Often, in frequency allocation problems for cellular networks, it is important to allot a unique frequency for each client, so that at least one frequency is unaffected by cancellation. Such problems can be theoretically formulated as a coloring problem on a set system, better known as conflict-free coloring . Formally, given a set system , a -conflict-free coloring is a function where for each set , there is an element such that for all , . In other words, each set has at least one element that is uniquely colored in the set. This variant of coloring has also been extensively studied for set systems induced by various geometric regions [2, 12, 20].
A natural step to study most coloring problems is to study them in graphs. Given a graph , denotes the set of vertices of while denotes the set of edges in . A -coloring of , for is a function . The most well-studied coloring problem on graphs is proper-coloring. A -coloring is called a proper-coloring if for each edge , . In this paper, we study two specialized variants of -conflict-free coloring on graphs, known as -ONCF-coloring and -CNCF-coloring, which are defined as follows.
Given a graph , a -coloring is called a -ONCF-coloring, if for every vertex , there is a vertex in the open neighborhood such that for all . In other words, every open neighborhood in has a uniquely colored vertex.
Given a graph , a -coloring is called a -CNCF-coloring, if for for every vertex , there is a vertex in the closed neighborhood such that for all . In other words, every closed neighborhood in has a uniquely colored vertex.
Observe that by the above definitions, the -ONCF-coloring (or -CNCF-coloring) problem is a special case of the conflict-free coloring of set systems. Given a graph , we can associate it with the set system , where consists of the sets given by open neighborhoods (respectively, closed neighborhoods ) for . A -ONCF-coloring (or -CNCF-coloring) of then corresponds to a -conflict-free coloring of the associated set system.
Notationally, let denote the minimum number of colors required for a conflict-free coloring of a set system . Similarly, we denote by and the minimum number of colors required for an ONCF-coloring and a CNCF-coloring of a graph , respectively. The study of conflict-free coloring was initially restricted to combinatorial studies. This was first explored in  and . Pach and Tardos  gave an upper bound of on for a set system when the size of is . In , it was also shown that for a graph with vertices . This bound was shown to be tight in . Similarly,  showed that .
However, computing or is NP-hard. This is because deciding whether a -ONCF-coloring or a -CNCF-coloring of exists is NP-hard . This motivates the study of the following decision problems under the lens of parameterized complexity.
-ONCF-Coloring Input: A graph . Question: Is there a -ONCF-coloring of ?
The -CNCF-Coloring problem is defined analogously.
Note that because of the NP-hardness for -ONCF-Coloring or -CNCF-Coloring even when , the two problems are para-NP-hard under the natural parameter . Thus, the problems were studied under structural parameters. Gargano and Rescigno  showed that both -ONCF-Coloring and -CNCF-Coloring have FPT algorithms when parameterized by (i) the size of a vertex cover of the input graph , (ii) and the neighborhood diversity of the input graph. Gargano and Rescigno also mention that due to Courcelle’s theorem, for a non-negative constant , the two decision problems are FPT with the treewidth of the input graph as the parameter.
Our Results and Contributions.
In this paper, we extend the parameterized study of the above two problems with respect to structural parameters. Our first objective is to provide both upper and lower bounds for FPT algorithms when using treewidth as the parameter (Section 3). We show that both -ONCF-Coloring and -CNCF-Coloring parameterized by treewidth can be solved in time . On the other hand, for , both problems cannot be solved in time under Strong Exponential Time Hypothesis (SETH). For , both problems cannot be solved in time under Exponential Time Hypothesis (ETH).
We also study the polynomial kernelization question (Section 4). Observe that both -ONCF-Coloring and -CNCF-Coloring cannot have polynomial kernels under treewidth as the parameter, as there are straightforward and-cross-compositions from each problem to itself.111This is true for a number of graph problems when parameterized by treewidth. For more information, see [6, Theorem 15.12] and the example given for Treewidth (parameterized by solution size) in [6, page 534]. Therefore, we will study the kernelization question by a larger parameter, namely the size of a vertex cover in the input graph. The kernelization complexity of the -Coloring problem (asking for a proper-coloring of the input graph) is very well-studied for this parameter, the problem admits a kernel of size  which is known to be tight unless . From this perspective however, -CNCF-Coloring and -ONCF-Coloring turn out to be much harder: -CNCF-Coloring for and -ONCF-Coloring for do not have polynomial kernels under the standard complexity assumptions, when parameterized by the size of a vertex cover. Interestingly, -CNCF-Coloring parameterized by vertex cover size does have a polynomial kernel and we obtain an explicit polynomial compression for the problem. Although this does not lead to a polynomial kernel of reasonable size, we study a restricted version called -CNCF-Coloring-VC-Extension (Section 4.4) and show that this problem has a kernel where is the vertex cover size. Therefore, -CNCF-Coloring behaves significantly differently from the other problems.
Finally, we obtain a number of combinatorial results regarding ONCF-colorings of graphs. Denote by the minimum for which a -proper-coloring for exists. While , the same upper bound does not hold for . For a graph , let , and denote the size of a minimum vertex cover, the size of a minimum feedback vertex set and the treewidth of , respectively. From the known result that , we could immediately obtain the fact that the same behavior holds for . However, to show that behaves similarly more work needs to be done. To the best of our knowledge no upper bounds on with respect to and were known, while a loose upper bound was provided with respect to in . We give a tight upper bound on with respect to and also provide the first upper bounds on with respect to and (Section 5).
Our main contributions in this work are structural results for the conflict-free coloring problem, which we believe gives more insight into the decision problems on graphs. Firstly, the gadgets we build for the ETH-based lower bounds could be useful for future lower bounds, but are also useful to understand difficult examples for conflict-free coloring which have not been known in abundance so far. We are able to reuse these gadgets in the constructions needed to prove the kernelization lower bounds. Secondly, our combinatorial results also give constructible conflict-free colorings of graphs and therefore provide more insight into conflict-free colored graphs. Finally, the kernelization dichotomy we obtain for -ONCF-Coloring and -CNCF-Coloring under vertex cover size as a parameter is a very surprising one.
For a positive integer , we denote the set in short with . For a graph , given a -coloring and a subset , we denote by the restriction of to the subset . For a graph that is -ONCF-colored by a coloring , for a vertex , suppose is such that for each ; then is referred to as the ONCF-color of . Similarly, for a graph that is -CNCF-colored by a coloring , for a vertex , a unique color in is referred to as the CNCF-color of .
An edge-star graph is a generalization of a star graph where there is a central edge and all other vertices have . A triangle is an example of an edge-star graph.
2.1 Tree decompositions and treewidth
We define treewidth and tree decompositions.
Definition 3 (Tree Decomposition )
A tree decomposition of a (undirected or directed) graph is a tuple , where is a tree in which each vertex has an assigned set of vertices (called a bag) such that the following properties hold:
For any , there exists a such that .
If and , then for all on the path from to in .
In short, we denote as .
The treewidth of a tree decomposition is the size of the largest bag of minus one. A graph may have several distinct tree decompositions. The treewidth of a graph is defined as the minimum of treewidths over all possible tree decompositions of . Note that for the tree of a tree decomposition, we denote a vertex of in bold font. If is rooted at a vertex , for a vertex , , where is the subtree rooted at .
A tree decomposition is called a nice tree decomposition if is a tree rooted at some node where , each node of has at most two children, and each node is of one of the following kinds:
Introduce node: a node that has only one child where and .
Forget vertex node: a node that has only one child where and .
Join node: a node with two children and such that .
Leaf node: a node that is a leaf of , and .
One can show that a tree decomposition of width can be transformed into a nice tree decomposition of the same width and with nodes, see e.g. .
We modify the definition of a nice tree decomposition slightly by ensuring that no bag in the tree decomposition is empty. This can easily be done by adding an arbitrary vertex to all bags of the current nice tree decomposition. This will ensure the non-emptiness property. Note that our nice tree decomposition will have width .
2.2 Parameterized complexity
Let be a finite alphabet. A parameterized problem is a subset of .
Definition 4 (Kernelization)
Let be two parameterized problems and let be some computable function. A generalized kernel from to of size is an algorithm that given an instance , outputs in time such that
(i) if and only if , and
(ii) and .
The algorithm is a kernel if . It is a polynomial (generalized) kernel if is a polynomial in .
Next, we describe a few methods that can be used to rule out the existence of polynomial kernels. One such method is by a polynomial parameter transformation  from a problem that is known to not admit a polynomial kernel. We repeat the necessary information here for completeness.
Definition 5 (Polynomial parameter transformation )
Let and be parameterized problems. A polynomial parameter transformation from to is an algorithm that takes an input and outputs such that the following hold.
if and only if , and
is bounded by a polynomial in .
We denote this as .
The following Theorem follows from [3, Prop. 2.16] and shows how to obtain lower bounds using polynomial parameter transformations.
Theorem 2.1 ()
Let and be parameterized problems with . If admits a polynomial generalized kernel, then admits a polynomial generalized kernel.
Another way to rule out the existence of polynomial kernels is using the framework of cross-compositions . We start by providing the necessary definitions.
Definition 6 (Polynomial equivalence relation )
An equivalence relation on is called a polynomial equivalence relation if the following two conditions hold:
There is an algorithm that given two strings decides whether and belong to the same equivalence class in time polynomial in .
For any finite set the equivalence relation partitions the elements of into a number of classes that is polynomially bounded in the size of the largest element of .
Definition 7 (Cross-composition )
Let be a language, let be a polynomial equivalence relation on , and let be a parameterized problem. An or-cross-composition of into (with respect to ) is an algorithm that, given instances of belonging to the same equivalence class of , takes time polynomial in and outputs an instance such that the following hold:
The parameter value is polynomially bounded in , and
The instance is a yes-instance for if and only if at least one instance is a yes-instance for .
The following theorem shows how cross-compositions are used to prove kernelization lower bounds.
Theorem 2.2 ()
If an NP-hard language or-cross-composes into the parameterized problem , then does not admit a (generalized) polynomial kernelization unless .
2.3 Fast Subset Convolution Computation.
Given a universe with elements, the subset convolution of two functions is a function such that for every , . Equivalently, .
Proposition 1 ()
For two functions , given all the values of and in the input, all the values of the subset convolution can be computed in arithmetic operations.
In fact, this result can be extended to subset convolution of functions that map to any ring, instead of . Consider the set , with the added relation that . The operator takes two elements from this set and outputs the maximum of the two elements. Notice that , along with as an additive operator and as a multiplicative operator, forms a semi-ring . We will call this semi-ring the integer max-sum semi-ring. The subset convolution of two functions , with and as the additive and multiplicative operators, becomes .
Proposition 2 ()
Given two functions , all the values of and in the input, and all the values of the subset convolution over the integer max-sum semiring can be computed in time .
3 Algorithmic results parameterized by treewidth
In this section, we state the algorithmic results obtained for the ONCF-Coloring and CNCF-Coloring problems parameterized by treewidth. On the algorithmic side, we have the following theorem.
-ONCF-Coloring and -CNCF-Coloring parameterized by treewidth admits a time algorithm.
We also obtain algorithmic lower bounds for the problems under standard assumptions.
The following algorithmic lower bounds can be obtained:
For , -ONCF-Coloring or -CNCF-Coloring parameterized by treewidth cannot be solved in time, under SETH.
-ONCF-Coloring or -CNCF-Coloring parameterized by treewidth cannot be solved in time, under ETH.
In the remainder of this section, we will prove the two theorems stated above.
In this section, we prove Theorem 3.1. In the following Lemma, we describe an algorithm for -ONCF-Coloring, parameterized by treewidth. The algorithm for -CNCF-Coloring parameterized by treewidth is very similar and has the same running time.
-ONCF-Coloring parameterized by treewidth admits a time algorithm.
We assume that a nice tree decomposition , rooted at a leaf , is given to us. Also, recall that no bag in empty, and that each leaf bag or the root bag has exactly one vertex in it. We proceed with the following treewidth dynamic programming. Given a bag corresponding to the vertex , a state for the bag is a tuple , where
determines the bag,
is a vertex coloring of . Intuitively, for a vertex , is the color receives in the conflict-free coloring we are after.
is a color assignment to each vertex of . For a vertex , should be the color that occurs exactly once in the neighborhood of .
is an indicator function for the vertices of . The idea is that indicates whether already has a neighbor of color in the subtree rooted at i.
Let be the set of all states associated with . A function is defined as follows: For a state , suppose there is a vertex coloring such that (i) its restriction to the vertices in is the coloring , (ii) for each , the color is used at most once in , (iii) for each , if there is a a vertex such that then and otherwise , (iv) for any vertex , has a uniquely colored vertex under coloring . Then . Otherwise, . In other words, is such that except for the vertices in the graph induced on is ONCF-colored and a state stores a snapshot of at the boundary of the graph seen so far.
Our dynamic programming will calculate the function for each bag . Note that for the root , if in there is a state such that and , then the graph has a -ONCF-coloring. We describe our dynamic programming in cases according to the types of nodes of the tree decomposition.
Let be a leaf node. Then,
This can be calculated in time. For the correctness, note that the uniquely colored neighbor of cannot appear in the graph seen so far as a leaf node only contains .
Let be a forget node with its child being . Also, let . Consider a state and a state . We say that is consistent with , or if (i) , (ii) and . Then,
This can be calculated in time. To prove correctness, first suppose and let be a coloring that is a witness to this. By definition of consistency, there is only one state such that . Since , the same coloring also witnesses the fact that . Conversely, suppose where is the unique state such that . Let be a coloring that is a witness to this. Since and by definition of consistency , the same coloring also witnesses the fact that . Thus, our recurrence correctly calculates .
Let be an introduce node with its child being . Also, let . Consider a state and a state . We say that is consistent with , or if (i) , (ii) if there is a such that then there is exactly one such and , otherwise there is no such and , (iii) If there is a such that then and , (iv) for all other , . Then,
This can be calculated in time. To prove correctness, first suppose and let be a coloring that is a witness to this. By definition of consistency, there is at least one state such that and the same coloring also witnesses the fact that . Conversely, suppose there is a state such that and . Then by definition of consistency, . Thus, our recurrence correctly calculates .
Let be a join node with its children being and . This means that . Consider a state , and states . We say that is consistent with , or if (i) , (ii) if there is a such that () then () and , (iii) for all other , . Then,
As before, the correctness of the recurrence follows from the definition of consistency.
It is straightforward to calculate this in time, we will further improve this to time as follows.
Notice that if we fix and , then get fixed for consistent states. Also, given , , and , consider the vertices . Then for each vertex , . Now consider . For consistent states, the following relations hold: (i) , (ii) and . Thus, if we are given the function , we can completely determine when we are looking at consistent states. Now, fix a function . We define functions in the following way. For a subset , define a function such that for any , and otherwise. Now, define and .
The correctness of this recurrence is same as the correctness of the previous recurrence. Due to fast subset convolution over the max-sum semi-ring , this can be calculated in time. ∎
3.2 Running time lower bounds
In this section, we given the proof of Theorem 3.2 by describing lower bounds on algorithmic running times for the ONCF-Coloring and CNCF-Coloring problems parameterized by treewidth.
We start by providing a running time lower bound on -ONCF-Coloring under ETH claimed in Theorem 3.2. The bound will be obtained by giving a reduction from -SAT, and in order to give the reduction we will need the following type of gadget.
An ONCF-gadget is a gadget on ten vertices, as depicted in Figure 1.
The objective of this gadget is the following. The vertices in Figure 1 will be the interaction points of the ONCF-gadget with the outside world. As will be proved in the following two lemmas, the gadget is designed so as to (i) disallow certain -ONCF-colorings and (ii) allow certain -ONCF-colorings on its interaction points.
Let be a ONCF-gadget with a coloring such that for all the neighborhood of is ONCF-colored by . If , then .
Suppose . Since , this implies . Similarly, we find . Since now has two blue vertices, we conclude that .∎
Let be a ONCF-gadget. Let be a partial -ONCF-coloring of . If there exists such that , then can be extended to a coloring satisfying
For every , the neighborhood of vertex is ONCF-colored by (contains at most one red, or at most one blue vertex), and
, , , and .
Let equal on vertex and and define , , , and . If or , define else define . If or , define , otherwise let . This completes the definition of . It is easy to verify that both requirements are satisfied by this coloring, refer to Figure 1 for an example coloring.∎
Now that we have introduced the necessary gadgets, we can prove the running time lower bound for -ONCF-Coloring.
-ONCF-Coloring parameterized by treewidth cannot be solved in time, under ETH.
We show this by giving a reduction from -SAT. Given an instance of -SAT with variables and clauses , create a graph as follows. Start by creating palette vertices , and , and edges and . For each variable , create vertices and add edges and . For the remainder of the construction we will reuse the ONCF-gadget as defined in Definition 8. For each , add an ONCF-gadget and connect of this gadget to . Add vertices , and and connect to in for . Let clause . Now if for some , connect to . Similarly, if , connect to . This concludes the construction of , it remains to show that is -ONCF-colorable if and only if the formula was satisfiable.
Suppose the satisfiability instance has satisfying assignment , we show how to color . Let , and . Let for all and define for all , . Finally, if , let and . Otherwise, let and . For each gadget , vertex for has neighbor . Let be the other neighbor of vertex . Define such that . Since the formula was satisfied by , for each there hereby exists such that . We use Lemma 3 to extend the partial coloring to color gadget , with and . It is straightforward to verify that is a -ONCF-coloring of .
Suppose has a -ONCF-coloring, we give a satisfying assignment . Assume without loss of generality that . Since for all , it follows that . We therefore define if and if . Let be a clause, we will show that satisfies to conclude the proof. Suppose for contradiction that does not satisfy . Then every vertex for had one neighbor in that is blue in . Thereby, its only other neighbor in gadget must be colored red. It follows from Lemma 2 that . Observe however that and that both these vertices are red, contradicting that is a -ONCF-coloring of . Thus, the formula is satisfied by .
Note that the graph induced by is a disjoint union of ONCF-gadgets and has treewidth two. As such, has treewidth at most .
In this reduction a -SAT formula on variables and clauses is reduced to a graph with treewidth at most . We proved that is satisfiable if and only if has a -ONCF-coloring. Since -SAT cannot be solved in time under ETH, this also implies that -CNCF-Coloring parameterized by treewidth cannot be solved in time, under ETH. ∎
Note that a reduction from -SAT to -ONCF-Coloring was given in Theorem 2 of . However, that reduction led to a quadratic blow-up in the input size. Hence, the need for the alternative reduction given above.
For , -ONCF-Coloring parameterized by treewidth cannot be solved in time, under SETH.
It was shown in  that for a constant , -Coloring cannot be solved in time, under SETH. For a graph , let be the graph obtained by subdiving every edge of once. It was shown in Theorem 3 of , that has a -coloring if and only if has a -ONCF-coloring. Also, note that since it is a subdivision of . Thus, for a constant , the lower bound of on the running time of any algorithm under SETH follows. ∎
-CNCF-Coloring parameterized by treewidth cannot be solved in time, under ETH.
In , a reduction of -CNCF-Coloring was given from -SAT. In this reduction a -SAT formula on variables and clauses is reduced to a graph with treewidth at most . It was shown that is satisfiable if and only if has a -CNCF-coloring. Since -SAT cannot be solved in time under ETH, this also implies that -CNCF-Coloring parameterized by treewidth cannot be solved in time, under ETH. ∎
For , -CNCF-Coloring parameterized by treewidth cannot be solved in time, under SETH.
It was shown in  that for a constant , -Coloring cannot be solved in time, under SETH. For a graph , Theorem 3.1 of  constructs a graph such that has a -coloring if and only if has a -CNCF-coloring. The construction of requires the graphs as described in Section 4.2, and first constructed in . Recall that the is defined recursively as in Definition 9.
Returning to the construction of , we obtain from in the following manner: (i) for each vertex we add two copies and of and make adjacent to all vertices of and , (ii) for each edge we add two copies and of and make the vertices and adjacent to all vertices of and . This completes the construction of . For the completion of our proof it remains to show that in order to obtain a lower bound of on the running time of any algorithm under SETH.
For a graph , .
We prove our statement by induction on . In the base case, it is true that and . Let the induction hypothesis be that for any