Parameterized Complexity of Maximum Edge Colorable Subgraph
Akanksha Agrawal, Madhumita Kundu, Abhishek Sahu, Saket Saurabh, Prafullkumar Tale
PParameterized Complexity of Maximum EdgeColorable Subgraph
Akanksha Agrawal , Madhumita Kundu , Abhishek Sahu , Saket Saurabh , ,and Prafullkumar Tale Ben Gurion University of the Negev, Israel. [email protected] Indian Statistical Institute, Kolkata, India. [email protected] The Institute of Mathematical Sciences, HBNI, Chennai, India. { asahu, saket } @imsc.res.in University of Bergen, Bergen, Norway. Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbr¨ucken,Germany. [email protected]
Abstract.
A graph H is p -edge colorable if there is a coloring ψ : E ( H ) → { , , . . . , p } , such that for distinct uv, vw ∈ E ( H ), we have ψ ( uv ) (cid:54) = ψ ( vw ). The Maximum Edge-Colorable Subgraph problemtakes as input a graph G and integers l and p , and the objective is to finda subgraph H of G and a p -edge-coloring of H , such that | E ( H ) | ≥ l . Westudy the above problem from the viewpoint of Parameterized Complex-ity. We obtain FPT algorithms when parameterized by: (1) the vertexcover number of G , by using Integer Linear Programming , and (2) l , a randomized algorithm via a reduction to Rainbow Matching , anda deterministic algorithm by using color coding, and divide and color.With respect to the parameters p + k , where k is one of the following: (1)the solution size, l , (2) the vertex cover number of G , and (3) l − mm ( G ),where mm ( G ) is the size of a maximum matching in G ; we show that the(decision version of the) problem admits a kernel with O ( k · p ) vertices.Furthermore, we show that there is no kernel of size O ( k − (cid:15) · f ( p )), forany (cid:15) > f , unless NP ⊆ coNP/poly . Keywords:
Edge Coloring · Kernelization · FPT Algorithms · KernelLower Bound.
Akanksha Agrawal:
Funded by the PBC Fellowship Program for Outstanding Post-Doctoral Researchers from China and India.
Saket Saurabh:
Funded by the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovation programme (grant agree-ment No 819416), and Swarnajayanti Fellowship (No DST/SJF/MSA01/2017-18).
Prafullkumar Tale:
Funded by the European Research Council (ERC) under the Eu-ropean Unions Horizon 2020 research and innovation programme under grant agree-ment SYSTEMATICGRAPH (No. 725978). Most parts of this work was completedwhen the author was a Senior Research Fellow at The Institute of MathematicalSciences, HBNI, Chennai, India. a r X i v : . [ c s . D M ] A ug Agrawal et al.
For a graph G , two (distinct) edges in E ( G ) are adjacent if they share an end-point. A p -edge coloring of G is a function ψ : E ( G ) → { , , . . . , p } such thatadjacent edges receive different colors. One of the basic combinatorial optimiza-tion problems Edge Coloring , where for the given graph G and an integer p , the objective is to find a p -edge coloring of G . Edge Coloring is a verywell studied problem in Graph Theory and Algorithm Design and we refer thereaders to the survey by Cao et al. [4], the recent article by Gr¨uttemeier et al.[12], and references with-in for various known results, conjectures, and practicalimportance of this problem.The smallest integer p for which G is p -edge colorable is called its chromaticindex and is denoted by χ (cid:48) ( G ). The classical theorem of Vizing [22] states that χ (cid:48) ( G ) ≤ ∆ ( G ) + 1, where ∆ ( G ) is the maximum degree of a vertex in G . (Noticethat by the definition of p -edge coloring, it follows that we require at least ∆ ( G )many colors to edge color G .) Holyer showed that deciding whether chromaticindex of G is ∆ ( G ) or ∆ ( G )+1 is NP -Hard even for cubic graphs [14]. Laven andGalil generalized this result to prove that the similar result holds for d -regulargraphs, for d ≥ Edge Coloring naturally leads to the question of finding the maximumnumber of edges in a given graph that can be colored with a given number ofcolors. This problem is called
Maximum Edge Colorable Subgraph whichis formally defined below.
Maximum Edge Colorable Subgraph
Input:
A graph G and integers l, p Output:
A subgraph of G with at least l edges and its p -edge coloring orcorrectly conclude that no such subgraph exits.Note that the classical polynomial time solvable problem, Maximum Match-ing , is a special case of
Maximum Edge Colorable Subgraph (when p = 1).Feige et al. [9] showed that Maximum Edge Colorable Subgraph is NP -hardeven for p = 2. In the same paper, the authors presented a constant factor ap-proximation algorithm for the problem and proved that for every fixed p ≥ (cid:15) >
0, for which it is NP -hard to obtain a (1 − (cid:15) )-approximation algo-rithm. Sinnamon presented a randomized algorithm for the problem [21]. To thebest of knowledge, Aloisioa and Mkrtchyan were the first to study this problemfrom the viewpoint of Parameterized Complexity [2] (see Section 2 for defini-tions related to Parameterized Complexity). Aloisioa and Mkrtchyana provedthat when p = 2, the problem is fixed-parameter tractable, with respect tovarious structural graph parameters like path-width, curving-width, and the di-mension of cycle space. Gr¨uttemeier et al. [12], very recently, obtained kernels,when the parameter is p + k , where k is one of the following: i) the number ofedges that needs to be deleted from G , to obtain a graph with maximum degreeat most p − , and ii) the deletion set size to a graph whose connected compo- Recall that any graph with maximum degree at most p −
1, is p -edge colorable [22],and thus, this number is a measure of “distance-from-triviality”.arameterized Complexity of Maximum Edge Colorable Subgraph nents have at most p vertices. Galby et al. [11] proved that Edge Coloring isfixed-parameter tractable when parameterized by the number of colors and thenumber of vertices having the maximum degree.
Our Contributions:
Firstly, we consider
Maximum Edge Colorable Sub-graph , parameterized by the vertex cover number, and we prove the followingtheorem.
Theorem 1.
Maximum Edge Colorable Subgraph , parameterized by thevertex cover number of G , is FPT . We prove the above theorem, by designing an algorithm that, for the giveninstance, creates instances of
ILP , and the resolves the
ILP instance using theknown algorithm ([16], [17]). Intuitively, for the instance (
G, l, p ), suppose (
H, φ )is the solution that we are seeking for, and let X be a vertex cover of G . (Wecan compute X by the algorithm of Chen et al. [6].) We “guess” H (cid:48) = H [ X ] and φ (cid:48) = φ | E ( H (cid:48) ) . Once we have the above guess, we try to find the remaining edges(and their coloring), using ILP .Next, we present two (different)
FPT algorithms for
Maximum Edge Col-orable Subgraph , when parameterized by the number of edges in the desiredsubgraph, l . More precisely, we prove following theorem. Theorem 2.
There exists a deterministic algorithm A and a randomized algo-rithm B with constant probability of success that solves Maximum Edge Col-orable Subgraph . For a given instance ( G, l, p ) , Algorithms A and B termi-nate in time O ∗ (4 l + o ( l ) ) and O ∗ (2 l ) , respectively. We remark that in the above theorem, the Algorithms A and B use differentsets of ideas. Algorithm A , uses a combination of the technique [7] of color-coding [3] and divide and color. Algorithm B uses the algorithm to solve Rain-bow Matching as a black-box. We note that the improvement in the runningtime of Algorithm B comes at the cost of de-randomization, as we do not knowhow to de-randomize Algorithm B .Next we discuss our kernelization results. We show that (the decision versionof) the problem admits a polynomial kernel, when parameteized by p + k , where k is one of the following: ( a ) the solution size, l , ( b ) the vertex cover number of G ,and ( c ) l − mm ( G ), where mm ( G ) is the size of a maximum matching in G ; admits akernel with O ( kp ) vertices. We briefly discuss the choice of our third parameter.By the definition of edge coloring, each color class is a set of matching edges.Hence, we can find one such color class, in polynomial time [19], by computinga maximum matching in a given graph. In above guarantee parameterizationtheme, instead of parameterizing, say, by the solution size ( l in this case), welook for some lower bound (which is the size of a maximum matching in G , forour case) for the solution size, and use a more refined parameter ( l − mm ( G )). Weprove the following theorem. Theorem 3.
Maximum Edge Colorable Subgraph admits a kernel with O ( kp ) vertices, for every k ∈ { (cid:96), vc ( G ) , l − mm ( G ) } . Agrawal et al.
We complement this kernelization result by proving that the dependency of k on the size of the kernel is optimal up-to a constant factor. Theorem 4.
For any k ∈ { (cid:96), vc ( G ) , l − mm ( G ) } , Maximum Edge ColorableSubgraph does not admit a compression of size O ( k − (cid:15) · f ( p )) , for any (cid:15) > and computable function f , unless NP ⊆ coNP /poly . For a positive integer n , we denote set { , , . . . , n } by [ n ]. We work with simpleundirected graphs. The vertex set and edge set of a graph G are denoted as V ( G )and E ( G ), respectively. An edge between two vertices u, v ∈ V ( G ) is denotedby uv . For an edge uv , u and v are called its endpoints . If there is an edge uv ,vertices u, v are said to be adjacent to one another. Two edges are said to be adjacent if they share an endpoint. The neighborhood of a vertex v is a collectionof vertices which are adjacent to v and it is represented as N G ( v ). The degree ofvertex v , denoted by deg G ( v ), is the size of its neighbhorhood. For a graph G , ∆ ( G ) denotes the maximum degree of vertices in G . The closed neighborhood ofa vertex v , denoted by N G [ v ], is the subset N G ( v ) ∪ { v } . When the context ofthe graph is clear we drop the subscript. For set U , we define N ( U ) as union of N ( v ) for all vertices v in U . For two disjoint subsets V , V ⊆ V ( G ), E ( V , V ) isset of edges where one endpoint is in V and another is in V . An edge in the set E ( V , V ) is said to be going across V , V . For an edge set E (cid:48) , V ( E (cid:48) ) denotes thecollection of endpoints of edges in E (cid:48) . A graph H is said to be a subgraph of G if V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ). In other words, any graph obtained from G by deleting vertices and/or edges is called a subgraph of G . For a vertex (resp.edge) subset X ⊂ V ( G ) (resp. ⊂ V ( G )), G − X ( G − Y ) denotes the graphobtained from G by deleting all vertices in X (resp. edges in Y ). Moreover, by G [ X ], we denote graph G − ( V ( G ) − X ).For a positive integer p , a p -edge coloring of a graph G is a function φ : E ( G ) → { , , . . . , p } such that for every distinct uv , wx ∈ E ( G ) s.t. { u, v } ∩{ w, x } (cid:54) = ∅ , we have φ ( uv ) (cid:54) = φ ( wx ). The least positive integer p for which thereexists a p -edge coloring of a graph G is called edge chromatic number of G andit is denoted by χ (cid:48) ( G ). Proposition 1 ([22] Vizing).
For any simple graph G , ∆ ( G ) ≤ χ (cid:48) ( G ) ≤ ∆ ( G ) + 1 . For a coloring function φ and for any i in { , , . . . , p } , the edge subset φ − ( i )is called the i th color class of φ . Notice that by the definition of p -edge coloring,every color class is a matching in G . We define a balanced p -edge coloring of agraph as a p -edge coloring in which the cardinality of any two color classes differby at most one. Lemma 1 ([9] Lemma . ). For a graph G , let φ be a p -edge coloring of G .Then, there exists a balanced p -edge coloring of G that can be derived from φ inpolynomial time. arameterized Complexity of Maximum Edge Colorable Subgraph Observation 21
There exists a subgraph H of G such that H is p -edge col-orable and | E ( H ) | ≥ l if and only if there exists p many edge disjoint matchings M , M , . . . , M p in G such that | M ∪ M ∪ · · · ∪ M p | ≥ l . For a graph G , a set of vertices W is called an independent set if no twovertices of W are adjacent with each other. A set X ⊆ V ( G ) is a vertex cover of G if G − S is an independent set. The size of a minimum vertex cover of graphis called its vertex cover number and it is denoted by vc ( G ). A matching of agraph G is a set of edges of G such that every edge shares no vertex with anyother edge of matching . The size of maximum matching of a graph G is denotedby mm ( G ). It is easy to see that mm ( G ) ≤ vc ( G ) ≤ · mm ( G ). Definition 1 (deg- -modulator). For a graph G , a set X ⊆ V ( G ) is a deg-1-modulator of G , if the degree of each vertex in G − X is at most 1.Expansion Lemma. Let t be a positive integer and G be a bipartite graph withvertex bipartition ( P, Q ). A set of edges M ⊆ E ( G ) is called a t -expansion of P into Q if (i) every vertex of P is incident with exactly t edges of M , and (ii)the number of vertices in Q which are incident with at least one edge in M isexactly t | P | . We say that M saturates the end-points of its edges. Note that theset Q may contain vertices which are not saturated by M . We need the followinggeneralization of Hall’s Matching Theorem known as expansion lemmas : Lemma 2 (See, for example, Lemma . in [8]). Let t be a positive integerand G be a bipartite graph with vertex bipartition ( P, Q ) such that | Q | ≥ t | P | and there are no isolated vertices in Q . Then there exist nonempty vertex sets P (cid:48) ⊆ P and Q (cid:48) ⊆ Q such that (i) P (cid:48) has a t -expansion into Q (cid:48) , and (ii) novertex in Q (cid:48) has a neighbour outside P (cid:48) . Furthermore two such sets P (cid:48) and Q (cid:48) can be found in time polynomial in the size of G .Integer Linear Programming. The technical tool we use to prove that
Max-imum Edge Colorable Subgraph is fixed-parameter tractable (defined innext sub-section) by the size of vertex cover is the fact that
Integer LinearProgramming is fixed-parameter tractable when parameterized by the num-ber of variables. An instance of
Integer Linear Programming consists of amatrix A ∈ Z m × q , a vector ¯ b ∈ Z m and a vector ¯ c ∈ Z q . The goal is to find avector ¯ x ∈ Z q which satisfies A ¯ x ≤ ¯ b and minimizes the value of ¯ c · ¯ x (scalarproduct of ¯ c and ¯ x ). We assume that an input is given in binary and thus thesize of the input instance or simply instance is the number of bits in its binaryrepresentation. Proposition 2 ([16], [17]). An Integer Linear Programming instance ofsize L with q variables can be solved using O ( q . q + o ( q ) · ( L +log M x ) · log( M x · M c )) arithmetic operations and space polynomial in L + log M x , where M x is an upperbound on the absolute value that a variable can take in a solution, and M c is thelargest absolute value of a coefficient in the vector ¯ c . Agrawal et al.
Parameterized Complexity.
The goal of parameterized complexity is to find waysof solving NP -hard problems more efficiently than brute force by associating a small parameter to each instance. Formally, a parameterization of a problemis assigning a positive integer parameter k to each input instance and we saythat a parameterized problem is fixed-parameter tractable ( FPT ) if there is analgorithm, that given an instance (
I, k ), resolves in time bounded by f ( k ) ·| I | O (1) ,where | I | is the size of the input I and f is an arbitrary computable functiondepending only on the parameter k .Such an algorithm is called an FPT algorithm and such a running time iscalled
FPT running time. Another central notion in the field of ParameterizedComplexity is kernelization . A parameterized problem is said to admit a h ( k )- kernel if there is a polynomial-time algorithm (the degree of the polynomialis independent of k ), called a kernelization algorithm, that, given an instance( I, k ) of the problem, outputs an instance ( I (cid:48) , k (cid:48) ) of the problem such that:( i ) | I (cid:48) | ∈ k (cid:48) ≤ h ( k ), and ( ii ) ( I, k ) and ( I (cid:48) , k (cid:48) ) are equivalent instances of theproblem i.e. ( I, k ) is a
Yes instance if and only if ( I (cid:48) , k (cid:48) ) is a Yes instance ofthe problem. It is known that a decidable problem admits an
FPT algorithm ifand only if there is a kernel. If the function h ( k ) is polynomial in k , then wesay that the problem admits a polynomial kernel. For more on parameterizedcomplexity, see the recent books [8,10].We say a parameter k is larger than a parameter k if there exists a com-putable function g ( · ) such that k ≤ g ( k ). In such case, we denote k (cid:22) k and say k is smaller than k . If a problem if FPT parameterized by k thenit is also FPT parameterized by k . Moreover, if a problem admits a kernel ofsize h ( k ) then it admits a kernel of size h ( g ( k )). For a graph G , let X beits minimum sized deg-1-modulator. By the definition of vertex cover, we have | X | ≤ vc ( G ). This implies | X | (cid:22) vc ( G ). In the following observation, we arguethat for “non-trivial” instances, vc ( G ) (cid:22) l and | X | (cid:22) l − mm ( G ). Observation 22
For a given instance ( G, l, p ) of Maximum Edge ColorableSubgraph , in polynomial time, we can conclude that either ( G, l, p ) is a Yes instance or vc ( G ) (cid:22) l and | X | (cid:22) ( l − mm ( G )) , where X is a minimum sizeddeg- -modulator of G .Proof. Let M be a maximum sized matching in graph G . Such matching can befound in polynomial time using the algorithm by Micali and Vazirani [19]. If l ≤| M | = mm ( G ) then we can conclude that ( G, l, p ) is a
Yes instance. Otherwise,we are working with an instance for which l > mm ( G ). As 2 mm ( G ) ≥ vc ( G ), wehave l > vc ( G ) / vc ( G ) (cid:22) l .Consider the graph G (cid:48) = G − M . Let M be a maximum sized matching in G (cid:48) . If | M | + | M | ≥ l then ( G, l, p ) is a
Yes instance. Otherwise, we are workingwith an instance for which | M | < l − | M | . This implies | V ( M ) | < l − mm ( G )).Consider the graph G − V ( M ). The only edges present in this graph are theones in M . Hence, every connected component in G − V ( M ) has degree at mostone. This implies | X | ≤ | V ( M ) | ≤ l − mm ( G )) where X is a minimum sizeddeg-1-modulator of G . (cid:117)(cid:116) arameterized Complexity of Maximum Edge Colorable Subgraph FPT
Algorithm Parameterized by the Vertex CoverNumber of the Input
In this section, we consider the problem
Maximum Edge Colorable Sub-graph , when parameterized the vertex cover number of the input graph. Let(
G, l, p ) be an instance of the problem, where the graph G has n vertices. Weassume that G has no isolated vertices as any such vertex is irrelevant for anedge coloring. We begin by computing a minimum sized vertex cover, X of G ,in time O (2 | X | n | X | ), using the algorithm of Chen et al. [5].We begin by intuitively explaining the working of our algorithm. We assumean arbitrary (but fixed) ordering over vertices in G , and let W = V ( G ) \ X .Suppose that we are seeking for the subgraph H , of G , with at least (cid:96) edgesand the coloring φ : E ( H ) → { , , . . . , p } . We first “guess” the intersectionof H with G [ X ], i.e., the subgraph H (cid:48) of G [ X ], such that V ( H ) ∩ X = H (cid:48) and V ( H ) ∩ E ( G [ X ]) = H (cid:48) . (Actually, rather than guessing, we will go overall possible such H (cid:48) s, and do the steps, that we intuitively describe next.) Let φ (cid:48) = φ E ( H (cid:48) ) . Based on ( H (cid:48) , φ (cid:48) ), we construct an instance of ILP , which will helpus “extend” the partial solution ( H (cid:48) , φ (cid:48) ), to the solution (if such an extendedsolution exists), for the instance ( G, (cid:96), p ). Roughly speaking, the construction ofthe
ILP relies on the following properties. Note that W is an independent setin G , and thus edges of the solution that do not belong to H (cid:48) , must have oneendpoint in X and the other endpoint in W . Recall that H has the partition(given by φ ) into (at most) p matchings, say, M , M , . . . , M p (cid:48) . The number ofdifferent neighborhoods in X , of vertices in W , is bounded by 2 k . This allowsus to define a “type” for M i − E ( H (cid:48) ), based on the neighborhoods, in X , of thevertices appearing in M i − E ( H (cid:48) ). Once we have defined these types, we cancreate a variable Y T ,α , for each type T and color class α (in { , , . . . , p (cid:48) } ). Thespecial color 0 will be used for assigning all the edges that should be coloredusing the colors outside { , , . . . , p (cid:48) } (and we will later see that it is enough tokeep only one such color). We would like the variable Y T ,α to store the numberof matchings of type T that must be colored α . The above will heavily rely onthe fact that each edge in H that does not belong to H (cid:48) , must be adjacent to avertex in X , this in turn will facilitate in counting the number of edges in thematching (via the type, where the type will also encode the subset of vertices in X participating in the matching). Furthermore, only for α = 0, the variable Y T ,α can store a value which is more than 1. Once we have the above variable set,by adding appropriate constraints, we will create an equivalent instance of ILP ,corresponding to the pair ( H (cid:48) , φ (cid:48) ). We will now move to the formal descriptionof the algorithm.For S ⊆ X , let Γ ( S ) be the set of vertices in W whose neighborhood in G is exactly S , i.e., Γ ( S ) := { w ∈ W | N G ( w ) = S } . We begin by defining a tuple,which will be a “type”, and later we will relate a matching (between W and X ),to a particular type. Agrawal et al.
Fig. 1.
The tuple T = (cid:104){ x , x , x } ; N ( w ) , N ( w ) , N ( w ) , ∅ , ∅(cid:105) is a type. The matching { x w , x w , x w } is of T . Definition 2 (Type). A type T = (cid:104) X (cid:48) = { x , x , . . . , x | X (cid:48) | } ; S , S , . . . , S | X | (cid:105) is a ( | X | + 1) sized tuple where each entry is a subset of X and which satisfyfollowing properties.1. The first entry, X (cid:48) , is followed by | X (cid:48) | many entries which are non-emptysubsets of X and the remaining ( | X | + 1 − | X (cid:48) | ) entries are empty sets.2. Any non-empty set S of X appears at most | Γ ( S ) | many times from thesecond entry onward in the tuple.3. For every i ∈ { , , . . . , | X (cid:48) |} , we have x i ∈ S i .See Figure 1 for an example. We note that the number of different types is atmost 2 | X | · | X | ∈ O ( | X | ) and it can be enumerated in time 2 O ( | X | ) · n O (1) . Weneed following an auxiliary function corresponding to a matching, which will beuseful in defining the type for a matching. Let M be a matching across X, W ( M has edges whose one endpoint is in X and the other endpoint is in W ). Define τ M : X ∩ V ( M ) → W ∩ V ( M ), as τ M ( x ) := w if xw is an edge in M . We dropwhen the context is clear. Definition 3 (Matching of type T ). A matching M = { xτ ( x ) | x ∈ X and τ ( x ) ∈ W } , is of type T = (cid:104) X (cid:48) ; S , S , . . . , S | X | (cid:105) if V ( M ) ∩ X = X (cid:48) ( := { x , x , . . . , x | X | (cid:48) } ), and S j = N ( τ ( x j )) for every j in { , , . . . , | X (cid:48) |} .We define some terms used in the sub-routine to construct an ILP in-stance. For a type T = (cid:104) X (cid:48) ; S , S , . . . , S | X | (cid:105) , we define | T | := | X (cid:48) | . Note that | T | is the number of edges in a matching of type T . For a vertex x ∈ X and a type T = (cid:104) X (cid:48) ; S , S , . . . , S | X | (cid:105) , value of is present ( x, T ) is 1 if x ∈ X (cid:48) , and otherwise it is 0. For w ∈ W , define false twins ( w ) as the num-ber of vertices in W which have the same neighborhood as that of w . That is, arameterized Complexity of Maximum Edge Colorable Subgraph false twins ( w ) = |{ ˆ w ∈ W | N ( w ) = N ( ˆ w ) }| . For a vertex w ∈ W and atype T = (cid:104) X (cid:48) ; S , S , . . . , S | X | (cid:105) , the value of nr nbr present ( w, T ) denotes thenumber of different j s in { , , . . . , | X (cid:48) |} for which S j = N ( w ). We remark thatthe values of all the functions defined above can be computed in (total) timebounded by 2 O ( | X | ) · n O (1) . Constructing ILP instances . Recall that G is the input graph and X is a(minimum sized) vertex cover for G . Let T be the set of all types. For everysubgraph H (cid:48) of G [ X ], a (non-negative) integer p ≤ p , and a p -edge coloring φ (cid:48) : E ( H (cid:48) ) → { , , , . . . , p } , we create an instance I ( H (cid:48) ,φ (cid:48) ) , of ILP as follows.Let [ p ] (cid:48) = { , , , . . . , p } . Define a variable Y T ,α for every type T and integer α ∈ [ p ] (cid:48) . (These variables will be allowed to take values from { , , . . . , p } ).Intuitively speaking, for α in [ p ] (cid:48) , the value assigned to Y T ,α will indicates thatthere is a matching of type T which is assigned the color α . Moreover, for α = 0,the value of Y T , will indicate that there are Y T , many matchings of type T ,each of which must be assigned a unique color which is strictly greater than p .Recall that for a type T ∈ T , | T | is the number of edges in a matching of T . Wenext define our objective function, which (intuitively speaking) will maximizethe number of edges in the solution.maximize (cid:88) T ∈ T ; α ∈ [ p ] (cid:48) Y T ,α · | T | We next discuss the set of constraints.For every vertex x in X , we add the following constraint, which will ensurethat x will be present in at most p matchings: (cid:88) T ∈ T ; α ∈ [ p ] (cid:48) Y T ,α · is present ( x, T ) ≤ p − deg H (cid:48) ( x ) . ( ConstSetI )For each x ∈ X , an edge x ˆ x incident on x in H (cid:48) , and T ∈ T , we add thefollowing constraint, which will ensure that no other edge incident on x and somevertex in W is assigned the color φ (cid:48) ( x ˆ x ): Y T ,φ (cid:48) ( x ˆ x ) · is present ( x, T ) = 0 . ( ConstSetII )We will next add the following constraint for each w ∈ W , which will helpus in ensuring that w is present in at most p matchings: (cid:88) T ∈ T ; α ∈ [ p ] (cid:48) Y T ,α · nr nbr present ( w, T ) ≤ p · false twins ( w ) . ( ConstSetIII )Notice that for two vertices w , w ∈ W , such that N ( w ) = N ( w ), theabove constraints corresponding to w and w is exactly the same (and we skipadding the same constraint twice). When α (cid:54) = 0, we want to ensure that at most one matching that is colored α . Thus, for α ∈ [ p ], add the constraint: (cid:88) T ∈ T Y T ,α ≤ . ( ConstSetIV )Note that we want at most p color classes, which will be ensured by our finalconstraint as follows. (cid:88) T ∈ T ; α ∈ [ p ] (cid:48) Y T ,α ≤ p. ( ConstSetV )This completes the construction of the
ILP instance of I ( H (cid:48) ,φ (cid:48) ) . Algorithm for Maximum Edge Colorable Subgraph:
Consider the giveninstance (
G, l, p ) of
Maximum Edge Colorable Subgraph . The algorithmwill either return a solution (
H, φ ) for the instance, or conclude that no suchsolution exists. We compute a minimum sized vertex cover, X of G , in time O (2 | X | n | X | ), using the algorithm of Chen et al. [5]. For every subgraph H (cid:48) of G [ X ], a (non-negative) integer p ≤ p , and a p -edge coloring φ (cid:48) : E ( H (cid:48) ) →{ , , , . . . , p } , we create the instance I ( H (cid:48) ,φ (cid:48) ) , and resolve it using Proposition 2.(In the above we only consider those φ (cid:48) : E ( H (cid:48) ) → { , , , . . . , p } , where eachof the color classes are non-empty.) If there exists a tuple ( H (cid:48) , φ (cid:48) ) for which theoptimum value of the corresponding ILP instance is at least ( l − | E ( H (cid:48) ) | ) thenalgorithm constructs a solution ( H, φ ) as specified in the proof of Lemma 4 andreturns it as a solution. If there is no such tuple then the algorithm concludesthat no solution exists for a given instance.For a solution (
H, φ : E ( H ) → [ p ]) for the instance ( G, l, p ), we say that(
H, φ ) is a good solution, if for some p ∈ [ p ], for each e ∈ E ( H ) ∩ E ( G [ X ]),we have φ ( e ) ∈ [ p ]. Note that if ( G, l, p ) has a solution, then it also has a goodsolution. We argue the correctness of the algorithm in the following two lemmas.
Lemma 3. If ( G, l, p ) has a good solution ( H, φ ) then the optimum value of the ILP instance I ( H (cid:48) ,φ (cid:48) ) is at least ( l −| E ( H (cid:48) ) | ) , where H (cid:48) = H [ X ] and φ (cid:48) : E ( H (cid:48) ) →{ , , . . . , p } , such that φ (cid:48) = φ | E ( H (cid:48) ) and p = max { φ ( e ) | e ∈ E ( H ) ∩ E ( G [ X ]) } .Proof. Let M , M , . . . , M p be the partition of edges in E ( H ) \ E ( H (cid:48) ) accordingto the colors assigned to them by φ , and M = { M i | i ∈ [ p ] } \ {∅} . Notice thateach M i is a matching, where the edges have one endpoint in X and the otherendpoint in W . We create an assignment asg : Var ( H (cid:48) ,φ (cid:48) ) → [ p ] (cid:48) , where Var ( H (cid:48) ,φ (cid:48) ) is the set of variables in the instance I ( H (cid:48) ,φ (cid:48) ) as follows. Initialize asg ( z ) = 0, foreach z ∈ Var ( I ( H (cid:48) ,φ (cid:48) ) ). For i ∈ [ p ], let T i be the type of M i and p i = φ ( e ), where e ∈ M i . For each i ∈ [ p ], we do the following. If p i > p , then increment asg ( Y T i , )by one, and otherwise increment value of asg ( Y T i ,p i ) by one. This completes theassignment of variables. Next we argue that asg satisfies all constraints in I ( H (cid:48) ,φ (cid:48) ) and the objective function evaluates to a value that is at least ( l − | E ( H (cid:48) ) | ). arameterized Complexity of Maximum Edge Colorable Subgraph As there are at most p matchings, we have (cid:80) T ∈ T ; α ∈ [ p ] (cid:48) Y T ,α ≤ p , and thus,the constraint in ConstSetV is satisfied.We will now argue that each constraint in
ConstSetI is satisfied. To this end,consider a variable x ∈ X , and let a x = (cid:80) T ∈ T ; α ∈ [ p ] (cid:48) asg ( Y T ,α ) · is present ( x, T ).Since H is p -edge colorable, deg H ( x ) ≤ ∆ ( H ) ≤ p . Hence, there are at most p edges incident on x in H (Proposition 1). For any T ∈ T and α ∈ [ p ] (cid:48) , if asg ( Y T ,α ) · is present ( x, T ) (cid:54) = 0, then there are asg ( Y T ,α ) many matchings oftype T in M , each of which contains an edge incident on x . Moreover, eachsuch matching contains a different edge incident on x . Since φ is a p -edgecoloring of H , we have a x + deg H (cid:48) ( x ) = deg H ( x ) ≤ p . This implies that a x = (cid:80) T ∈ T ; α ∈ [ p ] (cid:48) asg ( Y T ,α ) · is present ( x, T ) ≤ p − deg H (cid:48) ( x ). Thus we con-clude that all contraints in ConstSetI are satisfied.Now we argue that all constraints in
ConstSetII are satisfied. Consider x ∈ X ,an edge x ˆ x incident on x in H (cid:48) , and T ∈ T such that is present ( x, T ) = 1.Since x ˆ x ∈ E ( H (cid:48) ), there is no matching M i ∈ M , such that p i = φ (cid:48) ( x ˆ x ) and M contains an edge incident on x . Thus we can obtain that asg ( Y T ,φ (cid:48) ( x ˆ x ) ) =0 (recall that is present ( x, T ) = 1). From the above we can conclude that asg ( Y T ,φ (cid:48) ( x ˆ x ) ) · is present ( x, T ) = 0.Next we argue that all constraints in ConstSetIII are satisfied. To this end,consider a (maximal) subset W (cid:48) = { w , w , . . . , w r } ⊆ W , such that any twovertices in W (cid:48) are false twins of each other. Notice that for each j, j (cid:48) ∈ [ r ], (cid:80) T ∈ T ; α ∈ [ p ] (cid:48) Y T ,α · nr nbr present ( w j , T ) ≤ p · false twins ( w j ) is exactly thesame as (cid:80) T ∈ T ; α ∈ [ p ] (cid:48) Y T ,α · nr nbr present ( w j , T ) ≤ p · false twins ( w j (cid:48) ).Consider any w ∈ W (cid:48) , T ∈ T , and α ∈ [ p ] (cid:48) , such that we have asg ( Y T ,α ) · nr nbr present ( w, T ) (cid:54) = 0. There are asg ( Y T ,α ) many matchings in M eachof which contains nr nbr present ( w, T ) many edges incident vertices in W (cid:48) .Hence (cid:80) T ∈ T ; α ∈ [ p ] (cid:48) asg ( Y T ,α ) · nr nbr present ( w j , T ) is the number of edges in-cident on W (cid:48) in H . Note that p · false twins ( w ) is the maximum number ofedges in H which can be incident on vertices in W (cid:48) . Thus we can conclude that (cid:80) T ∈ T ; α ∈ [ p ] (cid:48) asg ( Y T ,α ) · nr nbr present ( w j , T ) ≤ p · false twins ( w ).For any α ∈ [ p ], there is at most one matching in M whose edges areassigned the color α . This implies that (cid:80) T ∈ T asg ( Y T ,α ) ≤
1. Hence all constraintsin
ConstSetIV are satisfied.There are at least ( l − | E ( H (cid:48) ) | ) many edges in E ( H ) \ E ( H (cid:48) ) and eachsuch edge has one endpoint in X and another in W . Every edge in match-ing contributes exactly one to the objective function. Thus we can obtain that (cid:80) T ∈ T ; α ∈ [ p ] (cid:48) asg ( Y T ,α ) · | T | ≥ ( l − | E ( H (cid:48) ) | ). This concludes the proof. (cid:117)(cid:116) Lemma 4.
If there is ( H (cid:48) , φ (cid:48) ) for which the optimum value of the ILP instance I ( H (cid:48) ,φ (cid:48) ) , is at least ( l − | E ( H (cid:48) ) | ) , then the Maximum Edge Colorable Sub-graph instance ( G, l, p ) admits a solution. Moreover, given asg : Var ( H (cid:48) ,φ (cid:48) ) → [ p ] (cid:48) , where Var ( H (cid:48) ,φ (cid:48) ) , we can be compute ( H, φ ) in polynomial time.Proof. We first describe an algorithm, which given an assignment asg : Var ( H (cid:48) ,φ (cid:48) ) → [ p ] (cid:48) for I ( H (cid:48) ,φ (cid:48) ) , such that the optimum value of objective functions is at least( l − | E ( H (cid:48) ) | ), constructs a solution ( H, φ ) for (
G, l, p ). We will construct (
H, φ ), such that (1) E ( H (cid:48) ) ⊆ E ( H ), (2) φ is a p -edge coloring of H which has atleast l edges, and (3) φ | E ( H (cid:48) ) is identical to that of φ (cid:48) . For every variable Y T ,α ∈ Var ( H (cid:48) ,φ (cid:48) ) , such that asg ( Y T ,α ) (cid:54) = 0, the algorithm will constructs a match-ing M α with | T | edges. At each step, the edges in M α are added to H , and φ assigns the color α to all the edges in M α . We will argue that by the end ofthis process, the number of edges in H is at least l . Recall that there is a fixedordering on vertices in X and W . and for a subset S i of X , Γ ( S i ) denotes thecollection of vertices in W whose neighborhood in G is exactly S i . We say that Γ ( S i ) is H -degree balanced set if for any two vertices w , w in the set, deg H ( w )and deg H ( w ) differs by at most one. Algorithm to construct ( H, φ ) : Initialize V ( H ) = V ( G ), E ( H ) = E ( H (cid:48) ), and φ | E ( H (cid:48) ) = φ (cid:48) . Consider α > T = (cid:104) X (cid:48) ; S , S , . . . , S | X | (cid:105) for which asg ( Y T ,α ) (cid:54) = 0. For the sake of clarity, assume X (cid:48) = { x , x , . . . , x | X (cid:48) | } . Thealgorithm constructs a matching M T ,α , of type T in the following way. Initialize M T ,α = ∅ . For i in { , , . . . , | X (cid:48) |} , let w i be a vertex in Γ ( S i ) such that ( a ) noedge incident on w i has already been added to M T ,α , ( b ) degree of w i in H isat most p −
1, and ( c ) Γ ( S i ) remains H -degree-balanced after increasing degreeof w i by one. If there are more than one vertices that satisfy these properties,select the lowest indexed vertex as w i . Add edge the edge x i w i to M T ,α beforemoving to next value of i . This completes the construction of M T ,α . Add all theedges in M T ,α to H and assign φ ( e ) = α , for every edge e in M .Now we will consider variables Y T , , such that asg ( Y T , ) (cid:54) = 0, for T ∈ T . Set α := p . Consider T ∈ T , and let asg ( Y T , ) = a T . For each (increasing) β ∈ [ a T ],do the following. We construct a matching M (cid:48) T ,β of type T similar to the onethat we diiscussed earlier. That is, initialize M (cid:48) T ,β = ∅ . For i in { , , . . . , | X (cid:48) |} ,let w i be a vertex in Γ ( S i ) such that ( a ) no edge incident on w i has already beenadded to M T ,β , ( b ) degree of w i in H is at most p −
1, and ( c ) Γ ( S i ) remains H -degree-balanced after increasing degree of w i by one. If there are more thanone vertices that satisfy these properties, select the lowest indexed vertex as w i .Add all the edges in M (cid:48) T ,β to H , set φ ( e ) = α + β , and move to the next choiceof β (if it exists). This completes the description of the algorithm.It is clear from the description of the algorithm that it can be executedin polynomial time. We next argue that: i) for each α ∈ [ p ], the algorithmconstructs M T ,α with | T | edges, for whenever asg ( Y T ,α ) = 1, and ii) for each β ∈ [ a T ] the algorithm constructs M T ,β with | T | edges, whenever asg ( Y T , ) (cid:54) = 0.We argue only the first statement, the proof of the second statement can beobtained by following similar arguments. For the sake of contradiction, assumethat there is α ∈ [ p ] and T = (cid:104) X (cid:48) ; S , S , . . . , S | X | (cid:105) , for which the algorithm couldnot construct M T ,α of size T , or in other words, by construction, the algorithmcould not construct M T ,α of type T . In the above, we consider the lowest iterationunder which ( α, T ) was under consideration and M T ,α of type T could not beconstructed. Thus, for some x i ∈ X (cid:48) , for every vertex w ∈ Γ ( S i ) at least oneof the following holds: ( a ) w has an edge incident on it which has already beenadded to M T ,α , ( b ) w has degree exactly p in H , or ( c ) Γ ( S i ) does not remainsan H -degree-balanced after increasing the degree of w in H by one. We consider arameterized Complexity of Maximum Edge Colorable Subgraph following two exhaustive cases: Case (1) There exists a vertex in Γ ( S i ) whosedegree in H is p . Case (2) Every vertex in Γ ( S i ) has degree at most ( p −
1) in H . We argue that Case (1) leads to the contradiction that the constraints in ConstSetIII is satisfied. We argue that in Case (2), T is not a type, again leadingto a contradiction.Consider Case (1). Since the algorithm failed for the first time, Γ ( S i ) isan H -degree balanced set and each vertex in it has a degree at most p , be-fore the algorithm started processing for the iteration for α and T . As Γ ( S i )at the current processing contains a vertex of degree p in H , every vertexin it has degree either ( p −
1) or p in H . Suppose there are n vertices in Γ ( S ) which have degree ( p − w be a vertex in Γ ( S ). By definition,we have false twins ( w ) = | Γ ( S ) | . Let L denote the summation on the lefthand side of the constraint of type ( ConstSetIII ) corresponding to w . Let L (cid:48) bethe summation of asg ( Y T (cid:48) ,α ) which have already been processed by the algo-rithm. Note that L ≥ L (cid:48) + asg ( Y T ,α ) · nr nbr present ( w, T ). Recall that for T (cid:48) ∈ T and integer α (cid:48) if both asg ( Y T (cid:48) ,α (cid:48) ) and nr nbr present ( w, T (cid:48) ) are non-zeros, the algorithm adds asg ( Y T (cid:48) ,α (cid:48) ) · nr nbr present ( w, T (cid:48) ) many edges inci-dent on vertices in Γ ( S i ). Since there are n vertices with degree ( p −
1) and Γ ( S i ) many vertices of degree p in H , there are p · ( | Γ ( S i ) | − n ) + ( p − · n many edges incident on vertices in Γ ( S i ). As Γ ( S i ) is a subset of W , whichis an independent set in G and hence in H , all these edges are across X, W .Hence, algorithm has added L (cid:48) = p · | Γ ( S i ) | + n · ( p −
1) many edges to H before it starts processing at α , T . Since n many edges, each of which incidenton vertices in Γ ( S i ) which has degree ( p − T , we have asg ( Y T ,α ) · nr nbr present ( w, T ) ≥ n + 1. This implies L ≥ L (cid:48) + asg ( Y T ,α ) · nr nbr present ( w, T ) ≥ p · ( | Γ ( S i ) |− n )+( p − · n + n +1 ≥ p · | Γ ( S i ) | + 1. As | Γ ( S i ) | = false twins ( w ), this contradicts the fact that con-straint in ConstSetIII are satisfied.Consider Case (2). Before the algorithm starts processing α , T , every vertexhas degree at most ( p −
1) and Γ ( S i ) is an H -degree balanced set. The algorithmcan select one edge incident on every vertex in Γ ( S i ) to add it to the matching.Note that the algorithm selects an edge incident on vertices in Γ ( S i ) if and onlyif the entry in the tuple is S i . Since the algorithm failed in this case, we canconclude that S i appears at least | Γ ( S i ) | + 1 many times from second placeonward in T . This contradicts the second property mentioned in Definition 2.As discussed in the previous two paragraphs, both Case (1) and Case (2)lead to contradictions. Hence our assumption that the algorithm is not able toconstruct a matching at certain steps is wrong. This implies the algorithm willalways return ( H, φ ).We now argue that φ is an edge coloring of H . Consider an arbitrary vertex x in X . Consider an edge x ˆ x in H (cid:48) which is incident on x . By the construction of ConstSetII , for any T if is present ( x, T ) = 1, then asg ( Y T ,φ (cid:48) ( x ˆ x ) ) = 0. Hence, atno step the algorithm modifies φ in a way that it assign color φ ( x ˆ x ) to a newlyadded edge which is incident on x . Moreover, by the constraints in ConstSetIV ,for α > T if asg ( Y T ,α ) (cid:54) = 0, then for any other T (cid:48) ∈ T , asg ( Y T (cid:48) ,α ) = 0. Hence, the algorithm does not add more that one edge of color α on any vertexin x . Consider an arbitrary vertex w in W . At the start of the process, there isno edge incident on w . At any stage, the algorithm adds at most one edge to H and assigns it a color that has not been used previously and will not be usedlater. Hence, every edge incident on w has been assigned to a different color. As x, w are arbitrary vertices in X, W , respectively, we can conclude that φ is anedge coloring of H .We argue that φ uses at most p colors. Consider a vertex x in X . Becauseof the constraints in ConstSetI , the algorithm adds at most p − deg H (cid:48) ( x ) manyedges incident on x . Hence there are at most p edges incident on any vertices in X . Consider a vertex w in W . As mentioned before, there are no edges incidenton w at the start of the process. By the constraint in ConstSetV , the processcreates at most p matchings. Hence there are at most p many edges incident on w . As x, w are arbitrary vertices in X, W , respectively, we can conclude that φ is a p -edge coloring of H .The algorithm adds Y T ,α · | T | many edges for every variable which has non-zero value. Since the objective function is at least ( l − | E ( H (cid:48) ) | ), we can concludethat H has at least l edges. This concludes the proof of the lemma. (cid:117)(cid:116) We are now in a position to state the main result of this section.
Proof (Proof of Theorem 1).
We prove that there exists an algorithm whichgiven a graph G on n vertices and integers l, p as input either outputs a subgraph H of G such that H is p -edge colorable and has at least l edges, or correctlyconcludes that no such subgraph exists. Moreover, the algorithm terminatesin time f ( vc ( G )) · n O (1) , where f ( vc ( G )) is some computable function whichdepends only on vc ( G ).We argue that the algorithm described in this section satisfy desired prop-erties. The correctness of the algorithm is implied by Lemma 3 and Lemma 4.We now argue that the algorithm runs in FPT time. The algorithm computes anoptimum vertex cover X in time 2 vc ( G ) ·| V ( G ) | O (1) . It then enumerates all tuplesof type ( H (cid:48) , φ (cid:48) ) where H (cid:48) is a subgraph of G [ X ] and φ (cid:48) is a p -edge coloring of H (cid:48) for some p ≤ p . There are 2 O ( | X | ) many possible choices for H (cid:48) . Recall thatthe algorithm only considers φ (cid:48) in which for every j in { , , . . . , p } there is anedge x ˆ x in E ( H (cid:48) ) such that φ (cid:48) ( x ˆ x ) = j . Hence p ≤ | X | . This implies that thetotal number of choices for φ is at most | X | O ( | X | ) . Hence the algorithm createsat most 2 O ( | X | log | X | ) many instances of ILP .The algorithm uses Proposition 2 to solve each instance of
ILP . To boundthe time taken for this step, we bound the number of variables in each instance of
ILP . As mentioned earlier, the number of different types is at most 2 | X | · | X | ∈ O ( | X | ) and all of them can be enumerated in time 2 O ( | X | ) · n O (1) . Since, α can have at most p + 1 ≤ O ( | X | ) different values, every instance has 2 O ( | X | ) many variables. By construction, upper bounds on the absolute value a variablecan take in a solution and the largest absolute value of a coefficient used islinearly bounded by n . By Proposition 2, this instance can be solved in time arameterized Complexity of Maximum Edge Colorable Subgraph O ( | X | · n O (1) . Hence the algorithm terminates in time FPT in vc ( G ) whichconcludes the proof. (cid:117)(cid:116) FPT
Algorithm Parameterized by the Number ofEdges in a Desired Subgraph
In this section, we prove Theorem 2. We say a randomized algorithm B solves Maximum Edge Colorable Subgraph problem with constant probability ofsuccess if given an instance (
G, l, p ) such that G contains a subgraph H which is p -edge colorable and | E ( H ) | ≥ l , the algorithm returns a solution with constantprobability. Our first algorithm uses the technique of color-coding combinedwith divide and color introduced in [7]. We present a randomized version of thisalgorithm which can be de-randomized using standard techniques (see for ex-ample [8]). For the second algorithm, we reduce a given instance of MaximumEdge Colorable Subgraph to an equivalent instance of
Rainbow Match-ing . This reduction along with the known algorithm for the later problem resultsin a different randomized
FPT algorithm for
Maximum Edge Colorable Sub-graph , with improved running time.
FPT
Algorithm
Given an instance (
G, l, p ) of
Maximum Edge Colorable Subgraph prob-lem, we can assume l ≡ p ). If it is not the case, then let l ≡ r (mod p )for some r ∈ [ p − G (cid:48) , l (cid:48) = l +( p − r ) , p ) where G (cid:48) isthe graph obtained obtained by adding ( p − r ) isolated edges. Formally, V ( G (cid:48) ) = V ( G ) ∪{ x , x , . . . , x p − r ) } and E ( G (cid:48) ) = E ( G ) ∪{ x i − x i | i ∈ { , , . . . , p − r }} .It is easy to see that ( G (cid:48) , l + ( p − r ) , p ) is a Yes instance if and only if (
G, l, p )is a
Yes instance. By Lemma 1, if ( G (cid:48) , l (cid:48) , p ) is a Yes instance of
MaximumEdge Colorable Subgraph problem, then there is a p -edge-coloring of G (cid:48) where exactly q = l (cid:48) /p edges are colored by every color. Hence, in the remainingsection, we assume that for a given instance ( G, l, p ), we have l ≡ p ).We present a randomized recursive algorithm (Algorithm 4.1) to solve theproblem and later specify how to de-randomize it. The central idea is to partitionthe edge set into two parts such that one part contains all the solution edgescolored by the first (cid:98) a/ (cid:99) colors and the other part contains all the solution edgescolored by the remaining (cid:100) a/ (cid:101) colors. We determine the answer to these sub-problems recursively and use them to return the answer to the original problem.To formalize these ideas, we define the term D ( a,q ) [ X ] for a ∈ N , X ⊆ E ( G (cid:48) ),where D ( a,q ) [ X ] is true if and only if there are a edge-disjoint matchings, each ofsize q , in X . Instead of computing D ( a,q ) [ X ], the algorithm computes D (cid:63) ( a,q ) [ X ].The relationship between these terms is as follows: if D (cid:63) ( a,q ) [ X ] is true then D ( a,q ) [ X ] is always true , but if D ( a,q ) [ X ] is true then D (cid:63) ( a,q ) [ X ] is true onlywith sufficiently high probability. Thus, we get a one-sided error Monte Carloalgorithm. We boost the success probability of correct partitions by repeatingthe partitioning process many times, to achieve constant success probability. We Input :
A subset X ⊆ E ( G ), integers 1 ≤ a ≤ p and q . Output: D a,q [ X ] ( D a,q [ X ] is true if and only if there are a edge disjointmatchings, each of size q , in G [ X ]) if a == 1 thenreturn true if there is a matching of size q in G [ X ], and otherwise false . end D (cid:63) ( a,q ) [ X ] = false ; for aq log (4 l ) many times do Partition X into L (cid:93) R uniformly at random; D (cid:63) ( (cid:98) a/ (cid:99) ,q ) [ L ] = Faster-Randomized-Algorithm( L, (cid:98) a/ (cid:99) , q ); D (cid:63) ( (cid:100) a/ (cid:101) ,q ) [ R ] = Faster-Randomized-Algorithm( R, (cid:100) a/ (cid:101) , q ); if D (cid:63) ( a,q ) [ X ] == false then D (cid:63) ( a,q ) [ X ] = D (cid:63) ( (cid:98) a/ (cid:99) ,q ) [ L ] ∧ D (cid:63) ( (cid:100) a/ (cid:101) ,q ) [ R ]; endendreturn D (cid:63) ( a,q ) [ X ] Algorithm 4.1:
Faster-Randomized-Algorithm(
X, a, q )note that the fact that each color class contains exactly q many edges ensuresthat at each partitioning step, two parts contain an almost equal number ofedges. This fact plays a crucial role while calculating the probability of successand the run time of the algorithm. Lemma 5.
There exists a randomized algorithm that given ( G, l, p ) either findsa subgraph H of G and its p -edge coloring such that | E ( H ) | ≥ l , or correctlyconcludes that no such subgraph exists in time O ∗ (4 l + o ( l + p ) ) . Moreover, if sucha subgraph exists in G , then the algorithm returns it with constant probability.Proof. Given an instance (
G, l, p ), the algorithm does the necessary modifications(as mentioned in the starting of this sub-section) to ensure that l ≡ p ).It then runs Algorithm 4.1 with X = E ( G ), a = p , and q = l/p as input. IfAlgorithm 4.1 return true then the algorithm returns Yes otherwise it returns No . It is easy to modify Algorithm 4.1, and hence the algorithm, to ensure thatthe algorithm returns a set of edges (and its coloring) instead of returning true .We argue the correctness of the algorithm using induction on a . The basecase occurs when a = 1. It is easy to see that in this case the algorithm cor-rectly concludes the value of D ( a,q ) [ E ( G )]. Assume that the algorithm is correctfor all values of a that are strictly less than p (cid:48) , for some 2 ≤ p (cid:48) < p . Thealgorithm returns Yes for input (
G, l, p ) only if Algorithm 4.1 has concluded D ( p,q ) [ E ( G )] = true . In this case, there exists a partition L (cid:93) R of E ( G ) suchthat D (cid:63) ( (cid:98) p/ (cid:99) ,q ) [ L ] and D (cid:63) ( (cid:100) p/ (cid:101) ,q ) [ R ] are set to true . By induction hypothesis,there exists (cid:98) p/ (cid:99) , (cid:100) p/ (cid:101) many edge-disjoint matchings, each containing q edges,in G [ L ] and G [ R ], respectively. This implies there are p edge-disjoint matchingseach containing q edges. By Observation 21, ( G, l, p ) is a
Yes instance.It remains to argue that given a
Yes instance (
G, l, p ), the algorithm returns
Yes with constant probability. Let E ( a ) denote the event that D ( a,q ) [ X ] = true arameterized Complexity of Maximum Edge Colorable Subgraph implies D (cid:63) ( a,q ) [ X ] = true . Notice that E ( p ) is exactly the event where our algo-rithm succeeds i.e. correctly determines D ( p,q ) [ E ( G )]. We present a lower boundon P r ( E ( a )) using following a recurrence equation. We say the algorithm cor-rectly partitions the solution edges if L and R contain the solution edges coloredwith first (cid:98) a/ (cid:99) colors and remaining [ (cid:100) a/ (cid:101) ] colors, respectively. The probabil-ity of success for the event depends on the following two independent events –( i ) the algorithm correctly partitions in at least one of the 2 aq log (4 l ) rounds,and ( ii ) the values D (cid:63) ( (cid:98) a/ (cid:99) ,q ) [ L ] and D (cid:63) ( (cid:100) a/ (cid:101) ,q [ R ] are computed correctly. Theprobability of a partition ( L, R ) failing to divide the solution edges ( aq many)correctly in any of the rounds can be upper bounded following expression: (cid:16) − aq (cid:17) aq · log (4 l ) ≤ l − P r ( E ( a )) ≥ (cid:16) − l − (cid:17) · P r ( E ( (cid:98) a/ (cid:99) )) · P r ( E ( (cid:100) a/ (cid:101) ))with P r ( E ( a )) = 1, when a = 1. The base case of the recurrence equation followsdirectly from the algorithm. The above recurrence implies P r ( E ( p )) ≥ / Yes instance, the algorithm returns
Yes with probability at least 1 / T ( a ) = 2 aq log(4 l ) · ( T ( (cid:98) a/ (cid:99) ) + T ( (cid:100) a/ (cid:101) )), where T (1) = | V ( G ) | O (1) .This recurrence equation solve to T ( p ) ≤ l + o ( l + p ) | V ( G ) | O (1) which gives us therunning time of our algorithm. (cid:117)(cid:116) We note that the algorithm mentioned in Lemma 5 can be de-randomizedusing ( E ( G ) , l )-perfect hash families [20]. FPT
Algorithm
In this subsection, we present a randomized
FPT algorithm running in time2 l · | V ( G ) | O (1) by reducing a given instance of Maximum Edge ColorableSubgraph to an instance of
Rainbow Matching . In
Rainbow Matching problem, the input is an edge-labeled graph G (cid:48) and a positive integer k and theobjective is to determine whether there exists a matching of size at least k suchthat all the edges in the matching have distinct labels. Such matching is calledas rainbow matching . We use the following known result. Proposition 3 (Theorem in [13]). There exists a randomized algorithmthat, given a
Rainbow Matching instance ( G (cid:48) , k ) , in time k · | V ( G (cid:48) ) | O (1) either reports a failure or finds a rainbow matching. Moreover, if the algorithmis given a Yes instance, it returns a rainbow matching with constant probability.
We use ‘colors’ for instances of
Maximum Edge Colorable Subgraph and‘labels’ for instances of
Rainbow Matching . Reduction :
Given an instance (
G, l, p ) of
Maximum Edge Colorable Sub-graph , the reduction algorithm returns an instance ( G (cid:48) , k ) of Rainbow Match-ing . To construct graph G (cid:48) , the algorithm creates p identical copies of G . For-mally, for every vertex u in V ( G ), it adds p vertices u i for i ∈ [ p ] in V ( G (cid:48) ). Forevery edge uv , it adds all the edges u i v i for i ∈ [ p ] in E ( G (cid:48) ). The algorithm ar-bitrary construct a one-to-one function ψ (cid:48) : E ( G ) → { , , . . . , | E ( G ) |} on edgesin G . It constructs an edge-labelling function ψ for edges in G (cid:48) in the followingway : for i ∈ [ p ], assign ψ ( u i v i ) = ψ (cid:48) ( uv ). Algorithm assigns k = l and returns( G (cid:48) , k ). Lemma 6.
Let ( G (cid:48) , k ) be the instance returned by the reduction algorithm wheninput is ( G, l, p ) . Then, ( G, l, p ) is a Yes instance of
Maximum Edge Col-orable Subgraph if and only if ( G (cid:48) , k ) is a Yes instance of
Rainbow Match-ing .Proof. ( ⇒ ) By Observation 21, there are p many edge disjoint matchings M , M ,. . . , M p in G such that | M ∪ M ∪ · · · ∪ M p | ≥ l . We construct a rainbow match-ing M (cid:48) in G (cid:48) in the following way: For i ∈ [ p ], if edge uv ∈ E ( G ) is in M i thenadd u i v i to M (cid:48) . By construction, M (cid:48) has at least k = l edges. Since M i is amatching, there is at most one edge in M i which is incident on any vertex in V ( G ). Hence, an edge u i v i is added to M (cid:48) then no other edge incident on u i or v i is added to M (cid:48) . This implies M (cid:48) is matching in G (cid:48) . We now argue thatall edges in M (cid:48) have distinct labels. Note that the only edges in G (cid:48) which hassame labels are u i v i and u j v j for some uv ∈ E ( G ) and i, j ∈ [ p ]. Since matchings M , M , . . . , M p are edge disjoint, if an edge uv is present in M i then it is notpresent in M j for any j ∈ [ p ] \ { i } . Hence, all edges in M (cid:48) have distinct labels.This implies ( G (cid:48) , k ) is a Yes instance.( ⇐ ) Let M (cid:48) be a matching in G (cid:48) such that | M (cid:48) | ≥ k and every edge in M (cid:48) has distinct label. By construction, every edge in E ( G (cid:48) ), and hence in M (cid:48) , isof the form u i v i for some i ∈ [ p ] and uv ∈ E ( G ). We construct p matchings M , M , . . . , M p in G in the following way: For i ∈ [ p ], if edge u i v i is in M (cid:48) thenadd uv to M i . Since M (cid:48) is a matching, if edges u i v i are in M (cid:48) then no other edgeincident on u i or v i is in M (cid:48) . Hence, for every i ∈ [ p ], set M i is a matching in G . We now argue that these constructed matchings are edge disjoints. Assume,for the sake of a contradiction, that for some i, j ∈ [ p ], matchings M i and M j intersect. Let uv be the edge in M i ∩ M j . The only reason edge uv is added to M i and to M j is because edges u i v i , u j v j are present in M (cid:48) . By construction,edges u i v i , u j v j have same label. This contradicts the fact that edges in M (cid:48) have distinct edges. Hence our assumption is wrong and the matchings in G arepairwise disjoint. This fact, along with the construction, implies that | M ∪ M ∪· · · ∪ M p | ≥ l = k . By Observation 21, ( G, l, p ) is a
Yes instance. (cid:117)(cid:116)
Proposition 3 and Lemma 6 implies that there exists a randomized algorithmthat given (
G, l, p ) either finds a subgraph H of G and its p -edge coloring suchthat | E ( H ) | ≥ l or correctly concludes that no such subgraph exists in time O ∗ (2 l ). Moreover, if such a subgraph exists in G , then the algorithm returns itwith constant probability. arameterized Complexity of Maximum Edge Colorable Subgraph In this section, we prove that
Maximum Edge Colorable Subgraph admitsa polynomial kernel when parameterized by the number of colors and | X | where X is a minimum sized deg-1-modulator. As discussed in Section 2, such resultimplies that the problem admits a polynomial kernel when parameterized by thenumber of colors together with one of the following parameters: (1) the number ofedges, l , in the desired subgraph, (2) the vertex cover number of the input graph vc ( G ), and (3) the above guarantee parameter ( l − mm ( G )). Our kernelizationalgorithm is based on the expansion lemma.Consider an instance ( G, p, l ) of
Maximum Edge Colorable Subgraph .We assume that we are given a deg-1-modulator X of G (see Definition 1).We justify this assumption later and argue that one can find a deg-1-modulatorwhich is close to a minimum sized deg-1-modulator in polynomial time. We startwith the following simple reduction rule. Reduction Rule 51
If there exists a connected component C of G − X suchthat no vertex of C is adjacent to a vertex in X , then delete all the vertices in C and reduce l by | E ( C ) | , i.e. return the instance ( G − V ( C ) , l − | E ( C ) | , p ) . Lemma 7.
Reduction Rule 51 is safe and given set X , it can be applied inpolynomial time. Let (
G, l, p ) be the instance obtained by exhaustively applying Reduction Rule 51.This implies that every connected component of G − X is adjacent to X . Let C be the set of connected components of G − X . We construct an auxiliary bi-partite graph B , with vertex bipartition X and C (each C ∈ C corresponds toa vertex, say b C of B ). There exists edge xb C in B for x ∈ X and b C ∈ C ifand only x is adjacent to at least one vertex in C in G . For C (cid:48) ⊆ C of connectedcomponents, V ( C (cid:48) ) ⊆ V ( G ) denotes the vertices in connected components in C (cid:48) and E ( C (cid:48) ) ⊆ E ( G ) denotes the edges that have both endpoints in V ( C (cid:48) ). Sinceevery connected component in C is adjacent to X , there are no isolated verticesin B . We can thus apply the following rule which is based on the ExpansionLemma. Reduction Rule 52 If |C| ≥ p | X | then apply Lemma 2 to find X (cid:48) ⊆ X and C (cid:48) ⊆ C such that (1) there exits a p -expansion from X (cid:48) to C (cid:48) ; and (2) no vertex in C (cid:48) has a neighbour outside X (cid:48) . Delete all the vertices in X (cid:48) ∪ V ( C (cid:48) ) from G andreduce l by p | X (cid:48) | + | E ( C (cid:48) ) | , i.e. return ( G − ( X (cid:48) ∪ V ( C (cid:48) )) , l − p | X (cid:48) | − | E ( C (cid:48) ) | , p ) . Lemma 8.
Reduction Rule 52 is safe and given set X , it can be applied inpolynomial time.Proof. Let M (cid:48) be the edges in p -expansion lemma from X (cid:48) to C (cid:48) for the bipartitegraph B . We construct a set M ⊆ E ( G ), corresponding to edges in as follows M (cid:48) . If there is an edge xb C in M (cid:48) then pick an edge whose one endpoint is x andanother endpoint is in C . If there are multiple such edges then arbitrarily pick one of them. Consider a subgraph H of G such that V ( H ) = X (cid:48) ∪ V ( C (cid:48) ) and E ( H ) = M ∪ E ( C (cid:48) ). A p -star graph is a tree on p + 1 vertices such that thereexists a vertex that is adjacent to all other vertices. Notice that each connectedcomponent of H is a tree. Since every connected component in C (cid:48) has at mostone edge, each tree in H can be obtained from a p -star by adding (at most one)new vertex and making it adjacent with one of its leaves. It is easy to see that H is a p -edge colorable graph and every vertex in X (cid:48) is of degree p in H .Suppose there exists exists a subgraph H (cid:48) of G − ( X (cid:48) ∪ V ( C (cid:48) ) such that H (cid:48) is p -edge colorable and has at least l − p | X | − | E ( C (cid:48) ) | edges. Then, H (cid:48) ∪ H is asubgraph of G which is p -edge colorable and has at least l edges. Here, H (cid:48) ∪ H denote the graph with vertex set ( V ( H ) ∪ V ( H ) and edges E ( H ) ∪ E ( H )).Suppose there exists a subgraph H of G which is p -edge colorable and has atleast l edges. By Proposition 1, the maximum degree of a vertex in H is p . Let H ◦ be the graph obtained from H by deleting all vertices in X (cid:48) ∪ V ( C (cid:48) ). Since H ◦ is a subgraph of H , it is p -edge colorable. Note that H ◦ is also a subgraph of G − ( X (cid:48) ∪ V ( C (cid:48) )). To complete the proof, we need to argue that H ◦ has at least | E ( H ) | − p | X (cid:48) | + | E ( C (cid:48) ) | edges. Since every vertex in H has degree at most p ,there are at most p | X (cid:48) | many edges across X, V ( C (cid:48) ) i.e. edges with one vertex in X and another in V ( C (cid:48) ). Moreover, there are at most | E ( C (cid:48) ) | edges in H whoseboth endpoints are in V ( C (cid:48) ). Since V ( C (cid:48) ) are adjacent with vertices only in X (cid:48) ,there are no other edges incident on V ( C (cid:48) ). This implies number of edges in H ◦ is at least | E ( H ) | − p | X (cid:48) | + | E ( C (cid:48) ) | which concludes the proof. (cid:117)(cid:116) In the following lemma, we argue that
Maximum Edge Colorable Sub-graph admits a polynomial kernel when parameterized by size of the givendeg-1-modulator.
Lemma 9.
Consider an instance ( G, l, p ) and let X be a deg- -modulator of G .Then, Maximum Edge Colorable Subgraph admits a kernel with O ( | X | p ) vertices.Proof. The algorithm then applies Reduction Rule 51 and 52 exhaustively. Itreturns the reduced instance as a kernel. We now argue the correctness andthe size bound on the reduced instance. Let ( G (cid:48) , l (cid:48) , p ) be the reduced instanceobtained by the algorithm after exhaustive application of reduction rules oninput instance ( G, l, p ). By Lemma 7 and 8, (
G, l, p ) is a
Yes instance if andonly if ( G (cid:48) , l (cid:48) , p (cid:48) ) is a Yes instance. Moreover, since reduction rules are notapplicable, the number of vertices in G (cid:48) is at most ( p + 1) | X | . (cid:117)(cid:116) Proof. (of Theorem 3) For an instance (
G, l, p ) of
Maximum Edge ColorableSubgraph the kernelization algorithm first uses Observation 22 to conclude thateither (
G, l, p ) is a
Yes instance or vc ( G ) (cid:22) l and | X opt | (cid:22) ( l − mm ( G )), where X opt is a minimum sized deg-1-modulator of G . In the first case, it returnsa vacuously true instance of constant size. If it can not conclude that giveninstance is a Yes instance then algorithm computes a deg-1-modulator, say X ,of G using the simple 3-approximation algorithm: there exists a vertex u whichis adjacent with two different vertices, say v , v then algorithm adds u, v , v to arameterized Complexity of Maximum Edge Colorable Subgraph the solution. It keeps repeating this step until every vertex is of degree at mostone. The algorithm uses the kernelization algorithm mentioned in Lemma 9 tocompute a kernel of size O ( p | X | ).The correctness of the algorithm follows from the correctness of Lemma 9.As X is obtained by using a 3-factor approximation algorithm, | X | ≤ | X opt | and hence | X | (cid:22) | X opt | . Since the algorithm was not able to conclude that( G, l, p ) is a
Yes instance, by Observation 22, we have | X opt | (cid:22) vc ( G ) (cid:22) l and | X opt | (cid:22) l − mm ( G ). This implies the number of vertices in the reduced instanceis at most O ( kp ) where k is one of the parameters in the statement of thetheorem. By Observation 22, Lemma 7 and Lemma 8, and the fact that everyapplication of reduction rules reduces the number of vertices in the input graph,the algorithm terminates in polynomial time. (cid:117)(cid:116) The objective of this section is to prove Theorem 4. Due to Observation 22, it issufficient to prove such result for the number of edges in the desired graph. Inother words, we prove that for any (cid:15) > f , MaximumEdge Colorable Subgraph does not admit a polynomial compression of size O ( l − (cid:15) · f ( p )) unless NP ⊆ coNP /poly . We obtain the above result by giving anappropriate reduction from Red Blue Dominating Set to Maximum EdgeColorable Subgraph . Red Blue Dominating Set ( RBDS , for short) takes as input a bipartitegraph G , with vertex bi-partitions as R, B and an integer k , and the objectiveis to decide if there is R (cid:48) ⊆ R of size at most k such that for each b ∈ B , R (cid:48) ∩ N ( b ) (cid:54) = ∅ . Without loss of generality, we can assume that there are noisolated vertices in the input graph. The problem
Dominating Set takes as aninput a graph G and an integer k , and the goal is to decide whether there exists X ⊆ V ( G ) of size at most k , such that for each v ∈ V ( G ), X ∩ N [ v ] (cid:54) = ∅ . Jansenand Pieterse proved that Dominating Set does not admit a compression of bitsize O ( n − (cid:15) ), for any (cid:15) > NP ⊆ coNP/poly , where n is the number ofvertices in the input graph [15]. This result directly implies the following (see,for instance [1], for a formal statement). Proposition 4.
Red Blue Dominating Set does not admit a compression ofbit size O ( n − (cid:15) ) , for any (cid:15) > , unless NP ⊆ coNP/poly . Here, n is the numberof vertices in the input graph. Holyer showed that it is NP -hard to distinguish whether the given cubic graphadmits a 3-edge coloring, or any edge coloring of it requires 4 colors) [14] (also seeProposition 1). Laven and Galil generalized this result to prove that for any fixed p , the problem of deciding whether the edge chromatic number of a regular graphof degree p is p or p + 1 [18]. We start with an inverting component presented in[14] (see Figure 2). We call this graph as a module . Note that a, b, c, d, and e are The sets R and B are referred as red and blue sets, respectively.2 Agrawal et al. Fig. 2.
Inverting components and its symbolic representation used in the reductionfrom
Red Blue Dominating Set to Maximum Edge Colorable Subgraph . labelings of the corresponding edges, and the other endpoints of these edges arenot shown in the figure. We state following two useful properties of modules. Claim 61 ([18, Lemma ]) For any -edge coloring φ of a module, either φ ( a ) = φ ( b ) or φ ( c ) = φ ( d ) . Moreover, if φ ( a ) = φ ( b ) then φ ( c ) , φ ( d ) , φ ( e ) are all different; else φ ( c ) = φ ( d ) , and then φ ( a ) , φ ( b ) , φ ( e ) are all different. Claim 62 ([18, Lemma ]) Consider a partial -coloring φ of edges in a mod-ule which satisfy either of two conditions: (1) φ ( a ) = φ ( b ) and φ ( c ) , φ ( d ) , φ ( e ) are all different; (2) φ ( c ) = φ ( d ) and φ ( a ) , φ ( b ) , φ ( e ) are all different. Then, φ can be extended to a -edge coloring of the module. We next present a polynomial time reduction from
RBDS to Maximum EdgeColorable Subgraph . Consider an instance (
G, R, B, k ) of
RBDS . We con-struct an instance ( G (cid:48) , l, p ) of Maximum Edge Colorable Subgraph , asfollows.
Reduction :
Initialize V ( G (cid:48) ) = V ( G ) = R ∪ B and E ( G (cid:48) ) = ∅ . For everyvertex r ∈ R , we construct a gadget using 2 · (deg G ( r ) + 1) modules as shownin Figure 3. This gadget has (deg G ( r ) + 1) many pairs of edges which acts asoutputs. Arbitrarily fix a pair edges in outputs and make r an endpoint of boththese edges. We call this gadget a red gadget corresponding to r . For every bluevertex b ∈ B , construct a cycle of length (2 · deg G ( r ) + 1). Arbitrarily fix avertex on this cycle and make it adjacent with b . Add deg G ( b ) many modulesto this cycle such that edges on the cycles are endpoints of pairs of edges whichare outputs of these modules (see Figure 3). Add these modules in such thatafter addition, the degree of every vertex on cycle is three. We call this gadgeta blue-gadget corresponding to b . For each edge rb in G , identify a pair of edgesin outputs of red-gadget corresponding to r with a pair of edges in inputs ofblue-gadget corresponding to b . In other words, the other endpoints of edges inred-gadget corresponding to r is in blue-gadget corresponding to b and vice-versa(see the edges e , e in Figure 3). This completes the construction of graph G (cid:48) . arameterized Complexity of Maximum Edge Colorable Subgraph Fig. 3. (Left Figure) A red-gadget made for a vertex of degree four. The gadget is madefrom ten modules and have four output pairs of edges. More generally, it can be madefrom 2( d + 1) modules and has d output pairs of edges. In the modified red-gadget,vertex r is replaced by two vertices r , r as shown in left-top corner. (Right Figure)A blue-gadget made for a vertex of degree three. More generally, it can be made from d modules and has d input pairs of edges. Assign p = 3, l = | E ( G (cid:48) ) | − k and return ( G (cid:48) , l, p ) as an instance of MaximumEdge Colorable Subgraph .Next we argue that thenumber of edges in G (cid:48) is at most constant times thenumber of edges in G . Lemma 10.
We have | E ( G (cid:48) ) | ≤ c · | E ( G ) | , where c is a (fixed) constant.Proof. Every module contains seven vertices each of which has degree three.Hence there are at most 21 edges which are incident on vertices in a module. Sincered-gadget corresponding to r uses 2 · (deg G ( r ) + 1) many gadgets, the numberof edges incident on this red-gadget is at most 42 · (deg G ( r ) + 1). Hence, the totalnumber of edges incident on red-gadgets is at most (cid:80) r ∈ R · (deg G ( r ) + 1) ≤ ·| E ( G ) | + | R | . Similarly, blue-gadget corresponding to b uses deg G ( b ) modules,a cycle with edges 2 · deg G ( r ) + 1, and an extra edge. Hence the total numberof edges incident on vertices in blue-gadgets is at most (cid:80) b ∈ B (21 · deg G ( b ) + 2 · deg G ( b ) + 2) ≤ | E ( G ) | + 2 | B | . Since there are no isolated vertices in G ( R, B ),we can conclude that total number of edges in G (cid:48) is at most 67 | E ( G ) | . Thisconcludes the proof of the lemma. (cid:117)(cid:116) To simplify our arguments in the proof of correctness of the reduction, weconstruct an auxiliary graph. We define the following process which takes thegraph G (cid:48) and a subset R (cid:48) of R and returns another graph G (cid:48)(cid:48) such that | V ( G (cid:48)(cid:48) ) | = | V ( G (cid:48) ) | + | R (cid:48) | and | E ( G (cid:48)(cid:48) ) | = | E ( G (cid:48) ) | . Modification of G (cid:48) at R (cid:48) : For every vertex r in R (cid:48) do the following process.Add two vertices r , r to G (cid:48) and delete r from G . Let x , x be the two verticesin red-gadget which were adjacent with r in G (cid:48) . Add edges r x and r x (seeFigure 3). Let R (cid:48) i = { r i | r ∈ R (cid:48) } for i ∈ { , } . In other words, G (cid:48)(cid:48) is obtainedfrom G by deleting all vertices in R (cid:48) and adding vertices in R (cid:48) ∪ R (cid:48) , and makingthem adjacent with the corresponding neighbors of vertices in R . If red-gadgetis modified at r in R (cid:48) then we call it the modified red-gadget at r .In any 3-edge-coloring of a red-gadget, edges incident on vertices in R areof different colors. With this observation, [18, Lemma 2] implies following twoproperties of red-gadgets. Claim 63
In any -edge coloring of a red-gadget, every pair of output edges arecolored with different colors. We modify the red-gadgets to ensure that every pair of output edges can becolored with the same color. Following claim is also implied by [18, Lemma 2].
Claim 64
There exists a -edge coloring of modified red-gadget such that everypair of output edges are colored with same colors. We mention following property of blue-gadget before mentioning a relation be-tween G and G (cid:48)(cid:48) . Claim 65 ([18, Lemma ]) In any -coloring of a blue-gadget at least one pairof input must be colored with the some color. Moreover, any -coloring of theinput edges which satisfied the previous condition can be completed to a -coloringof the gadget. Lemma 11.
Let ( G (cid:48) , l, p ) be the instance returned by the reduction algorithmwhen input is ( G, R, B, k ) . For a subset R (cid:48) ⊆ R , let G (cid:48)(cid:48) be the graph obtained bymodifying G (cid:48) at R (cid:48) . Then, R (cid:48) is adjacent with all vertices in B if and only if G (cid:48)(cid:48) is -edge colorable.Proof. ( ⇒ ) We construct a 3-edge coloring of graph G (cid:48)(cid:48) . We first color all edgesincident on red-gadgets and modified red-gadgets followed by edges in blue-gadgets. Note that the sets of edges incident on red-gadgets and modified red-gadgets do not intersect with each other. By Claim 61 and 62, there exists acoloring of red-gadgets and modified red-gadgets. Moreover, by Claim 64, it issafe to consider a 3-edge coloring of modified red-gadget in which every pair ofoutput edges are colored with the same colors. Since R (cid:48) is adjacent with everyvertex in B , every blue-gadget has at least one pair of input edges which arecolored with some color. By Claim 65, this coloring can be extended to otheredges incident on blue-gadgets. This completes a 3-edge coloring of G (cid:48)(cid:48) .( ⇐ ) Consider a 3-edge coloring of graph G (cid:48)(cid:48) . By Claim 65, for any blue-gadget, at least one pair of input must be colored with some color. By Claim 63,for a red-gadget, any pair of output edges are colored with different colors.Hence, for any blue-gadget, there is at least one input pair of edges that are arameterized Complexity of Maximum Edge Colorable Subgraph connected to an output pair of edges of a modified red-gadget. By construction,this implies that for any vertex in B , there exists an edge with some vertex in R (cid:48) . This implies that vertices in R (cid:48) are adjacent with every vertex in B . (cid:117)(cid:116) In the following lemma, we argue that the reduction is safe.
Lemma 12.
Let ( G (cid:48) , l, p ) be the instance returned by the reduction algorithmwhen ( G, R, B, k ) is given as input. Then, ( G, R, B, k ) is a Yes instance of
RBDS if and only if ( G (cid:48) , l, p ) is a Yes instance of
Maximum Edge Col-orable Subgraph .Proof.
The problem of determining whether ( G (cid:48) , l, p ) is a Yes instance of
Max-imum Edge Colorable Subgraph is equivalent to determining whether onecan delete at most ( | E ( G (cid:48) ) |− l ) many edges in G (cid:48) such that the resultant graph is p -edge colorable. We work with this formulation of the problem. Since ( G (cid:48) , l, p )is an instance returned by the reduction algorithm, | E ( G (cid:48) ) | − l ≤ k and p = 3.Let R (cid:48) be a subset of R . Define E R (cid:48) as the set of edges in E ( G (cid:48) ) formed byselecting exactly one edge incident every vertex in R (cid:48) . By construction, | E R (cid:48) | = | R (cid:48) | . Let G (cid:48)(cid:48) be the graph obtained from modification of G (cid:48) at R (cid:48) as specifiedabove. Note that the graph obtained from G (cid:48) by deleting all edges in E R (cid:48) isisomorphic to the graph obtained from G (cid:48)(cid:48) by deleting all vertices in R (cid:48) . Here, R (cid:48) is a set defined in modification process.In graph G (cid:48)(cid:48) , every vertex in R (cid:48) is pendant vertex and is adjacent with avertex of degree two. Hence, any 3-edge coloring of G (cid:48)(cid:48) − R (cid:48) can be triviallyextended to a 3-edge coloring of G (cid:48)(cid:48) . Also, as G (cid:48)(cid:48) − R (cid:48) is a subgraph of G (cid:48)(cid:48) , any3-edge coloring of G (cid:48)(cid:48) is also a 3-edge coloring of G (cid:48)(cid:48) − R (cid:48) . Hence, G (cid:48)(cid:48) − R (cid:48) is3-edge colorable if and only if G (cid:48)(cid:48) is 3-edge colorable. Lemma 11 implies that G (cid:48) − E R (cid:48) is 3-edge colorable if and only if R (cid:48) is adjacent with all vertices in B . Since graphs G (cid:48)(cid:48) − R (cid:48) and G (cid:48) − E R (cid:48) are isomorphic to each others, we get G (cid:48) − E R (cid:48) is 3-edge colorable if and only if R (cid:48) is adjacent with all vertices in B .Since | E R (cid:48) | = | R (cid:48) | , this concludes the proof of the lemma. (cid:117)(cid:116) We are now in a position to present a proof for Theorem 4.
Proof. (for Theorem 4) For the sake of contradiction, assume that there existsan (cid:15) > f such that Maximum Edge Col-orable Subgraph admits a compression of size O ( l − (cid:15) · f ( p )). This impliesthere is an algorithm A which takes an instance ( G (cid:48) , l, p ) of Maximum EdgeColorable Subgraph and in polynomial time returns an equivalent instancefor some problem which needs O ( l − (cid:15) · f ( p )) bits to encode.Let ( G, R, B, k ) be an instance of
RBDS , where G is a graph on n ver-tices. Using the reduction described, we create an instance ( G (cid:48) , l, p ) of Max-imum Edge Colorable Subgraph . It is easy to see from the descriptionof the reduction that this instance can be created in time polynomial in thesize of the given instance of
RBDS . By Lemma 12, instances (
G, R, B, k ) and( G (cid:48) , l, p ) are equivalent. On instance ( G (cid:48) , l, p ), we run the algorithm A mentionedin previous paragraph to obtain an equivalent instance of size O ( l − (cid:15) · p ). Note that this instance is equivalent to the given instance of RBDS . Since p = 3and l ≤ | E ( G (cid:48) ) | ∈ O ( | E ( G ) | ) ∈ O ( n ) (by Lemma 10), this instance is ofsize O ( n − (cid:15) ). This implies there exists an algorithm which in polynomial timereturns an equivalent instance of RBDS of size O ( n − (cid:15) ). This is a contradic-tion to Proposition 4. Hence our assumption was wrong and Maximum Edge-Colorable Subgraph does not admit a compression of size O ( l − (cid:15) · f ( p )). Theproof of the theorem follows from Observation 22. (cid:117)(cid:116) In this article, we studied the
Maximum Edge Colorable Subgraph prob-lem from the lense of Parameterized Complexity. We showed that the problemadmits a kernel with O ( k · p ) vertices where p is the number of colors and k isone of the following: ( a ) the number of edges, l , in a desired subgraph, ( b ) thevertex cover number of input graph, and ( c ) the difference between l and thesize of a maximum matching in the graph. Furthermore, we complimented theabove result by establishing that Maximum Edge Colorable Subgraph doesnot admit a polynomial compress of size O ( k − (cid:15) · f ( p )) for any (cid:15) > f , unless NP ⊆ coNP /poly . It will be interesting to closethe gap between the kernel lower bound and the size of the kernel. As a conse-quence of the above kernelization results, we can obtain that the problem hasa polynomial kernel when parameterized by (cid:96) . It will interesting to investigatewhether the problem has a polynomial kernel when parameterized by the vertexcover number, or the difference between l and the size of a maximum matchingin the input graph.We also designed FPT algorithms for the parameters, (cid:96) and the vertex covernumber. We leave it as an open question to determine whether the problemadmits an
FPT algorithm when parameterized by the difference between l andthe size of a maximum matching in the input graph. References
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