof. However one can improve the performance of the described algorithm by replacing the FPT-routine to obtain a vertex cover of size k by a polynomial 2-approximation algorithm which returns a vertex cover of size at most 2k (see, e.g., [26]). In addition, note that the vertex set of the graph G′ is formed by at most four disjoint subsets of vertices: {p}, C \ {p}, a set A containing the remaining vertices adjacent to p (A may be empty), and a set B containing the remaining vertices adjacent only to C \ {p}. Thus, we can reduce G′ as follows: (a) Set V ′ = {p} ∪ C ∪ A. (b) For each pair v, w ∈ C (v, w not necessarily distinct) and for each color ci do: select, if any, a vertex x ∈ B of color ci , neighbor of v and w , and set V (H) = V (H) ∪ {x}. (c) Remove from G′ all vertices that is not in V ′ . (d) Contract all neighbors of p with color ci into a single vertex of color ci in G′ . At this point, G′ will have at most k2 + k + 1 vertices per color. Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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2.1. Neighborhood diversity Definition 1. A graph G(V , E) has neighborhood diversity nd(G) = t if one can partition V into t sets V1 , . . . , Vt such that, for all v ∈ V and every i ∈ {1, . . . , t }, either v is adjacent to every vertex in Vi or it is adjacent to none of them. Note that each part Vi of G is either a clique or an independent set. The parameter neighborhood diversity is a natural generalization of the vertex cover number. In 2012, Lampis [21] showed that for every graph G we have nd(G) ≤ 2k + k, where k is the vertex cover number of G. The optimal neighborhood diversity decomposition of a graph G can be computed in O(n3 ) time [21]. Corollary 2. Flood-It is fixed-parameter tractable when parameterized by the neighborhood diversity. Proof. This proof is similar to the one presented in Theorem 1. Given the nd(G) parts of a graph G, we have that after flooding at least one vertex of each part, each remaining color is played once. At this point, we can construct a graph G′ from G as in the proof of Theorem 1. If two vertices with the same color occur in the same part then one of them can be removed. In addition, if two colors occur in exactly the same set of parts then they are like ‘‘twin’’ colors and we can remove one of them. After that, G′ has O(2nd(G) ) colors, each one containing at most nd(G) vertices. Finally, by exhaustively analyzing G′ we can construct an optimal solution for G in FPT time. □
3. Polynomial kernels Since every FPT problem admits a kernelization algorithm, it is interesting to study problems admitting kernelization algorithms that reduce instances to a size which is polynomially bounded by the parameter. Such problems are said to have a polynomial kernelization algorithm, or a polynomial kernel. Nice examples of kernelization include a polynomial kernel for the Vertex Cover Problem [10] and the meta-theorems for kernelization of problems on planar graphs [4]. On the other hand, Chromatic Number and Clique do not admit polynomial kernels with respect to the vertex cover number of the input graph unless there is a collapse in the polynomial hierarchy, in contrast to the fact that these problems are trivially fixed-parameter tractable for this parameter [6]. Theorem 3. Flood-It admits polynomial kernelization when parameterized by the neighborhood diversity (nd) and the number of colors (c). Proof. In Flood-It true (false) twin vertices with the same color work as if they were a single vertex. Therefore, each part of a partition of G described as Definition 1 needs only one vertex per color class, thus G admits a kernel of size nd(G) × c. □ Bodlaender et al. [3] and Fortnow and Santhanam [16] have developed a framework based on the notion of compositionality, to show that a problem does not admit a polynomial kernel unless coNP ⊆ NP /poly. Among the techniques used to demonstrate the infeasibility of polynomial kernels, the main ones are: Or-composition [3,5,8,11,16,25], Cross-composition [6], and Polynomial Parameter Transformation (or ppt-reduction) [3,7,11,25]. A useful type of polynomial parameter transformations was introduced by Bodlaender et al. [7]. Such transformations consist of reducing from a problem for which a kernelization lower bound is known to the problem in question, such that a polynomial kernel for the considered problem would transfer to a polynomial kernel for the problem one reduces from [11]. Definition 2. Let P and Q be parameterized problems. We say that P is polynomial parameter reducible to Q , written P ≤ppt Q , if there exists a polynomial-time computable function f : Σ ∗ × N → Σ ∗ × N and a polynomial p such that, for all (x, k) ∈ Σ ∗ × N: (a) (x, k) ∈ P if and only (x′ , k′ ) = f (x, k) ∈ Q , and (b) k′ ≤ p(k). The function f is called a ppt-reduction. To obtain a kernelization lower bound for Flood-It parameterized by the vertex cover number, we use a ppt-reduction from the following problem. Red/Blue Dominating Set (RBDS) Instance: A bipartite graph G = (R ∪ B, E) and an integer r. Goal: Determine whether G has a red/blue dominating set of size at most r. A vertex set R′ ⊆ R is considered a red/blue dominating set if every vertex in B has at least one neighbor in R′ . We note that RBDS is equivalent to Set Cover and Hitting Set, and thus NP-complete (see [17]). Lemma 4 ([11]). Red/Blue Dominating Set parameterized by (|B|, r) does not admit a polynomial kernel, unless coNP ⊆ NP /poly. Theorem 5. Flood-It parameterized by the vertex cover number does not admit a polynomial kernel, unless coNP ⊆ NP /poly, even restricted to bipartite or chordal graphs. Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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Proof. We use a ppt-reduction from RBDS parameterized by |B|. Given an instance (G, r) of RBDS parameterized by the size |B| of the blue set, we construct a board H for Food-it parameterized by the vertex cover number as follows.
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Set H = G. Add a new vertex p (the pivot) in H, and add edges between p and any vertex in R. Assign a distinct color to each vertex in V (G). For each vertex v in B, create a set of vertices Rv with the same size and the same color as R, and add edges between v and each vertex in Rv .
As we can observe, B ∪ {p} is a minimum vertex cover of H. Therefore, to complete the ppt-reduction, it is enough to show that G has a red/blue dominating set of size r if and only if H has a flooding of size at most r + |B| + |R|. If R has a subset S of size r which dominates B then we can flood H by first playing the colors in S ∩ H; next, playing once each color in B ∩ H; finally, playing one move for each color in R ∩ H. Conversely, if H admits an optimal flooding of size at most r + |B| + |R| then: (a) any color in B is played once; (b) immediately after flooding B any color in R is played once; (c) without loss of generality, we can assume that at most r moves using colors of R ∩ H are played before flooding any vertex in B, and such moves create links in R to flood B. Hence, such moves play colors that represent a subset of vertices in R of size at most r which dominates B. H is a bipartite graph. However, without loss of generality, any pair of neighbors of the pivot p could be adjacent, and in this case we obtain a chordal graph. □ A useful parameter to the Flood-It game is the number of bad moves used in a flooding. In [13] it is shown that Flood-It on a special kind of colored trees, called phylogenetic colored trees (see [13]), is fixed-parameter tractable when parameterized by the maximum number of bad moves to be played. Corollary 6. Flood-It does not admit a polynomial kernel, unless coNP ⊆ NP /poly, even when the vertex cover number and the maximum number of bad moves to be played are considered as aggregated parameters. Proof. Follows from Lemma 4 and the fact that r in the reduction above is exactly the number of bad moves played in H.
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4. Lower bounds based on the exponential-time hypothesis Let G be a board of the Flood-It game. In this section we obtain some lower bounds for the Flood-It game considering the following parameters: k, denoting the vertex cover number of G, and ic , denoting the minimum number of colors of a maximum independent set of G. The Exponential Time Hypothesis (ETH) is a conjecture stating that 3-SAT has no algorithm subexponential in the number of variables. ETH allows us to prove quantitative results of various forms. For example, one can prove that (assuming ETH) a problem cannot be solved in time 2o(n) , or a parameterized problem cannot be solved in time f (k)no(k) , or a fixed-parameter tractable parameterized problem cannot be solved in time 2o(k) nO(1) [10]. In addition it is known that, unless ETH fails, there exists a constant c > 0 such that no algorithm for 3-SAT can achieve running time O(2c(n+m) ) [19]. In particular, this implies that 3-SAT cannot be solved in 2o(n+m) time, unless ETH fails. A variant of ETH called the Strong Exponential Time Hypothesis (SETH) implies that CNF-SAT cannot be solved in O((2 − ε )n ) time for any ε ≥ 0. SETH can be used to give even more refined lower bounds; e.g., it can be used to prove that a parameterized problem cannot be solved in time (2 − ε )k nO(1) time for any ε ≥ 0 [10]. Based on ETH and SETH we obtain the following bounds for Flood-It. Theorem 7. Flood-It cannot be solved in 1. 2o(k+ic ) nO(1) time, unless ETH fails, and 2. (2 − ε )ic nO(1) time, unless SETH fails, even when the input graph is bipartite or chordal. Proof. Unless ETH fails, 3-SAT cannot be solved in 2o(n+m) time, where n is the number of variables and m is the number of clauses. A natural reduction from 3-SAT to Red/Blue Dominating Set consists of constructing an instance G of RBDS where R = O(n) and B = O(m), which implies that RBDS cannot be solved in 2o(|R|+|B|) time. Using the reduction presented in Theorem 5, from an instance (G, r) of RBDS we obtain a board H of Flood-It where |R| = ic and k = |B| + 1. Hence, Flood-It on graphs cannot be solved in 2o(k+ic ) nO(1) time, unless ETH fails. Item 2 follows from the reduction presented in Theorem 5 and the fact that RBDS cannot be solved in (2 − ε )|R| nO(1) time [10], unless SETH fails. □ Theorem 7 provides a strong bound for bipartite and chordal graphs. The next result gives us bounds for very restricted subclasses of bipartite and chordal graphs. Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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Theorem 8. Unless ETH fails, Flood-It cannot be solved in 2o(k+ic ) nO(1) time, even when the input graph G is a tree with height two or a split graph formed by a clique and a set of pendant vertices. Proof. Unless ETH fails, Vertex Cover cannot be solved in 2o(|V |+|E |) time (see [10]). Fellows, Souza, Protti, and Dantas da Silva in [13] showed that Flood-It remains NP-complete even on trees with height two, using a reduction from Vertex Cover, where ic and k in the resulting tree are of the order of the number of vertices and the number of edges of the instance of Vertex Cover, respectively. Moreover, since any pair of neighbors of a pivot p, without loss of generality, can be adjacent, by adding such edges we obtain a split graph formed by a clique and a set of pendant vertices. Therefore, assuming ETH, Flood-It cannot be solved in 2o(k+ic ) nO(1) time. □ 5. Exact algorithm Although Flood-It is trivially solvable when a vertex cover is already flooded (just play each remaining color once), by Theorem 8 one can conclude that this Flood-It remains NP-hard even when the following conditions are satisfied: (a) a minimum vertex cover C of G is known, (b) finding an optimal flooding of G[C ] is trivial, and (c) any vertex v of V (G) \ C , without loss of generality, is flooded only after its neighborhood is flooded. An example of a hard instance G meeting such conditions is a split graph formed by a clique and a set of pendant vertices, constructed as described in the proof of Theorem 8. In fact, for a graph G, the order in which each vertex of a vertex cover C is flooded may affect the number of remaining colors after flooding C . In addition, if C does not form a clique of G then some vertices of V (G) \ C may be used as links to flood vertices in C , and the subset of vertices used as links, as well as the order in which such vertices are flooded, can also affect the number of remaining colors after flooding C . Theorem 9. Flood-It can be solved in O(2O(k log(ic k)) nO(1) ) time, where k is the vertex cover number and ic is the minimum number of colors of a maximum independent set of G. Proof. Let G be a board of Flood-It. As we know, a minimum vertex cover C of G can be found in 2k nO(1) time, where k is the size of a minimum vertex cover. Also, any optimal flooding F of G has a minimal subset of moves FC ⊆ F which floods the vertices of G in C , and, without loss of generality, moves in FC are all played before moves in F \ FC . Note that |FC | ≤ 2k, because for each vertex v of C we may need one distinct move mv to flood v , and at most one preceding move m′v to create a link in V (G) \ C to play mv . Thus, we can focus on analyzing all possible floodings of FC . For each move m′v we have ic + 1 possibilities; either m′v is unnecessary or it plays a color of the maximum independent set, V (G) \ C . In addition, there are k! possible orderings to flood the vertices in C (if two vertices are flooded by the same move the partial order between them does not matter). Therefore, there are k! × (ic + 1)k possible configurations, where some of them are feasible floodings. Note that FC is one of such floodings, say F ′ , which minimizes |F ′ | + ℓ′ , where ℓ′ is the minimum number of moves needed to be played in order to extend F ′ to a flooding of G. Hence, since k! × (ic + 1)k ≤ 2k log((ic +1)k) , Flood-It can be solved in O(2O(k log(ic k)) nO(1) ) time. 6. Conclusions In contrast with the hardness of solving Flood-It on graphs with bounded clique-width, treewidth, pathwidth, or feedback vertex set, in this work we show the fixed-parameter tractability of Flood-It when parameterized by either the vertex cover number or the neighborhood diversity. A polynomial kernelization is presented when the neighborhood diversity and the number of colors are aggregate parameters, and a proof is given that Flood-It does not admit a polynomial kernel when the vertex cover number is a single parameter, unless coNP ⊆ NP /poly. Lower bounds based on the Exponential-Time Hypothesis (ETH), and a O(2O(k log(ic k)) nO(1) ) algorithm for Flood-It are also presented. By Theorem 8 one can conclude that Flood-It remains NP-hard even restricted to graphs that are simultaneously split, block, strongly chordal, P5 -free and distance-hereditary. On the other hand, it is known that Flood-it can be solved in polynomial time on cycles, co-comparability graphs, and 2 × n grids [9,15]. Therefore, it is interesting to identify new graph classes for which Flood-It can be non-trivially solvable in polynomial time. It is also interesting to identify other single parameters for which Flood-It is fixed-parameter tractable. Finally, we remark that determining whether Flood-It can be solved in 2O(k+ic ) nO(1) time, and determining whether FloodIt admits a polynomial kernel when the neighborhood diversity is a single parameter are both open questions. References [1] C. Aschwanden, Spatial simulation model for infectious viral diseases with focus on SARS and the common flu, in: 37th Annual Hawaii International Conference on System Sciences, Proceedings, IEEE, 2004, 5 pages. [2] P. Barone, P. Bonizzoni, G.D. Vedova, G. Mauri, An approximation algorithm for the shortest common supersequence problem: an experimental analysis, in: Proceedings of the 2001 ACM Symposium on Applied Computing, ACM, 2001, pp. 56–60. [3] H.L. Bodlaender, R.G. Downey, M.R. Fellows, D. Hermelin, On problems without polynomial kernels, J. Comput. System Sci. 75 (8) (2009) 423–434.
Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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[4] H.L. Bodlaender, F.V. Fomin, D. Lokshtanov, E. Penninkx, S. Saurabh, D.M. Thilikos, (Meta) kernelization, in: 50th Annual Symposium on Foundations of Computer Science, FOCS’09, Proceedings, IEEE, 2009, pp. 629–638. [5] H.L. Bodlaender, B.M.P. Jansen, S. Kratsch, Kernel bounds for path and cycle problems, Theoret. Comput. Sci. 511 (2013) 117–136. [6] H.L. Bodlaender, B.M.P. Jansen, S. Kratsch, Kernelization lower bounds by cross-composition, SIAM J. Discrete Math. 28 (1) (2014) 277–305. [7] H.L. Bodlaender, S. Thomassé, A. Yeo, Analysis of data reduction: Transformations give evidence for non-existence of polynomial kernels, in: Technical Report UU-CS-2008-030, Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands, 2008. [8] H.L. Bodlaender, S. Thomassé, A. Yeo, Kernel bounds for disjoint cycles and disjoint paths, Theoret. Comput. Sci. 412 (35) (2011) 4570–4578. [9] R. Clifford, M. Jalsenius, A. Montanaro, B. Sach, The complexity of flood filling games, Theory Comput. Syst. 50 (1) (2012) 72–92. [10] M. Cygan, F.V. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, S. Saurabh, Parameterized Algorithms, Springer, 2015. [11] M. Dom, D. Lokshtanov, S. Saurabh, Incompressibility through colors and IDs, in: S. Albers, A. Marchetti-Spaccamela, Y. Matias, S. Nikoletseas, W. Thomas (Eds.), Automata, Languages, and Programming, 36th Int. Colloquium, ICALP 2009, Proceedings - Part I, in: Lecture Notes in Computer Science, vol. 5555, Springer, 2009, pp. 378–389. [12] R.G. Downey, M.R. Fellows, Parameterized Complexity, Springer, 1999. [13] M.R. Fellows, U. dos Santos Souza, F. Protti, M.D. da Silva, Tractability and hardness of flood-filling games on trees, Theoret. Comput. Sci. 576 (2015) 102–116. [14] M. Fellows, M. Hallett, U. Stege, Analogs & duals of the MAST problem for sequences & trees, J. Algorithms 49 (1) (2003) 192–216. [15] R. Fleischer, G.J. Woeginger, An algorithmic analysis of the honey-Bee game, Theoret. Comput. Sci. 452 (2012) 75–87. [16] L. Fortnow, R. Santhanam, Infeasibility of instance compression and succinct PCPs for NP, in: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 133–142. [17] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide To the Theory of NP-Completeness, W. H. Freeman, New York, 1979. [18] M.T. Hallett, An integrated complexity analysis of problems from computational biology, (Ph.D. thesis), University of Victoria, 1998. [19] R. Impagliazzo, R. Paturi, F. Zane, Which problems have strongly exponential complexity? J. Comput. System Sci. 63 (4) (2001) 512–530. [20] A. Lagoutte, M. Noual, E. Thierry, Flooding games on graphs, Discrete Appl. Math. 164 (2014) 532–538. [21] M. Lampis, Algorithmic meta-theorems for restrictions of treewidth, Algorithmica 64 (1) (2012) 19–37. [22] K. Meeks, A. Scott, The complexity of flood-filling games on graphs, Discrete Appl. Math. 160 (7) (2012) 959–969. [23] K. Meeks, A. Scott, The complexity of free-Flood-It on 2 × n boards, Theoret. Comput. Sci. 500 (2013) 25–43. [24] K. Meeks, A. Scott, Spanning trees and the complexity of flood-filling games, Theory Comput. Syst. 54 (4) (2014) 731–753. [25] N. Misra, V. Raman, S. Saurabh, Lower bounds on kernelization, Discrete Optim. 8 (1) (2011) 110–128. [26] B. Monien, E. Speckenmeyer, Ramsey numbers and an approximation algorithm for the vertex cover problem, Acta Inform. 22 (1985) 115–123. [27] S. Rahmann, The shortest common supersequence problem in a microarray production setting, Bioinformatics 19 (suppl 2) (2003) ii156–ii161. [28] J.S. Sim, K. Park, The consensus string problem for a metric is NP-complete, J. Discrete Algorithms 1 (1) (2003) 111–117. [29] U.S. Souza, F. Protti, M.D. da Silva, Parameterized complexity of flood-filling games on trees, in: Proceedings of the 19th International Computing and Combinatorics Conference, in: Lecture Notes in Computer Science, vol. 7936, Springer, 2013, pp. 531–542. [30] U.S. Souza, F. Protti, M.D. da Silva, An algorithmic analysis of flood-It and Free-Flood-It on graph powers, Discrete Math. Theoret. Comput. Sci. 16 (3) (2014) 279.
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number✩ Michael Fellows a , Fábio Protti b , Frances Rosamond a , Maise Dantas da Silva b , Uéverton S. Souza b, * a b
University of Bergen, Bergen, Norway Fluminense Federal University, Niterói, Brazil
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Article history: Received 30 December 2015 Received in revised form 22 June 2017 Accepted 3 July 2017 Available online xxxx Keywords: Flood-It Flood-filling Vertex cover FPT Kernel ETH
a b s t r a c t Flood-It is a combinatorial problem on a colored graph whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a pivot vertex p. A flooding move consists of changing the color of the monochromatic component (maximal monochromatic connected subgraph) containing p. This problem generalizes a combinatorial game named alike which is played on m × n grids. It is known that FloodIt is NP-hard even for 3 × n grids and for instances with bounded number of colors, diameter, treewidth, or pathwidth. In [Fellows, Souza, Protti, Dantas da Silva, Tractability and hardness of flood-filling games on trees, Theoretical Computer Science, 576, 102-116, 2015] it is shown that Flood-It is W[1]-hard when played on trees with bounded number of colors, and the number of leaves is a single parameter. Contrasting with such results, in this work we show that Flood-It is fixed-parameter tractable when parameterized by either the vertex cover number or the neighborhood diversity. Additionally, we prove that Flood-It does not admit a polynomial kernel when the vertex cover number is a single parameter, unless coNP ⊆ NP /poly. Finally, lower bounds based on the (Strong) Exponential Time Hypothesis as well as an upper bound for the required time to solve Flood-It are also provided. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Let G be a vertex-colored graph and let p ∈ V (G) be a pivot vertex of G. A flooding move m = (p, c) in G consists of changing to c the color of p and of all vertices in the monochromatic component (maximal monochromatic connected subgraph) containing p in G. The problem of determining the minimum number of flooding moves to make the graph monochromatic is called Flood-It. Fig. 1 shows a sequence of moves to flood a graph. As shown in [13], Flood-It played on trees is analogous to a restricted case of the Shortest Common Supersequence Problem (SCS) [18], where no string has the same symbol in consecutive positions. Consequently, Flood-It inherits many applications from this special case of SCS, such as: microarray production [27], DNA sequence assembly [2], and multiple sequence alignment [28]. In particular, when each symbol occurs at most once in any path from the pivot to a leaf of the tree, each path is analogous to a phylogenetic sequence (see [14]). ✩ This project was partially supported by the FAPERJ-Brazil (grant no. E-26/010.001578/2016) and CNPq-Brazil (grant no. 459051/2014-8). Corresponding author. E-mail addresses: [email protected] (M. Fellows), [email protected] (F. Protti), [email protected] (F. Rosamond), [email protected] (M.D. da Silva), [email protected] (U.S. Souza).
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http://dx.doi.org/10.1016/j.dam.2017.07.004 0166-218X/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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Fig. 1. An optimal sequence of moves to flood a 6-colored Petersen graph.
As described in [13], Flood-It on trees can also be applied to scheduling. Each color corresponds to an operation in a sequential process of manufacturing an object. In the input tree T , paths from the pivot to the leaves correspond to the manufacturing sequences for a number of different objects that share the same production line. A flooding of T then corresponds to a schedule of operations for the production line that allows all of the different objects to be manufactured. Beyond these applications when the underlying graph is a tree, some disease spreading models described in [1] work in a similar way as the Flood-It game. The computational problem Flood-It is generalization of a combinatorial game named alike, which is originally played on a colored board consisting of an m × n grid, where each tile of the board has an initial color from a fixed color set. Many complexity issues on Flood-It have recently been investigated. In [9], Clifford, Jalsenius, Montanaro, and Sach show that Flood-It is NP-hard on n × n grids colored with at least three colors. Meeks and Scott [22] prove that Flood-It remains NP-hard on 3 × n grids colored with at least four colors. Clifford, Jalsenius, Montanaro, and Sach present in [9] a polynomialtime algorithm for Flood-It on 2 × n grids. Regarding the complexity of Flood-It played on general graphs, Fleischer and Woeginger [15] proved that Flood-It (denoted by Honey-Bee-Solitaire) remains NP-hard even when the game is restricted to trees or split graphs, but it is polynomial-time solvable on co-comparability graphs. In [13,29], Fellows, Souza, Protti, and Dantas da Silva show that Flood-It played on trees is analogous to an important subcase of SCS, as indicated earlier in this section. In [30], Souza, Protti and Dantas da Silva describe polynomial-time algorithms to play Flood-It on Cn2 or Pn2 (the second power of a cycle or a path on n vertices) and 2 × n circular grids, and Fellows, Souza, Protti, and Dantas da Silva [13] develop a multivariate investigation of the complexity of Flood-It when played on trees, analyzing the complexity consequences of parameterizing flood-filling problems in various ways. Besides that, a variant of Flood-It, where in each move the player can choose a new pivot vertex (so-called Free-Flood-It) was also studied in [9,13,20,23,24,30]. Flood-It remains NP-hard even assuming constant values for: number of colors [9]; diameter [29]; or treewidth [15]. In [13,29], Fellows, Souza, Protti, and Dantas da Silva show parameterized complexity results on Flood-It on trees; for instance, Flood-It on trees is W[1]-hard when parameterized by the aggregate parameter (number of leaves, number of colors). Therefore, finding interesting parameters for which Flood-It is fixed-parameter tractable seems to be a challenge. The main goal of this paper is to analyze the parameterized complexity of Flood-It when parameterized by the vertex cover number. Results. We describe an FPT-algorithm for Flood-It with either the vertex cover number or the neighborhood diversity as a single parameter, and we present a polynomial kernelization algorithm when the neighborhood diversity and the number of colors of the input graph form an aggregate parameter. In addition, we show the following results: Flood-It does not admit polynomial kernel when the vertex cover number is a single parameter, unless coNP ⊆ NP /poly; no 2o(k+ic ) nO(1) time algorithm for Flood-It is possible unless the Exponential Time Hypothesis (ETH) fails; and no (2 −ε )ic nO(1) time algorithm for Flood-It exists unless the Strong Exponential Time Hypothesis (SETH) fails, where k is the cardinality of a minimum vertex cover and ic is the number of colors of a maximum independent set. On the other hand, an O(2O(k log(ic k)) nO(1) ) algorithm for Flood-It is provided. Definitions and notation. - Two vertices a and b are m-connected when there is a monochromatic path between them. - A subgraph H is said to be flooded when H becomes monochromatic. Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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- A vertex v is flooded by a move m if the color of v is played in m and v becomes m-connected to p after playing m. We say that a move m floods a vertex v by a vertex w if v and w are neighbors, m changes the color of w to flood v , and w was the first neighbor of v to be flooded in G. - If v is a neighbor of p then v is flooded by p, and if v is flooded by w we say that w is the link to v . - A move m = (p, c) is played on subgraph H if p ∈ V (H). A maximal monochromatic connected subgraph H ′ of a subgraph H is abbreviated an mcs of H. - An island is a vertex v colored with a color c such that no neighbor of v is colored with c. - A good move for a color ca is a move that floods all non-flooded vertices having color ca . A move that is not good for some color is a bad move. - The graph G on which Flood-It is played is called the board of the game. 2. The vertex cover number and the neighborhood diversity as parameters In this section we present a parameterized complexity analysis of Flood-It with respect to the vertex cover number and with respect to the neighborhood diversity. Theorem 1. Flood-It is in FPT when parameterized by the vertex cover number (k). Proof. The proof of Theorem 1 is based in the following facts:
• Enumerating all the vertex covers of size at most k can be done in O(2k n) time [12]. • Given a vertex cover C of a graph G and a flooding F of G, in order to flood the vertices in C it is always possible to play a subset of moves in F .
• In any optimal flooding of G, after first flooding a vertex cover of G, each color is played at most one more time in order to complete the flooding. Since a minimum vertex cover C of G can be found in O(2k n) time, the main idea of this proof is to construct in f (k)nO(1) time a subgraph G′ of G, consisting of: (i) the pivot vertex p; (ii) a minimum vertex cover C of G; (iii) possible and nonredundant links to flood C in G. We need the following terminology: two distinct colors c1 and c2 are called twin colors if for every vertex v colored with ci , i ∈ {1, 2}, there is a vertex w colored with cj , j ∈ {1, 2} \ {i}, such that v and w have the same neighborhood. The construction of G′ is presented below. (a) Set G′ = G. (b) For each independent set R in V (G′ ) \ (C ∪{p}) such that all vertices in R have the same color and the same neighborhood, do: select one vertex v ∈ R and remove all the vertices in R \ {v}. (c) Let S be the set of colors not in C ∪ {p}; for each subset Q of S consisting of mutually twin colors do: select one color ci ∈ Q and remove all vertices colored with a color in Q \ {ci }. k
Graph G′ has O(2k+2 ) vertices: by (b) G′ has at most 2k + k + 1 vertices for each color; by (c) each color class that is k not the color of a vertex in C ∪ {p} represents a subset of P(C ) (the power set of C ), i.e., G′ has at most 22 + k + 1 colors. Thus, we can construct an optimal flooding S of G in FPT time as follows: (1) obtain a minimum vertex cover C in FPT time; (2) construct the graph G′ in polynomial time; (3) exhaustively analyze all the possible sequences of moves to flood C in G′ ; (4) for each possible flooding Fi′ of C in G′ , create a flooding Fi of G applying Fi′ , and next play each color at most one more time in order to flood G; and (5) finally set S as the flooding Fi with minimum number of moves. To show that such algorithm finds the optimal flooding, it is enough to verify that the removals made by Rules (b) and (c) are always safe. In fact, vertices with the same color and the same neighborhood work as a single vertex, thus Rule (b) can be applied without loss of generality. In addition, bad moves using colors that are not in C ∪ {p} are played to create links to flood some vertices in C ; hence the set of such bad moves can be played using no twin colors, because one can always play an already played color i instead of one of its twin colors. This means that Rule (c) can also be applied without loss of generality. □ To prove Theorem 1 we choose to present the simplest proof. However one can improve the performance of the described algorithm by replacing the FPT-routine to obtain a vertex cover of size k by a polynomial 2-approximation algorithm which returns a vertex cover of size at most 2k (see, e.g., [26]). In addition, note that the vertex set of the graph G′ is formed by at most four disjoint subsets of vertices: {p}, C \ {p}, a set A containing the remaining vertices adjacent to p (A may be empty), and a set B containing the remaining vertices adjacent only to C \ {p}. Thus, we can reduce G′ as follows: (a) Set V ′ = {p} ∪ C ∪ A. (b) For each pair v, w ∈ C (v, w not necessarily distinct) and for each color ci do: select, if any, a vertex x ∈ B of color ci , neighbor of v and w , and set V (H) = V (H) ∪ {x}. (c) Remove from G′ all vertices that is not in V ′ . (d) Contract all neighbors of p with color ci into a single vertex of color ci in G′ . At this point, G′ will have at most k2 + k + 1 vertices per color. Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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2.1. Neighborhood diversity Definition 1. A graph G(V , E) has neighborhood diversity nd(G) = t if one can partition V into t sets V1 , . . . , Vt such that, for all v ∈ V and every i ∈ {1, . . . , t }, either v is adjacent to every vertex in Vi or it is adjacent to none of them. Note that each part Vi of G is either a clique or an independent set. The parameter neighborhood diversity is a natural generalization of the vertex cover number. In 2012, Lampis [21] showed that for every graph G we have nd(G) ≤ 2k + k, where k is the vertex cover number of G. The optimal neighborhood diversity decomposition of a graph G can be computed in O(n3 ) time [21]. Corollary 2. Flood-It is fixed-parameter tractable when parameterized by the neighborhood diversity. Proof. This proof is similar to the one presented in Theorem 1. Given the nd(G) parts of a graph G, we have that after flooding at least one vertex of each part, each remaining color is played once. At this point, we can construct a graph G′ from G as in the proof of Theorem 1. If two vertices with the same color occur in the same part then one of them can be removed. In addition, if two colors occur in exactly the same set of parts then they are like ‘‘twin’’ colors and we can remove one of them. After that, G′ has O(2nd(G) ) colors, each one containing at most nd(G) vertices. Finally, by exhaustively analyzing G′ we can construct an optimal solution for G in FPT time. □
3. Polynomial kernels Since every FPT problem admits a kernelization algorithm, it is interesting to study problems admitting kernelization algorithms that reduce instances to a size which is polynomially bounded by the parameter. Such problems are said to have a polynomial kernelization algorithm, or a polynomial kernel. Nice examples of kernelization include a polynomial kernel for the Vertex Cover Problem [10] and the meta-theorems for kernelization of problems on planar graphs [4]. On the other hand, Chromatic Number and Clique do not admit polynomial kernels with respect to the vertex cover number of the input graph unless there is a collapse in the polynomial hierarchy, in contrast to the fact that these problems are trivially fixed-parameter tractable for this parameter [6]. Theorem 3. Flood-It admits polynomial kernelization when parameterized by the neighborhood diversity (nd) and the number of colors (c). Proof. In Flood-It true (false) twin vertices with the same color work as if they were a single vertex. Therefore, each part of a partition of G described as Definition 1 needs only one vertex per color class, thus G admits a kernel of size nd(G) × c. □ Bodlaender et al. [3] and Fortnow and Santhanam [16] have developed a framework based on the notion of compositionality, to show that a problem does not admit a polynomial kernel unless coNP ⊆ NP /poly. Among the techniques used to demonstrate the infeasibility of polynomial kernels, the main ones are: Or-composition [3,5,8,11,16,25], Cross-composition [6], and Polynomial Parameter Transformation (or ppt-reduction) [3,7,11,25]. A useful type of polynomial parameter transformations was introduced by Bodlaender et al. [7]. Such transformations consist of reducing from a problem for which a kernelization lower bound is known to the problem in question, such that a polynomial kernel for the considered problem would transfer to a polynomial kernel for the problem one reduces from [11]. Definition 2. Let P and Q be parameterized problems. We say that P is polynomial parameter reducible to Q , written P ≤ppt Q , if there exists a polynomial-time computable function f : Σ ∗ × N → Σ ∗ × N and a polynomial p such that, for all (x, k) ∈ Σ ∗ × N: (a) (x, k) ∈ P if and only (x′ , k′ ) = f (x, k) ∈ Q , and (b) k′ ≤ p(k). The function f is called a ppt-reduction. To obtain a kernelization lower bound for Flood-It parameterized by the vertex cover number, we use a ppt-reduction from the following problem. Red/Blue Dominating Set (RBDS) Instance: A bipartite graph G = (R ∪ B, E) and an integer r. Goal: Determine whether G has a red/blue dominating set of size at most r. A vertex set R′ ⊆ R is considered a red/blue dominating set if every vertex in B has at least one neighbor in R′ . We note that RBDS is equivalent to Set Cover and Hitting Set, and thus NP-complete (see [17]). Lemma 4 ([11]). Red/Blue Dominating Set parameterized by (|B|, r) does not admit a polynomial kernel, unless coNP ⊆ NP /poly. Theorem 5. Flood-It parameterized by the vertex cover number does not admit a polynomial kernel, unless coNP ⊆ NP /poly, even restricted to bipartite or chordal graphs. Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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Proof. We use a ppt-reduction from RBDS parameterized by |B|. Given an instance (G, r) of RBDS parameterized by the size |B| of the blue set, we construct a board H for Food-it parameterized by the vertex cover number as follows.
• • • •
Set H = G. Add a new vertex p (the pivot) in H, and add edges between p and any vertex in R. Assign a distinct color to each vertex in V (G). For each vertex v in B, create a set of vertices Rv with the same size and the same color as R, and add edges between v and each vertex in Rv .
As we can observe, B ∪ {p} is a minimum vertex cover of H. Therefore, to complete the ppt-reduction, it is enough to show that G has a red/blue dominating set of size r if and only if H has a flooding of size at most r + |B| + |R|. If R has a subset S of size r which dominates B then we can flood H by first playing the colors in S ∩ H; next, playing once each color in B ∩ H; finally, playing one move for each color in R ∩ H. Conversely, if H admits an optimal flooding of size at most r + |B| + |R| then: (a) any color in B is played once; (b) immediately after flooding B any color in R is played once; (c) without loss of generality, we can assume that at most r moves using colors of R ∩ H are played before flooding any vertex in B, and such moves create links in R to flood B. Hence, such moves play colors that represent a subset of vertices in R of size at most r which dominates B. H is a bipartite graph. However, without loss of generality, any pair of neighbors of the pivot p could be adjacent, and in this case we obtain a chordal graph. □ A useful parameter to the Flood-It game is the number of bad moves used in a flooding. In [13] it is shown that Flood-It on a special kind of colored trees, called phylogenetic colored trees (see [13]), is fixed-parameter tractable when parameterized by the maximum number of bad moves to be played. Corollary 6. Flood-It does not admit a polynomial kernel, unless coNP ⊆ NP /poly, even when the vertex cover number and the maximum number of bad moves to be played are considered as aggregated parameters. Proof. Follows from Lemma 4 and the fact that r in the reduction above is exactly the number of bad moves played in H.
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4. Lower bounds based on the exponential-time hypothesis Let G be a board of the Flood-It game. In this section we obtain some lower bounds for the Flood-It game considering the following parameters: k, denoting the vertex cover number of G, and ic , denoting the minimum number of colors of a maximum independent set of G. The Exponential Time Hypothesis (ETH) is a conjecture stating that 3-SAT has no algorithm subexponential in the number of variables. ETH allows us to prove quantitative results of various forms. For example, one can prove that (assuming ETH) a problem cannot be solved in time 2o(n) , or a parameterized problem cannot be solved in time f (k)no(k) , or a fixed-parameter tractable parameterized problem cannot be solved in time 2o(k) nO(1) [10]. In addition it is known that, unless ETH fails, there exists a constant c > 0 such that no algorithm for 3-SAT can achieve running time O(2c(n+m) ) [19]. In particular, this implies that 3-SAT cannot be solved in 2o(n+m) time, unless ETH fails. A variant of ETH called the Strong Exponential Time Hypothesis (SETH) implies that CNF-SAT cannot be solved in O((2 − ε )n ) time for any ε ≥ 0. SETH can be used to give even more refined lower bounds; e.g., it can be used to prove that a parameterized problem cannot be solved in time (2 − ε )k nO(1) time for any ε ≥ 0 [10]. Based on ETH and SETH we obtain the following bounds for Flood-It. Theorem 7. Flood-It cannot be solved in 1. 2o(k+ic ) nO(1) time, unless ETH fails, and 2. (2 − ε )ic nO(1) time, unless SETH fails, even when the input graph is bipartite or chordal. Proof. Unless ETH fails, 3-SAT cannot be solved in 2o(n+m) time, where n is the number of variables and m is the number of clauses. A natural reduction from 3-SAT to Red/Blue Dominating Set consists of constructing an instance G of RBDS where R = O(n) and B = O(m), which implies that RBDS cannot be solved in 2o(|R|+|B|) time. Using the reduction presented in Theorem 5, from an instance (G, r) of RBDS we obtain a board H of Flood-It where |R| = ic and k = |B| + 1. Hence, Flood-It on graphs cannot be solved in 2o(k+ic ) nO(1) time, unless ETH fails. Item 2 follows from the reduction presented in Theorem 5 and the fact that RBDS cannot be solved in (2 − ε )|R| nO(1) time [10], unless SETH fails. □ Theorem 7 provides a strong bound for bipartite and chordal graphs. The next result gives us bounds for very restricted subclasses of bipartite and chordal graphs. Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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Theorem 8. Unless ETH fails, Flood-It cannot be solved in 2o(k+ic ) nO(1) time, even when the input graph G is a tree with height two or a split graph formed by a clique and a set of pendant vertices. Proof. Unless ETH fails, Vertex Cover cannot be solved in 2o(|V |+|E |) time (see [10]). Fellows, Souza, Protti, and Dantas da Silva in [13] showed that Flood-It remains NP-complete even on trees with height two, using a reduction from Vertex Cover, where ic and k in the resulting tree are of the order of the number of vertices and the number of edges of the instance of Vertex Cover, respectively. Moreover, since any pair of neighbors of a pivot p, without loss of generality, can be adjacent, by adding such edges we obtain a split graph formed by a clique and a set of pendant vertices. Therefore, assuming ETH, Flood-It cannot be solved in 2o(k+ic ) nO(1) time. □ 5. Exact algorithm Although Flood-It is trivially solvable when a vertex cover is already flooded (just play each remaining color once), by Theorem 8 one can conclude that this Flood-It remains NP-hard even when the following conditions are satisfied: (a) a minimum vertex cover C of G is known, (b) finding an optimal flooding of G[C ] is trivial, and (c) any vertex v of V (G) \ C , without loss of generality, is flooded only after its neighborhood is flooded. An example of a hard instance G meeting such conditions is a split graph formed by a clique and a set of pendant vertices, constructed as described in the proof of Theorem 8. In fact, for a graph G, the order in which each vertex of a vertex cover C is flooded may affect the number of remaining colors after flooding C . In addition, if C does not form a clique of G then some vertices of V (G) \ C may be used as links to flood vertices in C , and the subset of vertices used as links, as well as the order in which such vertices are flooded, can also affect the number of remaining colors after flooding C . Theorem 9. Flood-It can be solved in O(2O(k log(ic k)) nO(1) ) time, where k is the vertex cover number and ic is the minimum number of colors of a maximum independent set of G. Proof. Let G be a board of Flood-It. As we know, a minimum vertex cover C of G can be found in 2k nO(1) time, where k is the size of a minimum vertex cover. Also, any optimal flooding F of G has a minimal subset of moves FC ⊆ F which floods the vertices of G in C , and, without loss of generality, moves in FC are all played before moves in F \ FC . Note that |FC | ≤ 2k, because for each vertex v of C we may need one distinct move mv to flood v , and at most one preceding move m′v to create a link in V (G) \ C to play mv . Thus, we can focus on analyzing all possible floodings of FC . For each move m′v we have ic + 1 possibilities; either m′v is unnecessary or it plays a color of the maximum independent set, V (G) \ C . In addition, there are k! possible orderings to flood the vertices in C (if two vertices are flooded by the same move the partial order between them does not matter). Therefore, there are k! × (ic + 1)k possible configurations, where some of them are feasible floodings. Note that FC is one of such floodings, say F ′ , which minimizes |F ′ | + ℓ′ , where ℓ′ is the minimum number of moves needed to be played in order to extend F ′ to a flooding of G. Hence, since k! × (ic + 1)k ≤ 2k log((ic +1)k) , Flood-It can be solved in O(2O(k log(ic k)) nO(1) ) time. 6. Conclusions In contrast with the hardness of solving Flood-It on graphs with bounded clique-width, treewidth, pathwidth, or feedback vertex set, in this work we show the fixed-parameter tractability of Flood-It when parameterized by either the vertex cover number or the neighborhood diversity. A polynomial kernelization is presented when the neighborhood diversity and the number of colors are aggregate parameters, and a proof is given that Flood-It does not admit a polynomial kernel when the vertex cover number is a single parameter, unless coNP ⊆ NP /poly. Lower bounds based on the Exponential-Time Hypothesis (ETH), and a O(2O(k log(ic k)) nO(1) ) algorithm for Flood-It are also presented. By Theorem 8 one can conclude that Flood-It remains NP-hard even restricted to graphs that are simultaneously split, block, strongly chordal, P5 -free and distance-hereditary. On the other hand, it is known that Flood-it can be solved in polynomial time on cycles, co-comparability graphs, and 2 × n grids [9,15]. Therefore, it is interesting to identify new graph classes for which Flood-It can be non-trivially solvable in polynomial time. It is also interesting to identify other single parameters for which Flood-It is fixed-parameter tractable. Finally, we remark that determining whether Flood-It can be solved in 2O(k+ic ) nO(1) time, and determining whether FloodIt admits a polynomial kernel when the neighborhood diversity is a single parameter are both open questions. References [1] C. Aschwanden, Spatial simulation model for infectious viral diseases with focus on SARS and the common flu, in: 37th Annual Hawaii International Conference on System Sciences, Proceedings, IEEE, 2004, 5 pages. [2] P. Barone, P. Bonizzoni, G.D. Vedova, G. 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Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.
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Please cite this article in press as: M. Fellows, et al., Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.07.004.