Parameterized and Approximation Algorithms for the Load Coloring Problem

Let $c, k$ be two positive integers and let $G=(V,E)$ be a graph. The $(c,k)$-Load Coloring Problem (denoted $(c,k)$-LCP) asks whether there is a $c$-coloring $\varphi: V \rightarrow [c]$ such that for every $i \in [c]$, there are at least $k$ edges with both endvertices colored $i$. Gutin and Jones (IPL 2014) studied this problem with $c=2$. They showed $(2,k)$-LCP to be fixed parameter tractable (FPT) with parameter $k$ by obtaining a kernel with at most $7k$ vertices. In this paper, we extend the study to any fixed $c$ by giving both a linear-vertex and a linear-edge kernel. In the particular case of $c=2$, we obtain a kernel with less than $4k$ vertices and less than $8k$ edges. These results imply that for any fixed $c\ge 2$, $(c,k)$-LCP is FPT and that the optimization version of $(c,k)$-LCP (where $k$ is to be maximized) has an approximation algorithm with a constant ratio for any fixed $c\ge 2$.


Introduction
Given a graph G = (V, E) and an integer k, the 2-Load Coloring Problem asks whether there is a coloring ϕ : V → {1, 2} such that for i = 1 and 2, there are at least k edges with both endvertices colored i. This problem is NP-complete [1], and Gutin and Jones studied its parameterization by k [6]. They proved that 2-Load Coloring is fixed-parameter tractable by obtaining a kernel with at most 7k vertices. It is natural to extend 2-Load Coloring to any number c of colors as follows. Henceforth, for a positive integer p, [p] = {1, 2, . . . , p}.
Definition 1 ((c, k)-Load Coloring). Given a positive integer c, a nonnegative integer k and graph G = (V, E), the (c, k)-Load Coloring Problem asks whether there is a c-coloring ϕ : V → [c] such that for every i ∈ [c], there are at least k edges with both endvertices colored i. We write G ∈ (c, k)-LCP if such a c-coloring exists.
Observe first that G ∈ (1, k)-LCP if and only if |E(G)| ≥ k. In this paper, we consider (c, k)-Load Coloring parameterized by k for every fixed c ≥ 2. Note that (c, k)-Load Coloring is NP-complete for every fixed c ≥ 2. Indeed, we can reduce (2, k)-Load Coloring to (c, k)-Load Coloring with c > 2 by taking the disjoint union of G with c − 2 stars K 1,k .
We prove that the problem admits a kernel with less than 2ck vertices. Thus, for c = 2 we improve the kernel result of [6]. To show our result, we introduce reduction rules, which are new even for c = 2. We prove that the reduction rules can run in polynomial time and we show that a reduced graph with at least 2ck vertices is in (c, k)-LCP.
While there are many parameterized graph problems which admit kernels linear in the number of vertices, usually only problems on classes of sparce graphs admit kernels linear in the number of edges (since in such graphs the number of edges is linear in the number of vertices), see, e.g., [2,4,7]. To the best of our knowledge, only trivial O(k)-edge kernels for general graphs have been described in the literature, e.g., the kernel for Max Cut parameterized by solution size. Thus, our next result is somewhat surprising: (c, k)-Load Coloring admits a kernel with O(k) edges for every fixed c ≥ 2. In fact, (2, k)-Load Coloring has a kernel with less than 8k edges and for every c ≥ 2, (c, k)-Load Coloring has a kernel with less than 16c 2 k − 6ck edges.
The optimization version of (c, k)-Load cColoring, called the c-Load Coloring Problem, is as follows: for a graph G and an integer c ≥ 2, find the maximum k such that G ∈ (c, k)-LCP. The above bounds on the number of edges in the kernel lead to approximation algorithms for this optimization problem: a (4 + ε)-approximation for c = 2 and a constant ratio approximation for c > 2.
The paper is organized as follows. In Section 2, we provide additional terminology and notation. In Section 3, we show that the problem admits a kernel with less than 2ck vertices. In Section 4, we prove an upper bound on the number of edges in a kernel for every c ≥ 2 and the corresponding approximation result for c-Load Coloring. We improve our bound for c = 2 in Section 5. The bound implies the approximation ratio of 4+ε for every ε > 0. We complete the paper with discussions in Section 6.

Terminology and Notation
Graphs. For a graph G, V (G) (E(G), respectively) denotes the vertex (edge, respectively) set of G, ∆(G) denotes the maximum degree of G, n its number of vertices, and m its number of edges. For a vertex x and vertex set X in G, and E(X, Y ) = {xy ∈ E(G) : x ∈ X, y ∈ Y }. A vertex u with degree 0 (1, respectively) is an isolated vertex (a leaf-neighbor of v, where uv ∈ E(G), respectively). For a coloring ϕ, we say that an edge uv is colored i if ϕ(u) = ϕ(v) = i.
Parameterized complexity. A parameterized problem is a subset L ⊆ * ×N over a finite alphabet . L is fixed-parameter tractable (FPT) if the membership of an instance (x, k) in * ×N can be decided in time f (k)|x| O(1) , where f is a computable function of the parameter k only. A kernelization of a parameterized problem L is a polynomial-time algorithm that maps an instance (x, k) to an instance (x ′ , k ′ ), the kernel, such that (x, k) ∈ L if and only if (x ′ , k ′ ) ∈ L, k ′ ≤ g(k), and |x ′ | ≤ g(k) for some function g of k only. We call g(k) the size of the kernel.
It is well-known that a parameterized problem L is FPT if and only if it is decidable and admits a kernelization. Due to applications, low degree polynomial size kernels are of main interest. Unfortunately, many FPT problems do not have kernels of polynomial size unless the polynomial hierarchy collapses to the third level [3,4]. For further background and terminology on parameterized complexity we refer the reader to the monographs [3,4,5,8].

Bounding Number of Vertices in Kernel
In this section, we show that (c, k)-Load Coloring admits a kernel with less than 2ck vertices. A matching with 2ck vertices suggests that this bound is likely to be optimal.
For τ ∈ {<, ≤, =, >, ≥} and integer i ≥ 1, K 1,τ i denotes a star K 1,j with j τ i and j ≥ 1. For example, K 1,≤p is a star with q edges such that q ∈ [p]. A K 1,τ i -graph is a forest in which every component is a star K 1,τ i , and a K 1,τ icover of G is a K 1,τ i -subgraph F of G such that V (F ) = V (G). We call any K 1,τ i -graph a star graph and any K 1,τ i -cover a star cover.
We first prove the bound for star graphs with small maximum degree.
Proof. The idea is to find for each color some induced subgraph with at least k edges and at most 2k vertices. If such subgraphs exist, it is possible to color at most 2ck vertices of the graph to obtain k edges for each of the c colors. We prove the lemma by induction on c. The base case of c = 1 holds since a K 1,<2k -graph G with at least 2k vertices has at least k edges (observe that a K 1,<2k -graph has no isolated vertices).
Observe now that because all components of G are trees, for each one the number of vertices is one more than the number of edges. If there is a component C, with k ≤ |E(C)| < 2k, color V (C) with the same color. Then we have used |V (C)| ≤ 2k vertices. Thus, we may assume that every component has less than k edges and let C 1 , C 2 , . . . , C t be the components of G. Let b be the minimum nonnegative integer for which there exists I ⊆ [t] such that Σ i∈I |E(C i )| = k + b ≥ k. Since there is no isolated vertex in a star graph, m ≥ n/2 ≥ ck, and thus such a set I exists. Observe that for any i ∈ I, |E(C i )| > b, as otherwise Σ j∈I\{i} |E(C j )| = k + b − |E(C i )| ≥ k, a contradiction to the minimality of b.
Since every component has less than k edges, b ≤ k − 2.
For a star (V, E), the ratio |V | |E| decreases when |E| increases. Thus, we ≥ 0, and so Σ j∈I |V (C j )| ≤ 2k. We may color the components C i , i ∈ I, by the same color. Observe that H = G − V ( i∈I C i ) has at least 2(c − 1)k vertices and so H ∈ (c − 1, k)-LCP by the induction hypothesis. Thus, G ∈ (c, k)-LCP.
Since G ∈ (c, k)-LCP whenever G has a subgraph H ∈ (c, k)-LCP, we have that any graph with n ≥ 2ck and a K 1,<2k -cover is in (c, k)-LCP.
We introduce now a family (O i,k ) i,k∈N of obstacles.
is incident to at least one of the i vertices in V 1 . Thus, an edge can only be colored with one of |V 1 | = i colors. From this observation, we deduce the following set of reduction rules.
Reduction rule R i,k . If an instance G for (c, k)-LCP contains an obstacle (V 1 , V 2 ) from O i,k , delete all the vertices of V 1 ∪ V 2 and decrease c by i. Now we will prove that Rules R i,k are safe and can be applied in time polynomial in n (recall that c is fixed).

Lemma 2.
Let G be a graph and G ′ be the graph obtained from G after applying Proof. For a positive integer p, we call a coloring of an instance G of (c, k)-LCP a good coloring with p colors if for at least p colors j ∈ [c], there are at least k edges colored with color j.
If G ′ ∈ (c − i, k)-LCP, then G ∈ (c, k)-LCP, since a good coloring of the obstacle with i colors together with a good coloring of G ′ with c − i colors gives a good coloring of G with c colors. On the other hand, if G ∈ (c, k)-LCP, then it has a good coloring with c colors. In this coloring, there are at least Proof. Generate all i-size subsets V 1 of V (G). For each V 1 , construct the set V 2 that includes every vertex outside V 1 whose only neighbors are in V 1 . If |V 2 | ≥ ik, construct the following bipartite graph B: the partite sets of B are V ′ 1 and V 2 , where V ′ 1 contains i copies of every vertex v of V 1 with the same neighbors as v. Observe that B has a matching covering V ′ 1 if and only if R i,k can be applied to G for the obstacle (V 1 , V 2 ). It is not hard to turn the above into an algorithm of runtime O(n i+O (1) ).
We say that a graph is reduced for (c, k)-LCP if it is not possible to apply any rule R i,k , i < c to the graph.
Lemma 4. Let G be a reduced graph for (c, k)-LCP and let G ∈ (c, k)-LCP. Then G has a K 1,≤max{3,k} -cover.
Proof. Let G be such a reduced graph. We first show that G has a star cover. Since it is not possible to apply R 0,k , G has no isolated vertex. By choosing a spanning tree of each component of G, we obtain a forest F . If a tree in F is not a star, it has an edge not incident to a leaf. As long as F contains such an edge, delete it from F . Observe that F becomes a star cover of G. However, the number of leaves in each star of F is only bounded by ∆(G). We will show that among the possible star covers of G, there exists a K 1,≤max(3,k) -cover.
For each star cover F , we define the F -sequence (n F,∆(G) ,n F,∆(G)−1 ,. . ., n F,1 ), where n F,i is the number of stars with exactly i edges, i ∈ [∆(G)]. We say a star cover F 1 is smaller than a star cover F 2 if and only if the F 1 -sequence is smaller than the F 2 -sequence lexicographically, i.e. there exists some i ∈ [∆(G)] such that n F1,i < n F2,i and for every j > i, n F1,j = n F2,j . We select a star cover S which has the lexicographically minimum sequence, that is, for any star cover F = S of G, the S-sequence is smaller or equal to the F -sequence. Suppose that ∆(S) > max{3, k}. Let C i (L i , respectively) be the set of all the centers (leaves, respectively) of all stars of S isomorphic to K 1,i . We also define L ≥i = ∪ j≥i L j . We will now prove two claims.
Indeed, suppose there exists one and let x, y be such that xu ∈ E(S), yv ∈ E(S). If v ∈ L ≥2 , then by deleting edges xu, yv and adding edge uv, we do not create any isolated vertex but we decrease the size of the stars centered at x and y, and thus we get a smaller star cover than S, a contradiction. Otherwise, v is an endvertex of an independent edge, and by deleting edge xu and adding edge uv, we decrease the size of the star centered at x, and create a star K 1,2 centered at v, which still induces a star cover smaller than S, a contradiction.
Claim 2 Suppose S contains a star isomorphic to K 1,i and centered at vertex x, and a star isomorphic to K 1,j and centered at vertex y, such that i − j ≥ 2.
There is no path from x to y in which the odd edges are in E(S) and go from a center to a leaf, and the even edges are in E(G) \ E(S) and go from a leaf to a center.
Suppose there exists such a path. Then by deleting the odd edges of the path and adding the even ones, we do not create isolated vertices because x still has leaf-neighbors, y gets a neighbor, every transitional center keeps the same number of leaf-neighbors and the transitional leaves always go to a new center.
This operation only decreases the size of star centered at x by 1 and increases the size of star centered at y by 1, giving us a lexicographically smaller star cover, a contradiction. Now, let S ′ be the subgraph of S containing all stars K 1,∆(S) of S. While there is an edge uv ∈ E(G)\E(S) such that u is a leaf of S ′ and v ∈ C ∆(S)−1 \S ′ , we add the star centered at v to S ′ . This procedure terminates because C ∆(S)−1 is finite.
Let C ′ (L ′ , respectively) be the centers (leaves, respectively) in S ′ . Assume now there is an edge uv Since v ∈ C ∆(S) ⊆ C ′ and since the above procedure has terminated, v ∈ C j for some j such that ∆(S) − j ≥ 2. Now, by construction, there is a alternating path from a vertex in C ∆(S) to a vertex in C j of the type described in Claim 2, which is impossible.
So, there is no edge uv ∈ E(G) \ E(S) such that u ∈ L ′ and v ∈ C ′ . This means that for any u ∈ L ′ , N (u) ⊆ C ′ . Furthermore, for each u ∈ C ′ , we can define V u to be the leaves of the star centered at u, for which we have Now we can prove the following: Theorem 1. For every fixed c, if G is reduced for (c, k)-LCP and has at least 2ck vertices, then G ∈ (c, k)-LCP. Thus, (c, k)-Load Coloring admits a kernel with less than 2ck vertices.
Proof. Observe that for every c, G ∈ (c, 0)-LCP, and G ∈ (c, 1)-LCP if and only if G has a matching with at least c edges. Thus, we may assume that k ≥ 2. By Lemmas 2 and 3, we can map, in polynomial time, any instance (G, c) into an instance (G ′ , c ′ ) such that c ′ ≤ c and G ′ is reduced for (c ′ , k)-LCP. We therefore may assume that G is reduced for (c, k)-LCP. Suppose that G ∈ (c, k)-LCP and n ≥ 2ck. By Lemma 4, G has a K 1,≤max(3,k) -cover which is a K 1,<2k -cover, since we assumed k ≥ 2. But then, Lemma 1 implies that G ∈ (c, k)-LCP, a contradiction.

Bounding Number of Edges in Kernel
In the previous section, we proved that (c, k)-Load Coloring admits a kernel with less than 2ck vertices. We would like to bound the number of edges in a kernel for the problem. Proof. We will prove the lemma by induction on i. For the base case, observe that any graph with at least k = b(1, k, n) edges is in (1, k)-LCP for every k and n. We now assume the claim holds for any j smaller than i + 1 and want to prove it for i + 1. Consider a bipartite graph G = (A ∪ B, E) with n vertices such that G ∈ (2 i+1 , k)-LCP. Let A 1 = A, B 1 = B and A 2 move an arbitrary vertex from A 1 to A 2 . Since we only move vertices from A 1 to A 2 or from B 1 to B 2 , we always have A = A 1 ∪A 2 and B = B 1 ∪ B 2 . Eventually, the partition of A ∪ B falls into one of two cases: Otherwise, let v be the last vertex moved from A 1 to A 2 . Observe that Thus, by taking a suitable 2 i -coloring of X and a suitable 2 i -coloring of Y , we have that G ∈ (2 i+1 , k)-LCP, a contradiction.
So we have proved that the claim also holds when Lemma 6. Let f (c, k, n) = (2c−1)ck +2n(c−1). For every nonnegative integer i and every graph G with n vertices, if m ≥ f (2 i , k, n) then G ∈ (2 i , k)-LCP.
Proof. We will prove the lemma by induction on i. For the base case, observe that any graph with at least k = f (1, k, n) edges is in (1, k)-LCP for every k and n. We now assume the claim holds for any j smaller than i+1 and want to prove it for i + 1. Consider a graph G with n vertices such that G ∈ (2 i+1 , k)-LCP and |E(G)| ≥ f (2 i , k, n). We will show that there exists a set A ⊆ V (G) such that f (2 i , k, |A|) ≤ |E(A)| ≤ f (2 i , k, |A|) + |A| (and thus G[A] ∈ (2 i , k)-LCP). We may construct the set A as follows: initially A = ∅ and while |E(A)| < f (2 i , k, |A|), add an arbitrary vertex of V (G) \ A to A. Let u be the last added vertex; we have , k, n), as otherwise we are done by Lemma 5. Finally, , k, n). The claim holds when j = i + 1, which completes the proof.
Proof. By Theorem 1, we can get a kernel with less than 2ck vertices. Let c ′ be the minimum power of 2 such that c ≤ c ′ . Observe that c ′ < 2c and thus by Lemma 6 we get a kernel with |E(G)| ≤ f (c ′ , k, 2ck) < f (2c, k, 2ck) = 16c 2 k − 6ck.
We now consider an approximation algorithm for the c-Load Coloring Problem: Given a graph G and integer c, we wish to determine the maximum k, denoted k opt , for which G ∈ (c, k)-LCP. We define the approximation ratio r(c) =  Proof. We prove the claim by induction on c. For c = 1, we have P (1) = 1. Assume the lemma is true for all c ′ < c.
Let G be an instance for c-Load Coloring with n vertices and m edges. We may assume that G has no isolated vertices. Clearly, k opt ≤ m c . Consider k = ⌊ m K(c) ⌋. If k = 0, then m < K(c) and we can find k opt in O(1) time. Now let k > 0. If n ≤ 2ck, then by the proof of Theorem 2, since m ≥ K(c)k, G ∈ (c, k)-LCP. So we return k, and . If n ≥ 2ck and G is reduced for (c, k)-LCP, then by Theorem 1, G ∈ (c, k)-LCP and we return k as above. If n ≥ 2ck and G is not reduced for (c, k)-LCP, we can use Lemma 3 to reduce (G, c) to (G ′ , c ′ ) with c ′ < c. By induction we may find k ′ such that k ′ opt ≤ 2 c ′ −1 P (c ′ )k ′ , where k ′ opt is the optimal solution for c ′ -Load Coloring on G ′ . Now consider three cases.
• k ′ ≥ k. Then G ′ ∈ (c ′ , k)-LCP and so G ∈ (c, k)-LCP. This is also a Yes-Instance case which leads to the same conclusion.
, therefore G ′ can be derived from G using a reduction rule for (c, The algorithm gives k ′ as an approximation of k opt . Then In every case, the approximation ratio is at most 2 c−1 P (c).

Number of Edges in Kernel for c = 2
In this section, we look into the edge kernel problem for the special case when c = 2. By doing a refined analysis, we will give a kernel with less than 8k edges for (2, k)-LCP, which is a better bound than the general one. Henceforth, we assume that G is reduced for (2, k)-LCP, and just consider the case when |V (G)| < 4k, as we have proved that if |V (G)| ≥ 4k then G ∈ (2, k)-LCP.
Lemma 7. If G has at least 3k − 2 edges and every component in G has less than k edges then G ∈ (2, k)-LCP.
Proof. We consider colorings of the graph such that vertices in the same component are colored with the same color. Thus every edge in the graph is colored with 1 or 2. Denote the set of edges colored i with E i , i = 1, 2. Among all possible colorings, choose a coloring of the graph such that |E 1 | ≥ |E 2 | and Changing the color of one component from 1 to 2, we get a new coloring of the graph. For the new coloring, denote the set of edges colored i with E ′ i , i = 1, 2. Since each component has less than k edges, If G has at least two components, each with at least k edges, it is obviously a Yes-instance. Therefore by Lemma 7, we may assume there is exactly one component C with at least k edges in the graph. Denote the total number of edges in G − V (C) with m ′ . Observe that if m ′ ≥ k, trivially G ∈ (2, k)-LCP. So assume that m ′ < k. Lemma 8. If G is a reduced graph for (2, k)-LCP, m ′ < k and ∆ = ∆(G) ≥ 3k − 2m ′ , then G ∈ (2, k)-LCP.
Proof. Let u be one of the vertices with degree ∆ and N (u) its neighbors. Because the graph is reduced by Reduction Rule R 1,k , u has at least 2k − 2m ′ neighbors which are not leaves. Arbitrarily select k − m ′ vertices among them and for each one, select any neighbor but u. Color the selected vertices and G − V (C) by 1. By construction, there are at least k edges colored 1 and there are at most 2k − 2m ′ colored vertices in N (u). So there are at least k uncolored vertices in N (u). We color them and u with 2. So G ∈ (2, k)-LCP.
The next lemma deals with the case ∆ = ∆(G) < 3k.
Proof. Because of Lemma 7, we may assume there exists a connected component C with at least k edges. In this component, choose a minimal set A ⊆ V (C) such that |A| ≤ k + 1 and |E(A)| = k + d ≥ k. We may find such a set A in the following way. Select arbitrarily a vertex in C and put it into A, then keep adding to this set some neighbor of some vertex in A until |E(A)| = k + d ≥ k. Since each time we select a neighbor of A we strictly increase |E(A)|, |A| ≤ k+1. If there is any vertex u ∈ A with |N A (u)| ≤ d, then A ′ = A \ {u} is a smaller vertex set such that |E(A ′ )| ≥ k. Thus, we may remove such vertices until |E(A)| = k + d and for each vertex u ∈ A, We may assume |E(B)| < k, as otherwise G ∈ (2, k)-LCP.
We now study the case max{|N B1 (u)| : u ∈ A} ≤ k. While there exists u ∈ B 1 such that |E(A, B 2 ∪ {u})| < 2k, move u from B 1 to B 2 . Then, (if and) max{|E(V 1 )|, |E(V 2 )|}. To see that Judicious Bipartition and 2-Load Coloring are different problems, following [1] consider 2nK 2 , the union of 2n disjoint edges, and observe that while the solution of 2-Load Coloring is n, that of Judicious Bipartition is zero.
To the best of our knowlege, we obtained the first linear-edge kernel for a nontrivial problem on general graphs. As we could see, such kernels can be used to obtain approximation algorithms. It would be interesting to obtain such kernels for other nontrivial problems.