Simultaneous Feedback Edge Set: A Parameterized Perspective

In this paper we consider Simultaneous Feedback Edge Set (Sim-FES) problem. In this problem, the input is an $n$-vertex graph $G$, an integer $k$ and a coloring function ${\sf col}: E(G) \rightarrow 2^{[\alpha]}$ and the objective is to check whether there is an edge subset $S$ of cardinality at most $k$ in $G$ such that for all $i \in [\alpha]$, $G_i - S$ is acyclic. Here, $G_i=(V(G), \{e\in E(G) \mid i \in {\sf col}(e)\})$ and $[\alpha]=\{1,\ldots,\alpha\}$. When $\alpha =1$, the problem is polynomial time solvable. We show that for $\alpha =3$ Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic graphs. The same reduction shows that the problem does not admit an algorithm of running time $O(2^{o(k)}n^{O(1)})$ unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time $O(2^{\omega k\alpha+\alpha \log k} n^{O(1)})$, where $\omega$ is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when $\alpha =2$. We also give a kernel for Sim-FES with $(k\alpha)^{O(\alpha)}$ vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph $G$, an integer $q$ and, a coloring function ${\sf col}: E(G) \rightarrow 2^{[\alpha]}$. The question is whether there is a edge subset $F$ of cardinality at least $q$ in $G$ such that for all $i\in [\alpha]$, $G[F_i]$ is acyclic. Here, $F_i=\{e \in F \mid i \in \textsf{col}(e)\}$. We give an FPT algorithm for running in time $O(2^{\omega q \alpha}n^{O(1)})$.


Introduction
Deleting at most k vertices or edges from a given graph G, so that the resulting graph belongs to a particular family of graphs (F), is an important research direction in the fields of graph algorithms and parameterized complexity.For a family of graphs F, given a graph G and an integer k, the F-deletion (Edge F-deletion) problem asks whether we can delete at most k vertices (edges) in G so that the resulting graph belongs to F. The F-deletion (Edge F-deletion) problems generalize many of the NP-hard problems like Vertex Cover, Feedback vertex set, Odd cycle transversal, Edge Bipartization, etc. Inspired by applications, Cai and Ye introduced variants of F-deletion (Edge Fdeletion) problems on edge colored graph [7].Edge colored graphs are studied in graph theory with respect to various problems like Monochromatic and Heterochromatic Subgraphs [15], Alternating paths [6,8,20], Homomorphism in edge-colored graphs [3], Graph Partitioning in 2-edge colored graphs [5] etc.One of the natural generalization to the classic F-deletion (Edge F-deletion) problems on edge colored graphs is the following.Given a graph G with a coloring function col : E(G) → 2 [α] , and an integer k, we want to delete a set S of at most k edges/vertices in G so that for each i ∈ [α], G i − S belongs to F. Here, G i is the graph with vertex set V (G) and edge set as {e ∈ E(G) | i ∈ col(e)}.These problems are also called simultaneous variant of F-deletion (Edge F-deletion).
Cai and Ye studied the Dually Connected Induced subgraph and Dual Separator on 2-edge colored graphs [7].Agrawal et al. [1] studied a simultaneous variant of Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set, in the realm of parameterized complexity.Here, the input is a graph G, an integer k, and a coloring function col : E(G) → 2 [α] and the objective is to check whether there is a set S of at most k vertices in G such that for all i ∈ [α], G i − S is acyclic.Here, G i = (V (G), {e ∈ E(G) | i ∈ col(e)}).In this paper we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set, in the realm of parameterized complexity.
In the Parameterized Complexity paradigm the main objective is to design an algorithm with running time f (µ) • n O (1) , where µ is the parameter associated with the input, n is the size of the input and f (•) is some computable function whose value depends only on µ.A problem which admits such an algorithm is said to be fixed parameter tractable parameterized by µ.Typically, for edge/vertex deletion problems one of the natural parameter that is associated with the input is the size of the solution we are looking for.Another objective in parameterized complexity is to design polynomial time pre-processing routines that reduces the size of the input as much as possible.The notion of such a pre-processing routine is captured by kernelization algorithms.The kernelization algorithm for a parameterized problem Q takes as input an instance (I, k) of Q, runs in polynomial time and returns an equivalent instance (I , k ) of Q.Moreover, the size of the instance (I , k ) returned by the kernelization algorithm is bounded by g(k), where g(•) is some computable function whose value depends only on k.If g(•) is polynomial in k, then the problem Q is said to admit a polynomial kernel.The instance returned by the kernelization is referred to as a kernel or a reduced instance.We refer the readers to the recent book of Cygan et al. [9] for a more detailed overview of parameterized complexity and kernelization.
Formally, the problem is stated below.

Simultaneous Feedback Edge Set (Sim-FES)
Parameter: k, α Input: An n-vertex graph G, an integer k and a coloring function col : E(G) → 2 [α]  Question: Is there a simultaneous feedback edge set of cardinality at most k in G Feedback Vertex Set (FVS) is one of the classic NP-complete [13] problems and has been extensively studied from all the algorithmic paradigms that are meant for coping with NP-hardness, such as approximation algorithms, parameterized complexity and moderately exponential time algorithms.The problem admits a factor 2-approximation algorithm [4], an exact algorithm with running time O(1.7217 n n O(1) ) [12], a deterministic parameterized algorithm running in O(3.619 k n O (1) ) time [16], a randomized algorithm running in O(3 k n O (1) ) time [10], and a kernel with O(k 2 ) vertices [24].Agrawal et al. [1] studied Simultaneous Feedback Vertex Set (Sim-FVS) and gave an FPT algorithm running in time 2 O(αk) n O (1)  and a kernel of size O(αk 3(α+1) ).Finally, unlike the FVS problem, Sim-FES is polynomial time solvable when α = 1, because it is equivalent to finding maximal spanning forest.
Our results and approach.In Section 3 we design an FPT algorithm for Sim-FES by reducing to α-Matroid Parity on direct sum of elongated co-graphic matroids of G i , i ∈ [α] (see Section 2 for definitions related to matroids).This algorithm runs in time O(2 ωkα+α log k n O (1) ).Unlike the vertex counterpart, we show that for α = 2 (2-edge colored graphs) Sim-FES is polynomial time solvable.This follows from the polynomial time algorithm for the Matroid parity problem.In Section 4 we show that for α = 3, Sim-FES is NP-hard.Towards this, we give a reduction from the Vertex Cover in cubic graphs which is known to be NP-hard [22].Furthermore, the same reduction shows that the problem cannot be solved in 2 o(k) n O (1) time unless Exponential Time Hypothesis (ETH) fails [14].We complement our FPT algorithms by showing that Sim-FES is W [1]-hard when parameterized by the solution size k (Section 5).When α = O(|V (G)|), we give a parameter preserving reduction from the Hitting Set problem, a well known W [2]-hard problem parameterized by the solution size [9].However, Sim-FES remains W [1]-hard even when α = O(log(|V (G)|)).We show this by giving a parameter preserving reduction from Partitioned Hitting Set problem, a variant of the Hitting set problem, defined in [1].In [1], Partitioned Hitting Set was shown to be W [1]-hard parameterized by the solution size.In Section 6 we give a kernel with O((kα) O(α) ) vertices.Towards this we apply some of the standard preprocessing rules for obtaining kernel for Feedback Vertex Set and use the approach similar to the one developed for designing kernelization algorithm for Sim-FVS [1].In Section 7 we give an FPT algorithm for the problem, when parameterized by the dual parameter.Formally, this problem is defined as follows.

Maximum Simultaneous Acyclic Subgraph (Max-Sim-Subgraph)
Parameter: q Input: An n-vertex graph G, a positive integer q and a function col is acyclic?
For solving Max-Sim-Subgraph we reduce it to an equivalent instance of the α-Matroid Parity problem.As an immediate corollary we get an exact algorithm for Sim-FES running in time O(2 ωnα 2 n O(1) ).

Preliminaries
We denote the set of natural numbers by N. For n ∈ N, by [n] we denote the set {1, . . ., n}.
For a set X, by 2 X we denote the set of all subsets of X.We use the term ground set/ universe I S A A C 2 0 1 6

5:4
Simultaneous Feedback Edge Set: A Parameterized Perspective to distinguish a set from its subsets.We will use ω to denote the exponent in the running time of matrix multiplication, the current best known bound for which is ω < 2.373 [25].
Graphs.We use the term graph to denote undirected graph.For a graph G, by V (G) and E(G) we denote its vertex set and edge set, respectively.We will be considering finite graphs possibly having loops and multi-edges.In the following, let G be a graph and let H be a subgraph of G.By d H (v), we denote the degree of the vertex v in H, i.e, the number of edges in H which are incident with v.A self-loop at a vertex v contributes 2 to the degree of v.For any non-empty subset by G − F we denote the graph obtained by deleting the edges in F .We use the convention that a double edge and a self-loop is a cycle.An α-edge colored graph is a graph G with a color function col : E(G) → 2 [α] .By G i we will denote the color i For an α-edge colored graph G, the total degree of a vertex v is α i=1 d Gi (v).We refer the reader to [11] for details on standard graph theoretic notations and terminologies.

Matroids and Representable Matroids.
A pair M = (E, I), where E is a ground set and I is a family of subsets (called independent sets) of E, is a matroid if it satisfies the following conditions: (I1) We refer the reader to [23] for more details.
Let A be a matrix over an arbitrary field F and let E be the set of columns of A. For A, we define matroid M = (E, I) as follows.A set X ⊆ E is independent (that is X ∈ I) if the corresponding columns are linearly independent over F. The matroids that can be defined by such a construction are called linear matroids, and if a matroid can be defined by a matrix A over a field F, then we say that the matroid is representable over F. A matroid M = (E, I) is called representable or linear if it is representable over some field F.

Direct Sum of Matroids
over F can be found in polynomial time [21,23].

Uniform Matroid.
A pair M = (E, I) over an n-element ground set E, is called a uniform matroid if the family of independent sets is given by where k is some constant.This matroid is also denoted as U n,k .Proposition 2.1 ([9, 23]).Uniform matroid U n,k is representable over any field of size strictly more than n and such a representation can be found in time polynomial in n.
Graphic and Cographic Matroid.Given a graph G, the graphic matroid M = (E, I) is defined by taking the edge set E(G) as universe and Let G be a graph and η be the number of components in G.The co-graphic matroid M = (E, I) of G is defined by taking the the edge set E(G) as universe and F ⊆ E(G) is in I if and only if the number of connected components in G − F is η.

Proposition 2.2 ([23]
). Graphic and co-graphic matroids are representable over any field of size ≥ 2 and such a representation can be found in time polynomial in the size of the graph.

5:5
Elongation of Matroid.Let M = (E, I) be a matroid and k be an integer such that rank(M ) ≤ k ≤ |E|.A k-elongation matroid M k of M is a matroid with the universe as E and S ⊆ E is a basis of M k if and only if, it contains a basis of M and |S| = k.Observe that the rank of the matroid M k is k.Proposition 2.3 ([18]).Let M be a linear matroid of rank r, over a ground set of size n, which is representable over a field F. Given a number ≥ r, we can compute a representation of the -elongation of M , over the field F(X) in O(nr ) field operations over F. α-Matroid Parity.In our algorithms we use a known algorithm for α-Matroid Parity.Below we define α-Matroid Parity problem formally and state its algorithmic result.

α-Matroid Parity
Parameter: α, q Input: A representation A M of a linear matroid M = (E, I), a partition P of E into blocks of size α and a positive integer q.Question: Does there exist an independent set which is a union of q blocks?Proposition 2.4 ([18, 21]).α-Matroid Parity can be solved in O(2 ωqα ||A M || O(1) ) time.

FPT Algorithm for Simultaneous Feedback Edge Set
In this section we design an algorithm for Sim-FES by giving a reduction to α-Matroid Parity on the direct sum of elongated co-graphic matroids associated with graphs restricted to different color classes.We describe our algorithm, Algo-SimFES, for Sim-FES.Let (G, k, col : E(G) → 2 [α] ) be an input instance to Sim-FES.Recall that for i ∈ [α], G i is the graph with vertex set as V (G) and edge set as . Let η i be the number of connected components in G i .To make G i acyclic we need to delete at least We need to delete at least k i edges from E(G i ) to make G i acyclic.Therefore, the algorithm Alg-SimFES for each i ∈ [α], guesses k i , where

By Proposition 3.1, for any basis
Let M i be the k i -elongation of the co-graphic matroid associated with G i .

7
Let M α+1 = U τ,k over the gound set Fake(G), where, For each e ∈ E(G), let Copies(e) be the block of elements of M .
) be a uniform matroid over the ground set Fake(G).That is, M α+1 = U τ,k .By Propositions 2.1 to Proposition 2.3 we know that M i s are representable over F p (X), where p > max(τ, 2) is a prime number and their representation can be computed in polynomial time.Let A i be the linear representation of Copies(e).Now we define an instance of α-Matroid Parity, which is the linear representation A M of M and the partition of ground set into Copies(e), e ∈ E(G).Notice that for all e ∈ E(G), |Copies(e)| = α.Also for each i ∈ Now Algo-SimFES outputs Yes if there is a basis (an independent set of cardinality αk) of M which is a union of k blocks in M and otherwise outputs No. Algo-SimFES uses the algorithm mentioned in Proposition 2.4 to check whether there is an independent set of M , composed of blocks.A pseudocode of Algo-SimFES can be found in Algorithm 1.
. This implies that Algo-SimFES will not execute Step 4. Consider the for loop for the choice (k 1 , . . ., k α ).We claim that the columns corresponding to S = e∈F Copies(e) form a basis in M and it is union of k blocks.Note that |S| = αk by construction.For all i ∈ [α], let F i = {e i | e ∈ F, i ∈ col(e)}, which is subset of ground set of M i .By Proposition 3.1, for all i ∈ [α], F i is a basis for M i .This takes care of all the edges in ∪ e∈F Original(e).Now let and thus is a basis since |S * | = k .Hence S is a basis of M .Note that S is the union of blocks corresponding to e ∈ F and hence is union of k blocks.Therefore, Algo-SimFES will output Yes.
In the reverse direction suppose Algo-SimFES outputs Yes.This implies that there is a basis, say S, that is the union of k blocks.By construction S corresponds to union of the sets Copies(e) for some k edges in G. Let these edges be F = {e 1 , . . ., e k }.We claim that F is a solution of (G, k, col : E(G) → 2 [α] ).Clearly |F | = k.Since S is a basis of M , for each Since α-Matroid Parity for α = 2 can be solved in polynomial time [19] algorithm Algo-SimFES runs in polynomial time for α = 2.This gives us the following theorem.Theorem 3.4.Sim-FES is in FPT and when α = 2 Sim-FES is in P.

4
Hardness results for Sim-FES In this section we show that when α = 3, Sim-FES is NP-Hard.Furthermore, from our reduction we conclude that it is unlikely that Sim-FES admits a subexponential-time algorithm.We give a reduction from Vertex Cover (VC) in cubic graphs to the special case of Sim-FES where α = 3.Let (G, k) be an instance of VC in cubic graphs, which asks whether the graph G has a vertex cover of size at most k.We assume without loss of generality that k ≤ |V (G)|.It is known that VC in cubic graphs is NP-hard [22] and unless the ETH fails, it cannot be solved in time [17].Thus, to prove that when α = 3, it is unlikely that Sim-FES admits a parameterized subexponential time algorithm (an algorithm of running time O (2 o(k) )), it is sufficient to construct (in polynomial time) an instance of the form (G , k = O(|V (G)| + |E(G)|), col : E → 2 [3] ) of Sim-FES that is equivalent to (G, k).Refer Figure 1 for an illustration of the construction.
To construct (G , k , col : E(G ) → 2 [3] ), we first construct an instance ( G, k) of VC in subcubic graphs which is equivalent to (G, k).We set the graph G is obtained from the graph G by subdividing each edge in E(G) twice.

Lemma 4.1 ( * ). G has a vertex cover of size k if and only if G has a vertex cover of size
Observe that in G every path between two degree-3 vertices contains an edge of the form {x v,u , x u,v }.Thus, the following procedure results in a partition For each degree-3 vertex v, let {v, x}, {v, y} and {v, z} be the edges containing v. We insert {v, x} into M 1 , {v, y} into M 2 , and {v, z} into M 3 (the choice of which edge is inserted into which set is arbitrary).Finally, we insert each edge of the form {x v,u , x u,v } into a set M i that contains neither {v, x v,u } nor {u, x u,v }.
We are now ready to construct the instance (G , k , col : E(G ) → 2 [3] ).Let V  set E(G ) and coloring col are constructed as follows.For each vertex v ∈ V ( G), add an edge {v, v } into E(G ) and its color-set is {1, 2, 3}.For each i ∈ [3] and for each {v, u} ∈ M i , add the edges {v, u} and {v , u } into E(G ) and its color-set is {i}.We set k = k.Clearly, the instance (G , k , col : E(G ) → 2 [3] ) can be constructed in polynomial time, and it holds that [3] ) is a Yes instance of Sim-FES.Observe that because of the above mentioned property of the partition (M 1 , M 2 , M 3 ) of E( G), we ensure that in G , no vertex participates in two (or more) monochromatic cycles that have the same color.By construction, each monochromatic cycle in G is of the form v − v − u − u − v, where {v, u} ∈ E( G), and for each edge {v, u} ∈ E(G ), where either v, u ∈ V ( G) or v, u ∈ V , G contains exactly one monochromatic cycle of this form.

Lemma 4.2 ( * ). ( G, k
) is a Yes instance of VC if and only if (G , k , col : E(G ) → 2 [3] ) is a Yes instance of Sim-FES.
We get the following theorem and its proof follows from Lemma 4.1 and Lemma 4.2.-hard (see, e.g., [9]).Thus, to prove the result, it is sufficient to construct (in polynomial time) an instance of the form (G, k, col : E(G) → 2 [α] ) of Sim-FES that is equivalent to (U, F, k), where α = O(|V (G)|).We construct a graph G such that V (G) = O(|U ||F|) and the number of colors used will be α = |F|.The intuitive idea is to have one edge per element in the universe which is colored with all the indices of sets in the family F that contains the element and for each F i ∈ F creating a unique monochromatic cycle with color i which passes through all the edges corresponding to the elements it contain.We explain the reduction formally in the next paragraph.
elongation of the co-graphic matroid associated with G i .Proposition 3.1 ( * 1 ).Let G be a graph with η connected components and M be an relongation of the co-graphic matroid associated with G, where r ≥ |E(G)| − |V (G)| + η.Then B ⊆ E(G) is a basis of M if and only if the subgraph G − B is acyclic and |B| = r.

Algorithm 1 : 1
and the elements of F restricted to the elements of M i form a basis for all i ∈ [α].For this we will construct an instance of α-Matroid Parity as follows.For each e ∈ E(G) and i ∈ col(e), we use e i to denote the corresponding element in M i .For each e ∈ E(G), by Original(e) we denote the set of elements {e j | j ∈ col(e)}.For each edge e ∈ E(G), we define Fake(e) = {e j | j ∈ [α] − col(e)}.Finally, for each edge e ∈ E(G), by Copies(e) we denote the set Original(e) ∪ Fake(e).Let Fake(G) = e∈E(G) Fake(e).Furthermore, let τ = |Fake(G)| = e∈E(G) |Fake(e)| and 1 Proofs of results marked with ( * ) can be found in the full version of the paper [2].Pseudocode of Algo-SimFES Input: A graph G, k ∈ N and col : E(G) → 2 [α] .Output: Yes if there is a simultaneous feedback edge set of size ≤ k and No otherwise.Let η i be the number of connected components in

10 if
there is an independent set of M composed of k blocks then 11 return Yes 12 return No

2 OFigure 1
Figure 1The construction given in the proof of Theorem 4.3.