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May 10, 2017 - Red/Blue-split graph and König-Egerváry graphs, SODA 2006]. .... complementary problem of Max R/B-split Subgraph, and in Theorem 3.6 we.
On Approximability of Optimization Problems Related to Red/Blue-split Graphs Sounaka Mishraa , Shijin Rajakrishnanb , Saket Saurabhc,∗ a Indian

Institute of Technology Madras, Chennai 600 036, India. Institute of Technology Madras, Chennai 600 036, India. c Institute of Mathematical Sciences, Chennai 600 113, India.

b Indian

Abstract An edge-bicolored graph G = (V, R ∪ B) is called Red/Blue-split graph if there exists a partition IB and IR of V such that IB and IR are independent sets in (V, B) and (V, R), respectively. Red/Bluesplit graphs generalize several well studied graph classes including split graphs, bipartite graphs and K¨ onig-Egerv´ ary graphs. In this paper we consider the algorithmic complexity of various optimization problems like minimum edge (or vertex) deletion and maximum edge (or vertex) induced subgraph related to Red/Blue split graphs. All these problems are N P -hard and thus we look at them from algorithmic paradigms that are meant for coping with N P -hardness. We obtain various hardness as well as algorithmic results for these problems in the realm of approximation algorithms and parameterized complexity. The main tool we use to obtain all our results is polynomial time transformations between appropriate problems. On the way, we also resolve some problems related to inapproximability about certain optimization problems mentioned by Korach, Nguyen and Peis [Subgraph Characterization of Red/Blue-split graph and K¨ onig-Egerv´ ary graphs, SODA 2006]. Keywords: Parametrized Complexity, Approximation Algorithms, K¨onig-Egerv´ary Graphs, Red/Blue-Split Graphs

1. Introduction Maximum matching and minimum vertex cover are two important graph optimization problems which have been studied extensively [20]. It is known that the maximum matching problem can be solved in polynomial time [8], whereas the minimum vertex cover problem is NP-complete [15]. In 1931, K¨onig and Egerv´ ary independently proved that, for bipartite graphs the size of the minimum vertex cover equals the size of the maximum matching. Interestingly this min-max equality relation holds for a larger class of graphs (which includes the bipartite graphs) known as K¨onig-Egerv´ary graphs (KE-graphs in short). This min-max relation leads to a polynomial-time algorithm for minimum vertex cover in KE-graphs [25, 5]. For this property, KE-graphs have been studied extensively [2, 5, 25, 19, 17, 22]. Red/Blue-split(R/Bsplit) graphs are a generalization of both KE-graphs and split graphs[13]. Interestingly, these three classes of graphs (R/B-split graphs, KE-graphs and split graphs) have excluded-subgraph characterizations. In this paper we consider algorithmic questions related to R/B-split graphs in the realm of approximation algorithms, exact exponential time algorithms and parameterized complexity. R/B-split graphs are simple graphs whose edges are colored by red, blue or both such that the vertex set can be partitioned into a red independent set and a blue independent set. A graph G = (V, E) with E = (R ∪ B) (the red edge set R and blue edge set B may not be disjoint) is called R/B-split if there exists a partition (IR , IB ) of V such that IR is an independent set in (V, R) and IB is an independent set in (V, B). If an edge e ∈ R ∩ B, then the edge e has color both red and blue. In this paper we shall represent a bicolored graph G as G = (V, R ∪ B ∪ RB), where R is the set of edges to which only red color is assigned, B is the set of edges to which only blue color is assigned and RB is the set of edges to which both red and blue colors are assigned. In this context, (V, R ∪ RB) and (V, B ∪ RB) are the red and blue subgraphs of G, respectively. ∗ Corresponding

author Email addresses: [email protected] (Sounaka Mishra), [email protected] (Shijin Rajakrishnan), [email protected] (Saket Saurabh) Preprint submitted to Elsevier

May 10, 2017

In [17], the authors mentioned some open problems about the approximability of several optimization problems related to R/B-split subgraph. The formal definitions of the decision versions of these optimization problems and the ones we consider in this paper are as follows. 1. R/B-split Subgraph: Given a graph G = (V, R ∪ B ∪ RB) and a positive integer k, does there exist a subset S ⊆ V with |S| ≥ k such that G[S] is an R/B-split graph. 2. R/B-split Deletion: Given a graph G = (V, R ∪ B ∪ RB) and a positive integer k, does there exist a subset S ⊆ V with |S| ≤ k such that G[V \ S] is an R/B-split graph. 3. R/B-split Edge Subgraph: Given a graph G = (V, R ∪ B ∪ RB) and a positive integer k, does there exist a subset S ⊆ (R ∪ B ∪ RB) with |S| ≥ k such that G0 = (V, S) is an R/B-split graph. 4. R/B-split Edge Deletion: Given a graph G = (V, R ∪ B ∪ RB) and a positive integer k, does there exist a subset S ⊆ (R ∪ B ∪ RB) with |S| ≤ k such that G0 = (V, (R ∪ B ∪ RB) \ S) is an R/B-split graph. 5. R/B-split (Blue) Edge Deletion: Given a graph G = (V, R ∪ B ∪ RB) and a positive integer k, does there exist a subset S ⊆ (B ∪ RB) with |S| ≤ k such that G0 = (V, (R ∪ (B ∪ RB \ S)) is an R/B-split graph. 6. R/B-split (Red) Edge Subgraph: Given a graph G = (V, R ∪ B ∪ RB) and a positive integer k, does there exist a subset S ⊆ B ∪ RB with |S ∪ R| ≥ k such that G0 = (V, R ∪ S) is an R/B-split graph. We define the optimization version of these decision problems in the natural way. For example optimization problems corresponding to R/B-split Subgraph will be called Max R/B-split Subgraph and the minimization problem corresponding to R/B-split Deletion will be called Min R/B-split Deletion. In Section 3, we consider the complexity of R/B-split Deletion and R/B-split Subgraph. In [21], the authors made a parameter preserving reduction from R/B-split Deletion to Above Guarantee Vertex Cover. In Above Guarantee Vertex Cover [7], the goal is to find a vertex cover S in a given graph G = (V, E) such that |S|−|M | is minimized, where M is a maximum matching in G. Based on this reduction they proved that it can be approximated within a factor of O(log n log log n). In [17], Korach et al. raised the question of approximability of Max R/B-split Subgraph. Max R/Bsplit Subgraph is known to be NP-complete, even for comparability graphs [9] and hard to approximate 32 , unless P=NP. Here, we improve this lower bound result in proving that, for any within a factor of 31  > 0, it is not approximable within a factor of O(n1− ), unless RP=co-RP. Next, we consider Min R/Bsplit Deletion, the complementary problem of Max R/B-split Subgraph, and in Theorem 3.6 we prove that it cannot be approximated within a factor of 1.3606, unless P=NP, but can be approximated √ within a factor of O(log n log log n) (O( log n) randomized). All our results are based on appropriate reductions and using known algorithms and hardness results they also imply corresponding results in the area of exact algorithms and parameterized complexity. We refer to Section 2 for basic definitions about fixed parameter tractability (FPT) and W -hardness. In Section 4, we consider the complexity of all the edge variants of problems related to R/B-split graphs. First we prove that Min R/B-split Edge Deletion is equivalent to Min 2SAT Deletion under an L-reduction, leading to the result that the former is approximable within a factor of √ O(log n log log n) (O( log n) randomized), and that assuming the Unique Games Conjecture(UGC), it is hard to approximate within a constant factor. These results give a complete answer to the question raised by Korach et al. in [17] about Min R/B-split Edge Deletion. The approximability of Min R/B-split (Blue) Edge Deletion is addressed in [17] and the authors remarked that this problem is as hard as the Min Vertex Cover problem. We prove that it is as hard as Above Guarantee Vertex Cover and thus cannot be approximated within a constant factor, √ unless UGC is false, and also prove that it can be approximated within a factor of O(log n log log n) (O( log n) randomized). Next, we show that Max R/B-split Edge Subgraph can be approximated within a factor of 1.07411. Finally, we consider Max R/B-split Edge (Red) Subgraph and prove that it is hard to approximate within 1 a factor of n 2 − , for any  > 0. Similar to the vertex variant of the problems, all our results are based on appropriate reductions and using known algorithms and hardness results they also imply corresponding results in the area of exact algorithms and parameterized complexity. Our results are summarized in Figure 1.

2

Problem Max R/B-split Subgraph Min R/B-split Deletion Max R/B-split Edge Subgraph Min R/B-split Edge Deletion Min R/B-split (Blue) Edge Deletion Max R/B-split (Red) Edge Subgraph

FPT W [1]-hard Corollary 3.4 O ∗ (2.3146k )1 OLD O ∗ (1.2835k ) Corollary 4.19 O ∗ (2.3146k ) Corollary 4.3 O ∗ (2.3146k ) Corollary 4.14

Approximability ? O(log n log log n) Proposition 2.3 1.07411 Corollary 4.17 O(log n log log n) Corollary 4.2 O(log n log log n) Corollary 4.13

W [1] hard Corollary 4.22

?

inapprox. O(n1− ) Thm. 3.1 1.3606 Thm. 3.6 ? not in APX Corr. 4.9 2.88854 Thm. 4.11 1

O(n 2 − ) Thm. 4.21

Exact Algo O(20.2576n ) Lemma 3.3 O ( 2.2576n ) OLD O(1.7315n ) Corollary 4.20 O(1.7315n ) Corollary 4.5 ? ?

Figure 1: Table containing results obtained in the paper. Results marked in bold are obtained in this paper and those with question mark are still open.

2. Preliminaries In this section, we define our notations, explain the reductions used in the paper, and list some known results which we rely upon. Reductions. A parameterized problem is a subset of Σ ∗ × Z≥0 , where Σ is a finite alphabet and Z≥0 is the set of nonnegative integers. An instance of a parameterized problem is a pair (I, k), where k is the parameter. In the framework of parameterized complexity, the running time of an algorithm is viewed as a function of two quantities: the size of the problem instance and the parameter. A parameterized problem is said to be fixed-parameter tractable (FPT) if there exists an algorithm that takes as input (I, k) and decides whether it is a YES or NO instance in time O(f (k) · |I|O(1) ), where f is a function depending only on k. The class FPT consists of all fixed parameter tractable problems. A parameterized problem π1 is fixed-parameter reducible to a parameterized problem π2 if there exist functions f, g : Z≥0 → Z≥0 , Π : Σ ∗ × Z≥0 → Σ ∗ and a polynomial p(·) such that for any instance (I, k) of π1 , (Φ(I, k), g(k)) is an instance of π2 computable in time f (k) · p(|I|) and (I, k) ∈ π1 if and only if (Φ(I, k), g(k)) ∈ π2 . Two parameterized problems are fixed-parameter equivalent if they are fixed-parameter reducible to each other. The basic complexity class for fixed-parameter intractability is W [1] as there is strong evidence that W [1]-hard problems are not fixed-parameter tractable. For more on parameterized complexity, refer to [7, 11, 23]. Approximation preserving reductions are used to establish the approximability or hardness of approximation of an optimization problem. One of the most commonly used reduction is the L-reduction [24]. Given two optimization problems π1 and π2 with objective functions c1 and c2 , we say that π1 ≤L π2 if there exists a pair of polynomial time computable functions f, g : Σ∗ → Σ∗ , such that f (·) maps instances, x, of the problem π1 to instances, f (x), of the problem π2 , g(·) maps a feasible solution y of an instance f (x) of π2 back to a feasible solution g(y) for an instance x of π1 , and there exist positive constants α and β such that OP Tπ2 (f (x)) ≤ α · OP Tπ1 (x), and |OP Tπ1 (x) − c1 (g(y))| ≤ β · |OP Tπ2 (f (x)) − c2 (y)|. It is easy to observe that if π1 ≤L π2 and π2 is c-approximable then π1 is approximable within a factor of cαβ. Known Results. We next mention a few known results about some problems which we will use in this paper. Proposition 2.1. [14] For any  > 0, Max Independent Set cannot be approximated within a factor of n1− , unless RP=co-RP. Proposition 2.2. [7] Max Independent Set is W [1]-hard Proposition 2.3. [22] Above Guarantee Vertex Cover can be approximated within a factor of O(log n log log n), where n is the number of √ vertices in the given graph. If we allow randomness, then it can be approximated within a factor of O( log n). Proposition 2.4. [21] R/B-split Deletion ≤L Above Guarantee Vertex Cover with α = 1 and β = 1. Proposition 2.5. [3, 22] It is NP-hard to approximate the Min-2Sat-Deletion problem to within any constant approximation factor less than 2.88854. If UGC is true, then Min-2Sat-Deletion does not admit a constant factor approximation algorithm. 3

Proposition 2.6. [1, 16] There exists a deterministic O(log n log log n) approximation and a randomized √ O( log n) approximation for Min-2SAT-Deletion, for an instance with n variables. Given a graph G = (V, R ∪ B ∪ RB), with red edges and blue edges, it can be decided in polynomial time whether G is R/B-split [13]. 3. Vertex Related Problems about Red/Blue-split Graphs In this section we consider the vertex versions of the optimization problems related to R/B-split graphs, namely Max R/B-split Subgraph and Min R/B-split Deletion. We start by proving a strong inapproximability result for Max R/B-split Subgraph. Theorem 3.1. For any  > 0, Max R/B-split Subgraph cannot be approximated within a factor of O(n1− ), unless RP=co-RP. Proof. We show that there is a polynomial time approximation preserving reduction from Max Independent Set to Max R/B-split Subgraph. Given an instance G = (V, E) of Max Independent Set, we construct an instance G0 = (V 0 , R ∪ B ∪ RB) of Max R/B-split Subgraph as follows: Let V = {v1 , . . . , vn } and let V 0 = V1 ∪ V2 where V1 = {u1 , . . . , un } and V2 = {w1 , . . . , wn }. The set of edges is defined as follows: 1. RB = {(ui , wj )|1 ≤ i, j ≤ n} ∪ {(ui , uj ), (wi , wj )|(vi , vj ) ∈ E}, 2. B = {(ui , uj )|(vi , vj ) ∈ / E} and 3. R = {(wi , wj )|(vi , vj ) ∈ / E}. The set of red edges induces a graph isomorphic to G in V1 and a complete graph in V2 . Similarly, the set of blue edges induces a graph isomorphic to G in V2 and a complete graph in V1 . The set of red (blue) edges across V1 and V2 induces a complete bipartite graph Kn,n . Claim 3.2. Graph G has an independent set of size k if and only if G0 has a subset S ⊆ V 0 of size at least 2k such that G0 [S] is an R/B-split graph. Proof. Let I be an independent set in G. Let IV1 = {ui : vi ∈ I} and IV2 = {wi : vi ∈ I}. For S = IV1 ∪ IV2 , G0 [S] is R/B-split with IB = IV2 and IR = IV1 and |S| = 2|I|. Conversely, let S be an induced R/B-split subgraph of G0 . Let IR be the red independent set and IB be the blue independent set in G0 [S]. Set IR can have at most one vertex from V2 , since the red edges induce a complete graph on V2 . Also if a vertex of V2 is in IR then it cannot have a vertex from V1 . A similar argument holds for IB . Hence we may assume that IR ∩ V2 = φ and IB ∩ V1 = φ. Observe that both IR and IB are independent sets in G and thus at least one of them is of size at least |S| 2 . Without . We take I as an independent set I of G. loss of generality assume that |IR | ≥ |S| R 2 Let So be an optimal solution for Max R/B-split Subgraph for the instance G0 , and let (IR , IB ) be a partition of So . We claim that |IR | = |IB |. Suppose |IR | > |IB |. Then S 0 = IR ∪ {wi | ui ∈ IR } induces an R/B-split subgraph in G0 with a larger size than So . From the proof of Claim 3.2, we have Io = {vi | ui ∈ IR } is a maximum independent set in G. Therefore, |Io | = 21 |So |. From the above discussion, it follows that G0 has an optimal solution So such that So = IR ∪ IB with ui ∈ IR if and only if wi ∈ IB . Since an optimal solution So of G0 has such a structural property, without loss of generality we assume that any maximal solution S of Max R/B-split Subgraph for o| the instance G0 has the same structure and therefore |S| = 2|I|, where I = {vi |ui ∈ S}. Thus |I|I|o | = |S |S| . By using Proposition 2.1 we have that Max R/B-split Subgraph cannot be approximated within a factor of O(n1− ), unless RP=co-RP. In [21], the authors establish a fixed-parameter reduction from R/B-split Deletion to Above Guarantee Vertex Cover. Given a graph G = (V, E) with n vertices and a positive integer k, the parametrized version of Above Guarantee Vertex Cover can be solved in O(2.3146k nO(1) ) time [18]. From this reduction and this algorithm it follows that R/B-split Deletion is fixed parameter tractable and can be solved in O(2.3146k nO(1) ) time. It is also known that Min R/B-split Deletion can be solved in O(20.2576n ) time, where n is the number of vertices in an instance, and this provides an exact algorithm for Max R/B-split Subgraph. 4

Lemma 3.3. Max R/B-split Subgraph can be solved in time O(20.2576n ), on input graphs with n vertices. Since the reduction in the proof of Theorem 3.1 is a parameter preserving reduction, by using the result in Proposition 2.2, we have Corollary 3.4. R/B-split Subgraph is W [1]-hard. For other results about Min R/B-split Deletion we need the following results about Min Vertex Cover. Proposition 3.5. [6] Unless P=NP, Min Vertex Cover is not approximable within a factor smaller than 1.3606. Theorem 3.6. Unless P=NP, Min R/B-split Deletion is not approximable within a factor of 1.3606. Proof. The result follows from Proposition 3.5, and the observation that the ratio preserving reduction in Theorem 3.1 can be construed as one from Min Vertex Cover to Minimum R/B-split Deletion. 4. Edge Related Problems about Red/Blue-split Graphs In this section we study approximability of edge variants of optimization problems related to Red/Bluesplit Graphs. Throughout this section, given a graph G = (V, R ∪ B ∪ RB) and F ⊆ E = R ∪ B ∪ RB, we say that F is a rb-deletion set if G0 = (V, E \ F ) is an R/B-split graph. Theorem 4.1. [13] Min R/B-split Edge Deletion ≤L Min 2SAT Deletion with α = 1 and β = 1. Proof. From an instance G = (V, R ∪ B ∪ RB) of Min R/B-split Edge Deletion we construct an instance φ of Min 2SAT Deletion as follows. Let ^ ^ ^ [(xu ∨ xv ) ∧ (xu ∨ xv )] φ= (xu ∨ xv ) (xu ∨ xv ) (u,v)∈R

(u,v)∈RB

(u,v)∈B

where xv is a binary variable associated with the vertex v ∈ V . We will show that G has rb-deletion set of size at most k if and only if there is a set of at most k clauses whose deletion from φ results in a satisfiable formula. Let F ⊆ E be a minimal rb-deletion set. Then, there exists a partition (IR , IB ) of G0 = (V, E \ F ) such that IR and IB are the independent sets in the red and blue edge subgraphs. From F and the given partition, we construct a clause deletion set [ F 0 = {(xu ∨ xv )|(u, v) ∈ R ∩ F } {(xu ∨ xv )|(u, v) ∈ B ∩ F } [ {(xu ∨ xv )|(u, v) ∈ RB ∩ F and u, v ∈ IR } [ {(xu ∨ xv )|(u, v) ∈ RB ∩ F and u, v ∈ IB }. By construction we have that |F 0 | ≤ |F |. We need to show that deleting clauses corresponding to F 0 from φ results in a satisfiable formula. From the partition (IR , IB ) of G0 = (V, E \ F ) we construct a truth assignment τ for φ as xv = 0 if v ∈ IR , xv = 1 if v ∈ IB . It is easy to observe that deletion of the clauses in F 0 makes φ satisfiable by the truth assignment τ . Conversely, let F 0 be a minimal clause deletion set for φ and let τ be a satisfying truth assignment for φ − F 0 . Since F 0 is a minimal set, F 0 contains at most one clause from (xu ∨ xv ) ∧ (xu ∨ xv ), for each edge (u, v) ∈ RB. From F 0 we construct the edge set F = {(u, v) | (xu ∨ xv ) ∈ F 0 or (xu ∨ xv ) ∈ F 0 }. Let IR = {v | τ (xv ) = 0} and IB = {v | τ (xv ) = 1}. It can be observed that (V, E \ F ) is an R/B-split graph with IR and IB as the red and blue independent sets respectively. Therefore, F is an edge deletion set for G and |F | = |F 0 |. From these observations it follows that this reduction is an L-reduction with α = 1 and β = 1. We can now extend the approximability of Min-2SAT-Deletion to Min R/B-split Edge Deletion. From Theorem 4.1 and Proposition 2.6, we have 5

x1

P1

x1

P1

x2

x2

P2

P2

x3

P3

x3

P3

Figure 2: A sketch of the graph G for the 2CNF formula φ = (x1 ∨ x2 ) ∧ (x1 ∨ x3 ) ∧ (x2 ∨ x3 )

Corollary 4.2. The Min R/B-split Edge Deletion problem is approximable within a factor of √ O(log n log log n) by a deterministic algorithm and within a factor of O( log n) by a randomized algorithm. The reduction in the proof of Theorem 4.1 is a parameter preserving reduction, i.e. in this reduction we can see that deletion of k edges from G makes it an R/B-split graph if and only if deletion of k clauses from φ makes it satisfiable. Since the parameterized version of 2-Sat Deletion, namely Almost-2SAT has O(2.3146k nO(1) ) algorithm [18], we have the following corollary. Corollary 4.3. There is an algorithm for R/B-split Edge Deletion running in time O(2.3146k nO(1) ). We can get a similar result for the complementary problem. Proposition 4.4. [26] Min-2SAT-Deletion can be solved in O(1.7315n ), where n is the number of variables in an instance. From the reduction in Theorem 4.1 and Proposition 4.4, we have the following corollary. Corollary 4.5. Min R/B-split Edge Deletion can be solved in O(1.7315n ), where n is the number of vertices in the input graph. We will show next that Min R/B-split Edge Deletion is equivalent to Min 2SAT Deletion. This will allow us to export the known lower bounds on Min 2SAT Deletion to Min R/B-split Edge Deletion. Theorem 4.6. Min 2SAT Deletion ≤L Min R/B-split Edge Deletion with α = 1 and β = 1. Proof. We exhibit a polynomial time cost preserving reduction from Min 2SAT Deletion to Min R/B-split Edge Deletion. Given an instance φ = c1 ∧ c2 ∧ · · · ∧ cm over n variables x1 , x2 , . . . , xn , of Min 2SAT Deletion, we construct an instance G = (V, R ∪ B ∪ RB) of Min R/B-split Edge Deletion in polynomial time as follows. For each variable xi , we construct a complete bipartite graph i K(m+1),(m+1) (variable gadget for xi ) with vertex set Vi = Pi ∪ P i , where Pi = {xi , xi1 , . . . , xim } and i P i = {xi , xi1 , . . . , xim }. All the edges in K(m+1),(m+1) are colored with both red and blue. For each clause ck = (li ∨ lj ), we introduce a red edge (li , lj ) (clause gadget for ck ). This completes the construction of G. For an illustration of this construction we refer to Figure 2. Next, we prove the following claim which will be useful in proving our final statement. Claim 4.7. φ is satisfiable if and only if G is R/B-split. Proof. Suppose φ is satisfiable. Consider a satisfying truth assignment τ for φ. With respect to this truth assignment we define IR = [∪i:τ (xi )=1 Pi ] ∪ [∪i:τ (xi )=0 P i ] and IB = V \ IR . From the construction of this partition of V , it is easy to observe that all the red and i blue edges that are present in all the variable gadgets K(m+1),(m+1) are cross edges with respect to this vertex partition. In the construction of G, all the blue edges are from the variable gadgets, therefore IB is an independent set in the blue graph. In order to show that G is R/B-split, it is enough to show that IR is an independent set in the red graph. Suppose there is a red edge with both end vertices in IR . Then this edge cannot be an edge from any variable gadget and must be representing a clause of φ. 6

Then it is of the form (li , lk ) for some i and k, which implies that (¯li ∨ ¯lk ) is a clause in φ which is not satisfied as both the literals li and lk receive value 1 under τ . For the converse part, assume that G is R/B-split and let V = IR ] IB , where IR and IB are red and blue independent sets in G. Since G is R/B-split, all the blue edges (appearing in variable gadgets) must be cross edges. Otherwise, if a partition contains a blue edge then it must contain a red edge (with the same end vertices) hence that partition can be neither a red independent set nor a blue independent set in G. Again, it implies that all the edges (both red and blue) appearing in variable gadgets must be cross edges in the vertex partition. From this vertex partition, we define a truth assignment τ for φ as τ (xi ) = 1 if and only if xi ∈ IR . Since all the edges corresponding to a clause is a cross edge in this vertex partition, at least one of the end vertex is in IB . Therefore, atmost one of the literals in each clause receives value 0 under this truth assignment, implying that φ is satisfied by this truth assignment. This completes the proof of Claim 4.7. Let E 0 ⊆ R ∪ B ∪ RB be a minimal rb-deletion set. Without loss of generality, we can assume that E 0 does not contain any edge from any of the variable gadgets. Otherwise for some variable xi , E 0 contains at least one edge, say (p, q), where p ∈ Pi and q ∈ P i . Since both p and q are in IR or IB , the number of edges in G[(Pi ∪ P i ) ∩ IR ] or G[(Pi ∪ P i ) ∩ IB ] is at least m. In a variable gadget there are two edges (one red and one blue in color) between any pair of vertices u, v, with u ∈ Pi and v ∈ P i . Therefore, if G[(Pi ∪ P i ) ∩ IR ] (or G[(Pi ∪ P i ) ∩ IB ]) contains at least m edges then it contains at least m blue and at least m red edges. Therefore it will be necessary to delete at least m edges to get a R/B-split graph. At the same time, φ has m clauses and all the m edges corresponding to the clauses in φ form a rb-deletion set of G. Therefore, an optimal rb-deletion set consists of only the edges corresponding to the clauses gadgets. Hence, without loss of generality, we shall assume that any rb-deletion set E 0 ⊆ R ∪ B ∪ RB for the graph G is a subset of the set of edges corresponding to the clause gadgets. Thus given a minimal rb-deletion set E 0 , we form a set of clauses C 0 by taking the clauses corresponding to the edges in E 0 . Using Claim 4.7 one can easily observe that deleting clauses corresponding to C 0 from φ results in a satisfying formula. In the reverse direction, let C be any minimal set of clauses whose deletion makes φ satisfiable. Then by Claim 4.7, the deletion of the corresponding set of edges from G makes the graph R/B-split. Hence, the above reduction is a cost preserving reduction. This completes the proof. From Theorem 4.6 and Proposition 2.5, we have the following lower bound result for Min R/B-split Edge Deletion. Corollary 4.8. If UGC is true then Min R/B-split Edge Deletion does not admit a constant factor approximation algorithm. The same inapproximability result holds even when B = ∅ in the input graph G. However, it is NP-hard to approximate the Min R/B-split Edge Deletion problem to within any constant approximation factor less than 2.88854. Corollary 4.9. If UGC is true then Min R/B-split Edge Deletion does not admit a constant factor approximation algorithm, even when only red edges are deleted from the input graph G. Even, in this case it is NP-hard to approximate the problem to within any constant approximation factor less than 2.88854. Now, we consider the deletion of edges only from B ∪ RB. We show that Min R/B-split (Red) Edge Deletion is at least as hard as Above Guarantee Vertex Cover and use a hardness result on the latter to prove a hardness result on Min R/B-split (Blue) Edge Deletion. Proposition 4.10. [3, 22] Above Guarantee Vertex Cover for graphs with perfect matching cannot be approximated within a factor of 2.88854 unless P=NP. Assuming UGC, Above Guarantee Vertex Cover cannot be approximated within any constant factor for graphs with perfect matching. Theorem 4.11. Min R/B-split (Blue) Edge Deletion cannot be approximated within a factor of 2.88854, unless P=NP. Assuming UGC, it cannot be approximated within any constant factor. Proof. Let (G = (V, E), M ) be an instance of Above Guarantee Vertex Cover with perfect matching M . We construct a graph G0 = (V, R ∪ B) with B = M and R = E \ M . We prove that G has a vertex cover of size µ(G) + k if and only if removal of k blue edges from G0 makes G0 R/B-split. Let S be a vertex cover in G with |S| = µ(G) + k. Let S 0 = {(u, v) ∈ M | u, v ∈ S}. That is, S 0 is the set of edges from M with both end vertices in S. Therefore, |S 0 | = k. We show that G∗ = (V, (R ∪ B) \ S 0 ) is an R/B-split. Based on the vertex cover S, we define a partition (IR , IB ) of V 7

with IB = S and IR = V \ S. Since S is a vertex cover in G, IR is an independent set in G. Hence IR is a red independent set in (V, R \ S 0 ). The set IB is a blue independent set in (V, R \ S 0 ). If IB is not an independent set in this subgraph, then there must be a blue edge (u, v) with both u and v in IB = S. Since, (u, v) is a blue edge in G0 with both end vertices in IB this blue edge (u, v) is in M , which is a contradiction. Thus G∗ is an R/B-split graph. Conversely, let S 0 ⊆ B be a minimal blue edge deletion set for G0 . That is, G∗ = (V, (R ∪ B) \ S 0 ) is an R/B-split. Let (IR , IB ) be a partition of G0 \ S 0 into red independent set and blue independent set, respectively. We also assume that there is no blue edge of G0 with both its end vertices in IR . If there exists such a blue edge (u, v) then, without loss of generality, we can remove u from IR and add it to IB , and this will not violate the blue independence property of IB . This process is possible as the set of blue edges in G0 forms a maximum matching in G. It can be observed that each blue edge in S 0 must have its both end vertices in IB (since S 0 is minimal red edge deletion set). Hence, IB is a vertex cover for G as IR is a red independent set and it has no blue edge also. Therefore, |IB | = µ + |S 0 |. Theorem follows by using the lower bound result in Proposition 4.10 and the above cost preserving reduction. Next we give an approximation algorithm for Min R/B-split (Blue) Edge Deletion by giving an appropriate reduction to Min 2SAT Deletion. Theorem 4.12. Min R/B-split (Blue) Edge Deletion ≤L Min 2SAT Deletion. Proof. Given a graph G = (V, E = R ∪ B ∪ RB), we construct a 2CNF formula ^ ^ ^ φ= φ(eR ) φ(eB ) φ(eRB ) e∈R

e∈B

e∈RB

in polynomial time as follows. Vt i i i ) ∧ (xv ∨ ) ∧ (xu ∨ z ieu ) ∧ (xu ∨ zeu • For each edge e = (u, v) ∈ R, φ(eR ) = (xu ∨ xv ) i=1 [(zeu ∨ zev i i i i z ev ) ∧ (xv ∨ zev )], where zeu , zev are new variables and t = |B| + |RB|. • For each edge e = (u, v) ∈ B, φ(eB ) = (xu ∨ xv ). • For each edge e = (u, v) ∈ RB, φ(eRB ) = (xu ∨ xv ) ∧ (xu ∨ xv ). i i |e = (u, v) ∈ R, 1 ≤ i ≤ t} , zev It can be observed that the set of variables in φ is X = {xv |v ∈ V } ∪ {zeu and |X| = n + 2t|R| = O(n4 ). Using an argument analogous to the proof of Theorem 4.1, we can prove that G has an edge deletion set (subset of B ∪ RB) of size k if and only if deletion of k clauses (not from φ(eR )) makes φ satisfiable. It can be observed that it is an L-reduction with α = 1 and β = 1.

From Proposition 2.6 and Theorem 4.12 we have the following result. Corollary 4.13. Min R/B-split (Blue) Edge Deletion is approximable within a factor of O(log n log log n) √ by a deterministic algorithm and within a factor of O( log n) by a randomized algorithm. Since the parametrized version of Min 2-Sat Deletion – Amost-2-Sat is FPT and can be solved in O(2.3146k nO(1) ) time [18] and the reduction in Theorem 4.12 is parameter preserving, we have the following Corollary 4.14. There is an algorithm for R/B-split (Red) Edge Deletion running in time O(2.3146k nO(1) ). We now consider the approximability of the complementary problem of Min R/B-split Edge Deletion, namely, Max R/B-split Edge Subgraph. The reduction mentioned in the proof of Theorem 4.1 is also a ratio preserving reduction from Max R/B-split Edge Subgraph to Max 2Sat. Therefore, we have the following result. Theorem 4.15. Max R/B-split Edge Subgraph ≤L Max 2Sat with α = 1 and β = 1. Based on a semidefinite programming formulation of Max 2Sat, Goemans and Williamson [10] proved that it can be approximated within a factor of 1/0.931 (≈ 1.07411). 8

x1

x2

xm ...

y1

H

y2

ym

G Figure 3: A sketch of the construction of G from the graph H.

Proposition 4.16. [10] Max 2Sat problem can be approximated within a factor of 1/0.931 (≈ 1.07411). From Theorem 4.15 and Proposition 4.16 we have the following corollary. Corollary 4.17. Max R/B-split Edge Subgraph can be approximated within a factor of 1.07411. Next we describe an algorithm for Atleast 2-SAT – a decision version of Max 2Sat problem. In this problem we are given as input a 2-SAT formula φ and a positive integer k and the objective is to check whether there is an assignment satisfying at least k clauses of φ. Lemma 4.18. Atleast 2-SAT can be solved in time O(1.2835k nO(1) ). Proof. Let φ be an input to Atleast 2-SAT. We first apply the kernelization algorithm described in [4, Theorem 2.16] and in polynomial time either obtain an assignment satisfying at least k claues or obtain an equivalent instance (φ0 , k 0 ) of Atleast 2-SAT with at most k 0 ≤ k variables and at most 2k 0 ≤ 2k clauses. Given this instance we can apply the algorithm of Gaspers and Sorkin [12] for Max 2-CSP that runs in time O(29m/50 ), where m is the number of constraints. Since m in our case is upper bounded by 2k, we have that the algorithm for Atleast 2-SAT runs in time O(218k/50 ) = O(1.2835k ). This completes the proof. By Lemma 4.18 and the reduction in Theorem 4.15, we have the following result. Corollary 4.19. R/B-split Edge Subgraph can be solved in time O(1.2835k nO(1) ). Also from Proposition 4.4 and the reduction in Theorem 4.15 we have Corollary 4.20. Max R/B-split Edge Subgraph can be solved in O(1.7315n ), where n is the number of vertices in the input graph. Next we consider approximability of Max R/B-split Edge (Red) Subgraph which is a variant of Max R/B-split Edge Subgraph. Given a graph G = (V, R ∪ B ∪ RB), in Max R/B-split Edge (Red) Subgraph we are asked to find an R/B-split subgraph in G of maximum number of edges containing all the red edges of G. We prove that it is as hard as approximating Max Independent Set, by establishing an approximation preserving reduction from Max Independent Set. Theorem 4.21. For any  > 0, Max R/B-split Edge (Red) Subgraph cannot be approximated 1 within a factor of n 2 − , unless RP=co-RP. Proof. Let H = (V, E) be an instance of Max Independent Set with |V | = n and |E| = m. We assume that the size of a largest independent set I o in H is at least n2 . From H, we construct a graph G = (V 0 , R ∪ B ∪ RB), an instance of Max R/B-split Edge (Red) Subgraph as follows. First we make a copy of H with all its edges of color red. Then we add m independent red edges (xi , yi ), 1 ≤ i ≤ m, consisting of a set of 2m new vertices. We denote the set of these new vertices as P = {xi , yi |1 ≤ i ≤ m}. Finally, we introduce 2nm blue edges (u, v) with u ∈ V and v ∈ P . In G, V 0 = V ∪ P , R = E ∪ {(xi , yi )|1 ≤ i ≤ m}, B = {(u, v)|u ∈ V and v ∈ P } and RB = ∅. Note that |V 0 | = O(n2 ) as |E| = O(n2 ). For a sketch of this construction we refer to Figure 3. Let (V 0 , S) be a maximal R/B-split subgraph of G having all the red edges. Let IR and IB the respective independent sets in (V 0 , S). For any 1 ≤ i ≤ m, both xi and yi cannot be in IR as S contains 9

all the red egdes of G. Here we assume that, for 1 ≤ i ≤ m, both xi and yi are not in IB . Suppose for some i, both xi and yi are in IB , then we construct a new R/B-split subgraph with vertex partition as IR ∪ {xi } and IB \ {yi }. It can be observed that the new R/B-split subgraph has |S| + n − |V \ IR |, which is larger than |S|. Therefore, without loss of generality, we assume that {xi |1 ≤ i ≤ m} ⊂ IR and {yi |1 ≤ i ≤ m} ⊂ IB . From this structural assumption, it follows that, if IS = IR ∩ V then f (|IS |) = m(|IS | + n + 2) = |S|. Let I be a maximal independent set in H. From I we construct a vertex partition with IR = I ∪ {xi |1 ≤ i ≤ m} and IB = (V \ I) ∪ {yi |1 ≤ i ≤ m}. This partition induces an R/B-split subgraph with f (|I|) = 2m + m|I| + nm = m(|I| + n + 2) edges, it contains 2m red edges and m|I| + nm blue edges. Therefore, it follows that I o is a maximum independent set in H if and only if IR = I o ∪{xi |1 ≤ i ≤ m} and IB = (V \ I o ) ∪ {yi |1 ≤ i ≤ m} induces a largest R/B-split subgraph with f (|I o |) = m(|I o | + n + 2) edges. For any R/B-split edge subgraph of G, we have  m(|I o | + n + 2) |I o | 1 + f (|I o |) = = f (|I|) m(|I| + n + 2) |I| 1 +

n+2 |I o | n+2 |I|



 |I o | 1 + ≥ |I| 1 +

n+2 n n+2 n/2



o

 |I o | 2 + = |I| 3 +

2 n 4 n

 ≥

2 |I o | . 7 |I|

o

|) Therefore, for any maximal independent set I in H, we have |I|I|| ≤ 27 ff(|I (|I|) . From this inequality, Proposition 2.1 and since this reduction constructs an instance G with O(n2 ) vertices, it follows that 1 Max R/B-split Edge (Red) Subgraph is hard to approximate within a factor of n 2 − , unless RP=coRP, for any  > 0.

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