Multivariate Analysis of Scheduling Fair Competitions
MMultivariate Analysis of Scheduling Fair Competitions
Siddharth Gupta ∗ Meirav Zehavi † Abstract A fair competition , based on the concept of envy-freeness, is a non-eliminating competi-tion where each contestant (team or individual player) may not play against all other contes-tants, but the total difficulty for each contestant is the same: the sum of the initial rankingsof the opponents for each contestant is the same. Similar to other non-eliminating competi-tions like the Round-robin competition or the Swiss-system competition, the winner of thefair competition is the contestant who wins the most games. The Fair Non-EliminatingTournament ( Fair-NET ) problem can be used to schedule fair competitions whose in-frastructure is known. In the
Fair-NET problem, we are given an infrastructure of atournament represented by a graph G and the initial rankings of the contestants representedby a multiset of integers S . The objective is to decide whether G is S -fair , i.e., there existsan assignment of the contestants to the vertices of G such that the sum of the rankingsof the neighbors of each contestant in G is the same constant k ∈ N . We initiate a studyof the classical and parameterized complexity of Fair-NET with respect to several centralstructural parameters motivated by real world scenarios, thereby presenting a comprehensivepicture of it. ∗ Ben-Gurion University of the Negev, Israel. [email protected] † Ben-Gurion University of the Negev, Israel. [email protected] a r X i v : . [ c s . D S ] F e b Introduction
Various real life situations require to conduct fair competitions. For illustration, suppose wewant to schedule a non-eliminating sports competition in which there are n contestants and n grounds located on the circumference of a circle. As organizers, we want to assign a homeground to each contestant in such a way that every contestant c plays only against r contestantswhose home ground is nearest to c ’s home ground rather than all the contestants. The underlyingrationale can be time constraints and also to minimize the travel time for each contestant (similarto the Traveling Tournament problem, see e.g. [21, 22]). However, the total difficulty foreach contestant should be the same, i.e., the sum of the initial rankings of the opponents foreach contestant is the same. We can model this problem as an instance of the
Fair Non-Eliminating Tournament ( Fair-NET ) problem, where we are given an infrastructure of atournament represented by a graph G and the initial rankings of the contestants representedby a multiset of integers S . The objective is to decide whether G is S -fair , i.e., there existsan assignment of the contestants to the vertices of G such that the sum of the rankings of theneighbors of each contestant in G is the same constant k ∈ N . Here, k is called the S -fairnessconstant , or simply fairness constant if S is clear from the context, of G . Clearly, the aboveproblem is equivalent to having an r -regular graph G with n vertices, one for each ground,and edges connecting each vertex to r/ r/ G is S -fair where S is the multiset ofthe rankings of the contestants (see Figure 1). As the total difficulty for each contestant in thecompetition is the same, we refer to such a competition as a fair competition .In general, if we have the infrastructure of the competition (implicitly, like the above ex-ample, or explicitly) and we want to schedule a fair competition, we can model the problemas that of determining whether the graph representing the infrastructure of the competition is S -fair where S is the multiset of the rankings of the contestants. This situation is very fre-quently observed in on-line games or in other recurring competitions - in such competitions, theinfrastructure of the competition is fixed and the set of contestants keeps changing.Scheduling competitions and tournaments with different objectives is a well studied problemin the literature. There are, mainly, two fundamental competition designs, with all other designsconsidered as variations and hybrids. The first one is the elimination (or knockout) competition,in which the contestants are mapped to the leaf nodes of a complete binary tree. Contestantsmapped to nodes with same parent compete against each other in a match, and the winnerof the match moves up the tree. The contestant who reaches the root node is the winner ofthe tournament. The second one is the non-eliminating competition, in which no contestantis eliminated after one or few loses, and the winner is decided at the end of all the games byselecting the contestant with largest number of wins.In recent years, algorithmic perspectives of scheduling both kinds of competitions havereceived significant attention by the computational social choice community. We will first discusselimination competitions, followed by non-eliminating competitions. With respect to eliminationcompetitions, the design of a fair elimination competition under various definitions of being fairhas received notable attention [23, 38, 46, 47]. In this context, it is also relevant to mention the Tournament Fixing problem. Here, we are given n contestants, an encoding of the outcomeof each potential match between every two contestants as a digraph D , and a favorite contestant v : the goal is to design an elimination tournament so that v wins the tournament. This problemwas introduced by Vu et al. [45]. After this, it was extensively studied from both combinatorialand algorithmic (as well as parameterized) points of view [3, 20, 27, 28, 29, 34, 40, 41, 49].With respect to non-eliminating competitions, a round-robin tournament (RRT) is one ofthe most popular forms, in which each contestant plays every other contestant [37]. A wellstudied problem regarding RRTs is the Traveling Tournament problem, where the goal is1igure 1: Example of a 4 regular graph where every vertex is connected to 2 vertices on the leftand 2 on the right.to design a fair RRT by minimizing the total travel distance for every team [19, 21, 22, 30, 43, 50].Another related problem is to design a fair RRT by minimizing the number of “breaks” duringthe tournament [35, 44, 51]. Despite of being a popular non-eliminating competition, RRTshave some disadvantages. The first disadvantage of this format is the long tournament length,as each contestant plays against all other contestants. From the fairness point of view, a seconddisadvantage of this format, also mentioned in [37], is that it favors the strongest contestants(i.e., the contestants with the highest initial ranking). To see this, let R = { r , r , . . . , r n } bethe initial rankings of the n contestants, where r i is the initial ranking of contestant i , and let R be the total sum of the rankings. In RRT, the total difficulty faced by contestant i is R − r i ,which shows that the total difficulty faced by a contestant increases as we go from the strongestcontestants to the weakest contestants. In light of the above disadvantage, motivated by oneof the definitions proposed for fairness in [37, 38, 47], and based on a popular fairness conceptcalled envy-freeness introduced by Foley [13] in the study of fair division and allocation problemsin multi-agent systems (see, e.g. [5, 7, 8, 17]), we define a fair competition in an attempt toaddress both the above disadvantages with RRTs. Here, a fair competition is one where eachcontestant plays with a subset of all other contestants, yet the total difficulty for each contestantin the competition is the same. Similar to an envy-free division where no agent feels envy ofanother agent’s share, in a fair competition no contestant feels envy about another contestant’sschedule as the total difficulty for each contestant is the same.Apart from scheduling fair competitions, Fair-NET can be used to model other compu-tational problems in social choice. For example, suppose we have m candidates and n jobs,and every job is associated with an integer “reward”. Every candidate can choose r jobs andevery job is chosen by exactly one candidate. Now, we want to get an assignment of the jobsto the candidates such that the total reward collected by every candidate is k , for some integer k . Then, this is equivalent of having a graph G that is a collection of m stars, each having r leaves, with n total leaves, and the objective is to determine whether G is S -fair with thefairness constant k where S is the union of (i) the multiset of rewards and (ii) the multiset S (cid:48) = { k, . . . , k } containing the element k m times. Intuitively, S (cid:48) represents the multiset ofrewards collected by every candidate. Every star vertex corresponds to a candidate c , and itsleaves correspond to the jobs c is assigned to.The Fair-NET problem can also be used to design semi-magic and magic squares [48]defined as follows. A semi-magic square is an n × n grid ( n ≥
3) filled with positive integersfrom a multiset I such that each cell contains a distinct integer occurrence in I and the sum ofintegers in each row and each column is the same. A magic square is a semi-magic square withthe additional constraint that the sum of the integers in both the diagonals is also the same andequal to the sum of integers in each row and each column.We can model a semi-magic square (and similarly a magic square) as an instance of Fair- G corresponding to a 3 × NET as follows. Let G be a graph with a vertex for each cell in the grid and a vertex foreach row and each column, and edges between every cell vertex and its corresponding row andcolumn vertices (see Figure 2). Let the required sum be k and let S be the union of the multiset I and the multiset containing k − n times. The following observationshows that an n × n grid can have a semi-magic square filled with integers from the multiset I if and only if G is S -fair. Observation 1.1.
Given an n × n grid G ( n ≥ , and a multiset of n positive integers I , let k be the required sum of any semi-magic square on G , and G and S be the corresponding graphand the multiset respectively. Then G can have a semi-magic square, filled with integers fromthe multiset I if and only if G is S -fair with fairness constant k .Proof. First, assume that there exists a semi-magic square M on G filled with integers from themultiset I . Then, the sum of integers in each row and each column is k . We define an assignmentfrom V ( G ) to S as follows. Every cell vertex gets the same integer label as the integer it isfilled with in M . Clearly, the sum of neighbors for every row vertex and every column vertexin G is k . Every row is labeled k − k as it is adjacent to exactly one row vertex and exactly onecolumn vertex.Conversely, let G be S -fair. Then, for every vertex in G , the sum of the neighbors is k .From the construction of G , the degree of any row and any column vertex is n and the degreeof any cell vertex is 2. Let v be a vertex having a neighbor whose label is k −
1. As the sumof the neighbors of v is k and every integer in S is positive, the degree of v must be 2 and thelabel of the other neighbor of v is 1. This implies that v can only be a cell vertex, and all andonly the labels k − n row and n column vertices. So, the labels assignedto cell vertices belong to I . As for every row and column vertex, the sum of the neighbors is k , by filling every cell in G with the label of its corresponding cell vertex, we get a semi-magicsquare. To the best of our knowledge, while
Fair-NET has been studied extensively from a combinato-rial point of view (Section 1.2), close to nothing is known from an algorithmic point of view. Weinitiate a systematic algorithmic study of
Fair-NET . On the one hand, we show NP -hardnessresults on special graph classes, which imply para-NP -hardness for the problem with respectto several combinations of structural graph parameters. (For basic notions in parameterizedcomplexity, see Section 2). On the other hand, we show that the problem is fixed-parametertractable ( FPT ) for four different combinations of these parameters.3able 1: Summary of our results. Here ∆ , tw , fvs and vc denote the maximum degree, treewidth,feedback vertex set number and vertex cover number of the input graph, respectively; α denotesthe number of distinct elements in S . Note that tw ≤ fvs ≤ vc . Parameters Parameterized Complexity tw + ∆ NP -hard for tw = 3 , ∆ = 3 (also forregular graphs) [Theorem 3.1] α + ∆ NP -hard for α = 3 , ∆ = 6 (also forregular graphs) [Theorem 3.3] fvs + ∆ NP -hard for fvs = 0 , ∆ = 3 [Theo-rem 3.2] fvs + ∆ + α FPT [Theorem 4.1] fvs
FPT (for regular graphs) [Theo-rem 4.3] vc + α FPT [Theorem 4.2]The choice of our parameters is motivated by the real world examples from the introduction.In the example of fair competition, we may want every contestant to play only a fraction of thetotal possible games, which in turn means that the maximum degree ∆ of the infrastructuregraph is small compared to the total number of contestants. Similarly, it is likely to happenthat a lot of contestants have the same rankings or a lot of jobs have the same rewards, whichimplies that the number α of unique elements in S is small compared to the total number ofcontestants or jobs. In the case of job assignment, the underlying graph is a set of stars, whichhas treewidth 1 and the size of minimum feedback vertex set is 0. Moreover, treewidth, feedbackvertex set and vertex cover are central parameters in the field of parameterized complexity.Our main results are as follows (summarized in Table 1). First, we show that Fair-NET is NP -hard for three different graph classes: disjoint unions of K , ’s, disjoint unions of K , ’sand 6-regular graphs with 3 distinct labels. Consequently, it is para-NP -hard parameterized by(i) treewidth plus maximum degree, (ii) maximum degree plus feedback vertex set number, and(iii) maximum degree plus the number of distinct labels. The para-NP -hard results hold evenfor regular graphs when parameterized by either treewidth plus maximum degree or maximumdegree plus the number of distinct labels.Second, we show that Fair-NET is FPT parameterized by (i) maximum degree plus feedbackvertex set number plus the number of distinct labels, (ii) vertex cover number plus the numberof distinct labels, and (iii) feedback vertex set number for regular graphs. We derive some ofthese results by using insights into
Fair-NET itself when the input graph is a cycle, a disjointunion of stars, or a connected graph with minimum degree 1, and Integer Linear Programming.Our choice of parameters also shows several borders of (in-)tractability. For example, theproblem is para-NP -hard when parameterized by either ∆ + α or by fvs + ∆, but becomes FPT when parameterized by fvs + ∆ + α . Similarly, it is para-NP -hard by fvs + ∆, but becomes FPT by fvs for regular graphs. Overall, we give a comprehensive picture of the classicaland parameterized complexity of Fair-NET . For lack of space, some results and proofsmarked with an asterisk ( ∗ ) are omitted or sketched. They are included in the fullversion, which is available as a supplementary material. The
Fair-NET problem was first introduced by Vilfred [25] when S = { , , . . . , n } . Such alabeling is called sigma-labeling in that paper. The concept of fair scheduling was independently4tudied by Miller et al. [31] in 2003 under the name 1 -vertex magic and by Sugeng et al. [42] underthe name distance magic labeling . For recent surveys on distance magic labeling, see [1, 36].The Fair-NET problem for a general multiset S was first studied by O’Neal and Slater [32]. Inthe same paper, they also proved that if a graph G is S -fair, then the S -fairness constant of G is unique. In [39], Slater proved that Fair-NET is NP -hard. More recently, Godinho et al. [18]studied the special case of Fair-NET where S is a set and not a multiset. They gave a simplerproof for the uniqueness of S -fairness constant and also exhibited several families of S -fairgraphs. Recently, the same set of authors studied a measure called distance magic index relatedto S -fair labeling and determined the distance magic index of trees and complete bipartitegraphs in [2]. There has also been a long line of studies on other kinds of graph labeling, like { } - Fair-NET , where we consider the closed neighborhood of every vertex instead of openneighborhood (i.e., the vertex itself is also considered in its neighbor set). Another exampleis vertex-bimagic labeling , in which there exists two constants k and k such that the sum ofneighbors of every vertex is either k or k . For more information, see the recent survey [15]. Sets and Functions.
Given two multisets A = { a , a , . . . , a n } and B = { b , b , . . . , b m } ,their disjoint union is the multiset S = A (cid:93) B = { a , a , . . . , a n , b , b , . . . , b m } . For example, let A = { , , , , } and B = { , , , } . Then, A (cid:93) B = { , , , , , , , , } . Given a multiset S , α ( S ) denotes the number of distinct elements in S , and for every a ∈ S, α S ( a ) denotes thenumber of times a appear in S . Given a multiset A , (cid:80) A denotes the sum of its elements (incase they are integers), and | A | denotes its size. For any t ∈ N , [ t ] denotes the set { , , . . . , t } .Given a function f defined on a multiset A , f ( A ) = { f ( a ) : a ∈ A } . Let f : A → B be afunction from a multiset A to a multiset B . Then the restriction of f to a multiset A (cid:48) ⊆ A isthe function f | A (cid:48) : A (cid:48) → B given as f | A (cid:48) ( x ) = f ( x ) for every x ∈ A (cid:48) . Graphs.
In this paper, we consider only undirected graphs. Given a graph G , we denote itsvertex set and edge set by V ( G ) and E ( G ), respectively. For a vertex v ∈ V ( G ), the set of allthe neighbors of v in G is denoted by N G ( v ), i.e. N G ( v ) = { u ∈ V ( G ) | { u, v } ∈ E ( G ) } . Thedegree of a vertex v ∈ V ( G ) in G is denoted by deg G ( v ). When G is clear from the context,we drop the subscript. Given an induced subgraph H of G , the set of neighbors of vertices in H which are not in H is denoted by N G ( H ), i.e., N G ( H ) = (cid:0) (cid:83) v ∈ V ( H ) N G ( v ) (cid:1) \ V ( H ). Themaximum and minimum degree of G are denoted by ∆( G ) and δ ( G ), respectively. Given aset V (cid:48) ⊆ V ( G ), the subgraph of G induced by V (cid:48) is denoted by G [ V (cid:48) ]. A path on n verticesis denoted by P n . A cycle on n vertices is denoted by C n . A complete bipartite graph withbipartition A and B such that | A | = m, | B | = n is denoted by K m,n ( A, B ). If A and B areclear from the context, we write K m,n ( A, B ) as K m,n . Given a forest F , the set of leaves of F is denoted by leaves( F ). Given a rooted tree T , for a vertex v ∈ V ( T ), the set of children of v in T is denoted by children T ( v ).An r -regular graph is a regular graph where every vertex has degree r . An n -star graph (on n + 1-vertices) is the complete bipartite graph K ,n . Given an n -star graph where n ≥
2, the star-vertex is the unique vertex with degree n . The disjoint union of two graphs G and G ,denoted by G + G , is the graph with vertex set V ( G ) (cid:93) V ( G ) and edge set E ( G ) (cid:93) E ( G ).For any m ∈ N , we denote the disjoint union of m copies of a graph G by mG . Note that,the disjoint union of two or more nonempty graphs is always a disconnected graph. For otherstandard notations not explicitly defined here, we refer to the book [11].The treewidth, vertex cover number and feedback vertex set number of a graph G are definedas follows. 5 efinition 2.1 ( Treewidth). A tree decomposition of a graph G is a tree T whose nodes,called bags , are labeled by subsets of vertices of G . For each vertex v , the bags containing v must form a nonempty contiguous subtree of T , and for each edge { u, v } , at least one bag mustcontain both u and v . The width of the decomposition is one less than the maximum cardinalityof any bag, and the treewidth tw ( G ) of G is the minimum width of any of its tree decompositions. Based on the definition of treewidth, we have the following observation about disjoint union.
Observation 2.1.
The treewidth of the disjoint union of two vertex-disjoint graphs G and G is max { tw ( G ) , tw ( G ) } . Definition 2.2 ( Vertex Cover). A vertex cover of a graph G is a set of vertices in G suchthat every edge in G has at least one endpoint in the set. We denote the minimum size of avertex cover of G by vc ( G ) . Definition 2.3 ( Feedback Vertex Set). A feedback vertex set of a graph G is a set of verticeswhose removal results in an acyclic graph. We denote the minimum size of a feedback vertexset of G by fvs ( G ) . We will show hardness results from 3 -Partition and a variant of SAT called 3-XSAT (which were proved to be strongly NP -complete and NP -complete in [16] and [33], respectively),defined as follows. Definition 2.4 (3 -Partition). Given a multiset W of n = 3 m positive integers, for some m ∈ N , can W be partitioned into m triplets W , W , . . . , W m ( i.e., W = (cid:85) i ∈ [ m ] W i and | W i | = 3 for every i ∈ [ m ]) such that for every i ∈ [ m ] , (cid:80) W i = (cid:80) W/m ? Definition 2.5 (3 -XSAT ). Given a formula in conjunctive normal form (CNF) where allliterals are positive, each clause has size exactly , and each variable occurs exactly times,does there exist a truth assignment to the variables so that each clause has exactly one truevariable? The following lemma about a 3-XSAT formula will be useful throughout the paper. Lemma 2.1.
Let ρ be a -XSAT formula with n variables and m clauses. Then m = n .Moreover, ρ is satisfiable only if n is divisible by .Proof. Let A be the set of n vertices corresponding to n variables and B be the set of m verticescorresponding to m clauses. Consider the bipartite graph G with bipartition A and B and edgesbetween a vertex x ∈ A and a vertex c ∈ B if and only if the variable corresponding to x is inclause corresponding to c . As every variable appears in exactly 3 clauses and every clause hasexactly 3 variables, G is 3-regular. Because G is bipartite, the number of edges in G must beequal to both 3 n and 3 m which means that m = n .Now, assume that ρ is satisfiable, i.e. there exists a truth assignment such that every clausehas exactly one true variable. Let k be the number of true variables. Let v ans v be two verticesin A corresponding to two different true variables. Then N G ( v ) ∩ N G ( v ) = ∅ , otherwise thereis a clause c that has two true variables. Therefore, as G is 3-regular, total number of clausessatisfied is 3 k . As this number is also m (which equals n ), this means k = n/
3. So, ρ is satisfiableonly if n is divisible by 3.From the above lemma, it is easy to see that 3-XSAT remains NP -complete even when n is divisible by 3. So, in the rest of the paper, we assume that given a 3-XSAT formula with n variables and m clauses, m = n and n is divisible by 3.6 air Non-Eliminating Tournament. Given an infrastructure of a tournament representedby a graph G and the initial rankings of the contestants represented by a multiset of integers S , G is called S -fair if there exists an assignment of the contestants to the vertices of G such that thesum of the rankings of the neighbors of each contestant in G is the same. Equivalently, given theinfrastructure graph G and the multiset of contestants’ rankings S with | S | = | V ( G ) | , G is S -fair if there exists a bijection f : V ( G ) → S such that for every vertex v ∈ V ( G ) , (cid:80) f ( N ( v )) = k ,where k is a constant called S -fairness constant .For any vertex v ∈ V ( G ) , f ( v ) is called the label of v . We denote the set of all bijectivefunctions that satisfy the above property by M ( G, S ). In [32], O’Neal and Slater showed thatif a graph G is S -fair, then its S -fairness constant is unique.Since the infrastructure graph of a tournament is an undirected graph and the initial rankingof several players can be the same, we define the Fair Non-Eliminating Tournament ( Fair-NET ) problem as follows.
Definition 2.6 ( Fair-NET Problem).
Given an undirected graph G and a multiset of positiveintegers S with | S | = | V ( G ) | , is G S -fair ( i.e. M ( G, S ) (cid:54) = ∅ ) ? The following observations about S -fair graphs follow directly from its definition. Observation 2.2 ( Label Swap).
Let G be an S -fair graph. Let f ∈ M ( G, S ) . Let u, v ∈ V ( G ) such that N G ( u ) = N G ( v ) . Consider f (cid:48) : V ( G ) → S , defined as follows. For all w ∈ V ( G ) \{ u, v } , f (cid:48) ( w ) = f ( w ); f (cid:48) ( u ) = f ( v ); f (cid:48) ( v ) = f ( u ) . Then f (cid:48) ∈ M ( G, S ) . Observation 2.3.
Let G be an S -fair graph. Let u, v ∈ V ( G ) such that N G ( u ) = N G ( v ) and { u, v } ∈ E ( G ) . Then, for all f ∈ M ( G, S ) , f ( u ) = f ( v ) . Observation 2.4.
Let G be an r -regular S -fair graph. Then the S -fairness constant is equal to r · (cid:80) S/ | V ( G ) | . Observation 2.5.
Given two graphs G and G and a multiset of positive integers S , G + G is S -fair if and only if G is S -fair and G is S -fair with the same fairness constant for some S , S ⊆ S such that S (cid:93) S = S . Observation 2.6.
Given a complete bipartite graph K m,n ( A, B ) and a multiset of positiveintegers S , K m,n is S -fair if and only if there exists a bijection f : V ( K m,n ) → S such that (cid:80) f ( A ) = (cid:80) f ( B ) = (cid:80) S/ . Integer Linear Programming.
In the
Integer Linear Programming Feasibility (ILP)problem, the input consists of p variables x , x , . . . , x p and a set of m inequalities of the fol-lowing form: a , x + a , x + · · · + a ,p x p ≤ y a , x + a , x + · · · + a ,p x p ≤ y ... ... ... ... a m, x + a m, x + · · · + a m,p x p ≤ y m where every coefficient a i j and y i is required to be an integer. The task is to check whetherthere exists an assignment of integer values for every variable x i so that all inequalities aresatisfiable. The following theorem about the tractability of the ILP problem will be usefulthroughout Section 4. Theorem 2.1 ([26, 24, 14]) . The ILP problem with p variables is FPT parameterized by p . arameterized Complexity. A problem Π is a parameterized problem if each problem in-stance of Π is associated with a parameter k . For simplicity, we denote a problem instance of aparameterized problem Π as a pair ( I, k ) where the second argument is the parameter k asso-ciated with I . The main objective of the framework of Parameterized Complexity is to confinethe combinatorial explosion in the running time of an algorithm for an NP -hard parameterizedproblem Π to depend only on k . In particular, a parameterized problem Π is fixed-parametertractable ( FPT ) if any instance (
I, k ) of Π is solvable in time f ( k ) ·| I | O (1) , where f is an arbitrarycomputable function of k . Moreover, a parameterized problem Π is para-NP -hard if it is NP -hard for some fixed constant value of the parameter k . For more information on ParameterizedComplexity, we refer the reader to books such as [12, 10]. Para-NP -hardness Results
In this section, we exhibit the para-NP -hardness of the
Fair-NET problem with respect toseveral structural graph parameters. We start with a para-NP -hardness result with respect tothe parameter tw + ∆. Theorem 3.1.
The
Fair-NET problem is NP -hard for -regular graphs with treewidth . Inparticular, it is para-NP -hard parameterized by tw + ∆ , even for regular graphs.Proof. We present a reduction from 3 -Partition . Given a multiset W of n = 3 m positiveintegers, for some m ∈ N , we create two instances of Fair-NET based on the value of m asfollows (see Figure 3). Case 1 [When m is a multiple of ]: In this case, we create an instance (
G, S ) of
Fair-NET where G = ( m/ K , and S = W . Note that G is a 3-regular graph. Since tw ( K , ) = 3, byObservation 2.1, tw ( G ) = 3. Let V ( G ) = (cid:85) i ∈ [ m/ V i where V i = A i ∪ B i is the vertex set ofthe i -th copy of K , with bipartition A i and B i . We now prove that W is a Yes instance of3 -Partition if and only if G is S -fair.Assume first that W is a Yes instance of 3 -Partition . Let W , W , . . . , W m be a corre-sponding partition of W . Then, by Definition 2.4, for every i ∈ [ m ] , (cid:80) W i = (cid:80) W/m . Let f : V ( G ) → S be a bijective function defined as follows. For every i ∈ [ m/ , let f ( A i ) = W i and f ( B i ) = W m/ i (the internal labeling within A i and B i is arbitrary). So, for every i ∈ [ m/ , (cid:80) f ( A i ) = (cid:80) f ( B i ) = (cid:80) W/m . Thus, by Observations 2.5 and 2.6, G = ( m/ K , is S -fair.Conversely, let G = ( m/ K , be S -fair. Then, by Observations 2.5 and 2.6, there existsa bijection f : V ( G ) → S such that for every i ∈ [ m/ , (cid:80) f ( A i ) = (cid:80) f ( B i ) = (cid:80) S/m = (cid:80) W/m . Thus, { f ( A ) , f ( A ) , . . . , f ( A m/ ) , f ( B ) , . . . , f ( B m/ ) } is a partition of W satisfyingthe required property, so W is a Yes instance of 3 -Partition . Case 2 [When m is not a multiple of ]: Without loss of generality, we can assume thatevery element in W is greater than 1 as otherwise we can get an equivalent instance of 3 -Partition by adding 1 to all the elements of W . Let sum = (cid:80) W/m be the required sum ofevery subset. As every element in W is greater than 1, sum ≥
6. In this case, we create aninstance (
G, S ) of
Fair-NET where G = (cid:0) ( m + 1) / (cid:1) K , and S = W (cid:93) { sum − , , } . Notethat G is a 3-regular graph and tw ( G ) = 3. Let V ( G ) = (cid:85) i ∈ [ m +1 / V i where V i = A i ∪ B i is thevertex set of the i -th copy of K , with bipartition A i and B i . We now prove that W is a Yes instance of 3 -Partition if and only if G is S -fair.Assume first that W is a Yes instance of 3 -Partition . Let W , W , . . . , W m be the corre-sponding partition of W . Then, by Definition 2.4, for every i ∈ [ m ] , (cid:80) W i = (cid:80) W/m = sum .Let W m +1 = { sum − , , } . Clearly, (cid:80) W m +1 = sum . As S = W (cid:85) { sum − , , } , S = (cid:85) i ∈ [ m +1] W i . Let f : V ( G ) → S be a bijective function defined as follows. For every8 B A B A t B t Figure 3: Example of the graph G built in the reduction of Theorem 3.1. t = m/ t = ( m + 1) / i ∈ [( m + 1) / , let f ( A i ) = W i and f ( B i ) = W ( m +1) / i (the internal labeling within A i and B i is arbitrary). So, for every i ∈ [( m + 1) / , (cid:80) f ( A i ) = (cid:80) f ( B i ) = sum . Thus, byObservations 2.5 and 2.6, G = (cid:0) ( m + 1) / (cid:1) K , is S -fair.Conversely, let G = ( m + 1) / K , be S -fair. Then, by Observations 2.5 and 2.6, thereexists a bijection f : V ( G ) → S such that for every i ∈ [( m + 1) / , (cid:80) f ( A i ) = (cid:80) f ( B i ) = (cid:80) S/ ( m + 1) = (cid:80) W/m . Without loss of generality, let A s be the set containing sum −
2, for some s ∈ [( m + 1) / (cid:80) A s = sum, | A s | = 3 and all the elements in W aregreater than 1, necessarily A s = { sum − , , } . As S = W (cid:85) { sum − , , } , we get that { f ( A ) , . . . , f ( A s − ) , f ( A s +1 ) , . . . , f ( A ( m +1) / ) , f ( B ) , . . . , f ( B ( m +1) / ) } is a partition of W sat-isfying the required property, so W is a Yes instance of 3 -Partition .We now proceed with the para-NP -hardness result with parameter fvs + ∆.
Theorem 3.2.
The
Fair-NET problem is NP -hard for forests with ∆ = 3 . Since forests have fvs = 0 , Fair-NET is para-NP -hard parameterized by fvs + ∆ .Proof. We present a simple reduction from 3 -Partition . Given a multiset W of n = 3 m positive integers, for some m ∈ N , let sum = (cid:80) W/m be the required sum of every subset.We create an instances (
G, S ) of
Fair-NET where G = mK , and S = W (cid:85) { s = sum, s = sum, . . . , s m = sum } . Note that G is a forest with ∆( G ) = 3. Let V ( G ) = (cid:85) i ∈ [ m ] V i where V i = { v i } ∪ B i is the vertex set of the i -th copy of K , with B i being the set of leaves. SeeFigure 4. We now prove that W is a Yes instance of 3 -Partition if and only if G is S -fair.Assume first that W is a Yes instance of 3 -Partition . Let W , W , . . . , W m be the cor-responding partition of W . Then, by Definition 2.4, for every i ∈ [ m ] , (cid:80) W i = sum . Let f : V ( G ) → S be a bijective function defined as follows. For every i ∈ [ m ], let f ( B i ) = W i and f ( v i ) = sum (the internal labeling of B i is arbitrary). So, for every i ∈ [ m ] , f ( v i ) = (cid:80) f ( B i ) = sum . Thus, by Observations 2.5 and 2.6, G = mK , is S -fair.Conversely, let G = mK , be S -fair. Then, by Observations 2.5 and 2.6, there exists abijection f : V ( G ) → S such that for every i ∈ [ m ] , f ( v i ) = (cid:80) f ( B i ) = (cid:80) S/ ( m + 1) = sum . As S = W (cid:85) { s = sum, s = sum, . . . , s m = sum } , we get that { f ( B ) , . . . , f ( B m ) } is a partitionof W satisfying the required property, so W is a Yes instance of 3 -Partition .Finally, we give the para-NP -hardness result with parameter α + ∆. (Recall that α is thenumber of distinct integers in the input multiset.) Theorem 3.3.
The
Fair-NET problem in NP -hard for -regular graphs with distinct labels.In particular, it is para-NP -hard parameterized by α + ∆ , even for regular graphs.Proof. We present a reduction from 3-XSAT . Given a 3-XSAT formula ρ with n variablesand n clauses, we create an instance ( G, S ) of
Fair-NET as follows. Suppose that the variablesare indexed by 1 , , . . . , n and so do the clauses. For every i ∈ [ n ], the variable gadget in G v v m B B B m Figure 4: Example of the graph G built in the reduction of Theorem 3.2.consists of a single vertex x i called a variable vertex. Let A be the set of all the variable vertices.For every i ∈ [ n ], the clause gadget in G consists of 15 vertices c i , c i , . . . , c i . For every i ∈ [ n ],we add the following edges between these 15 vertices in the clause gadget in G (see Figure 5): • ∀ j ∈ [3] , { c i , c ji } . [Edge set of K , ( { c i } , { c i , c i , c i } )]. • ∀ j ∈ [3] , { c i , c ji } . [Edge set of K , ( { c i } , { c i , c i , c i } )]. • { c i , c i } , { c i , c i } , { c i , c i } . [Edge set of complete graph on { c i , c i , c i } ]. • { c i , c i } , { c i , c i } , { c i , c i } . [Edge set of complete graph on { c i , c i , c i } ]. • ∀ j ∈ { , , } , ∀ k ∈ { , , } , { c ji , c ki } . [Edge set of K , ( { c i , c i , c i } , { c i , c i , c i } )]. • ∀ j ∈ { , , } , ∀ k ∈ { , , } , { c ji , c ki } . [Edge set of K , ( { c i , c i , c i } , { c i , c i , c i } )]. • ∀ j ∈ { , , , , , } , { c i , c ji } . [Edge set of K , ( { c i } , { c i , c i , c i , c i , c i , c i } )].We now explain how we connect the variable and the clause gadgets. For every variable vertex x i , let j, k and l be the indices of the clauses where the i -th variable appears. Then, we addthe 6 edges { x i , c j } , { x i , c j } , { x i , c k } , { x i , c k } , { x i , c l } and { x i , c l } to G . This completes theconstruction of G . Note that | V ( G ) | = n + 15 n = 16 n . We now define S as the multisetcontaining 3 distinct labels 1 , α S (1) = 2 n/ , α S (2) = 15 n and α S (4) = n/
3. Fromthe above construction, it is easy to see that G is a 6-regular graph and S contains 3 distinctlabels. By Observation 2.4, the S -fairness constant k = 12.In what follows, we will set a variable to true if and only if the label of the correspondingvariable vertex is 4 and false otherwise. We now prove that ρ is a satisfiable if and only if G is S -fair.Assume first that ρ is satisfiable. Let A (cid:48) be the subset of variable vertices for which thecorresponding variables are true. From Lemma 2.1, | A (cid:48) | = n/
3. Let f : V ( G ) → S be abijective function defined as follows: (i) for all v ∈ A (cid:48) , f ( v ) = 4; (ii) for all v ∈ A \ A (cid:48) , f ( v ) = 1;(iii) for all v ∈ V ( G ) \ A, f ( v ) = 2. For every i ∈ [ n ], let B be the set containing verticesthe c i and c i . From the construction of G , only vertices from B have neighbors in A , so forall v ∈ V ( G ) \ B, f ( N G ( v )) = { , , , , , } . As every clause has exactly one true variableand two false variables, for every vertex v ∈ B, f ( N G ( v )) = { , , , , , } . So, for all v ∈ V ( G ) , (cid:80) f ( N G ( v )) = 12. Hence, G is S -fair.Conversely, let G be S -fair. Then, there exists a bijection f : V ( G ) → S such that forall v ∈ V ( G ) , (cid:80) f ( N G ( v )) = 12. Note that, for every i ∈ [ n ] , N G ( c i ) = N G ( c i ) = N G ( c i )and { c i , c i } , { c i , c i } , { c i , c i } ∈ E ( G ), so by Observation 2.3, f ( c i ) = f ( c i ) = f ( c i ). Similarly, f ( c i ) = f ( c i ) = f ( c i ), f ( c i ) = f ( c i ) = f ( c i ) and f ( c i ) = f ( c i ) = f ( c i ).Consider the vertex c i , for any i ∈ [ n ]. As (cid:80) f ( N G ( c i )) = 12 and N G ( c i ) = { c i , c i , c i , c i ,c i , c i } , we have that 3 f ( c i ) + 3 f ( c i ) = 12 (by the equalities above). Therefore, f ( c i ) +10 j x k x l c i c i c i c i c i c i c i c i c i c i c i c i c i c i c i Figure 5: Example of the clause gadget in G for a clause c i = x j ∨ x k ∨ x l built in the reductionof Theorem 3.3. f ( c i ) = 4 which necessarily implies that f ( c i ) = f ( c ) = 2. Now, consider the vertex c i .As (cid:80) f ( N G ( c i )) = 12 and N G ( c i ) = { c i , c i , c i , c i , c i , c i } , we have that 3 f ( c i ) + f ( c i ) = 8(by equalities above). Therefore, necessarily f ( c i ) = f ( c i ) = 2. Similarly, f ( c i ) = 2. Now,consider the vertex c i . As (cid:80) f ( N G ( c i )) = 12 and N G ( c i ) = { c i , c i , c i , c i , c i , c i } , necessarily f ( c i ) = 2. Similarly, f ( c i ) = 2.So far, we conclude that all and only the occurrences of integer 2 in S are used to label thevertices of the clause gadgets. Finally, consider the vertex c i . Let c i is adjacent to variablevertices x j , x k and x l . As (cid:80) f ( N G ( c i )) = 12 and N G ( c i ) = { c i , c i , c i , x j , x k , x l } , we get that f ( x j ) + f ( x k ) + f ( x l ) = 6. The only solution to this equation is { , , } as the remaining labelsare 4 and 1. Without loss of generality, let f ( x j ) = 4 , f ( x k ) = f ( x l ) = 1. Recall that, we seta variable to true if and only if the label of the corresponding variable vertex is 4 and falseotherwise, so we assign variable corresponding to x j as true and variables corresponding to x k and x l as false. It is easy to see that a clause has exactly one true variable. As the numberof times 4 appear in S is n/ n . Hence ρ is satisfiable. In this section, we develop
FPT algorithms for the
Fair-NET problem with respect to severalstructural graph parameters. We begin by giving a observation for a graph G to be S -fair when G contains isolated vertices. Observation 4.1.
Let G be a graph with δ ( G ) = 0 and S be a multiset of positive integers.Then, G is S -fair if and only if for every vertex v ∈ V ( G ) , deg G ( v ) = 0 (i.e., G contains onlyisolated vertices). Due to the above observation, in the rest of this section, we assume that G does not containany isolated vertices. We now give conditions that a graph G must satisfy to be S -fair when G is a cycle or δ ( G ) = 1. Lemma 4.1.
Let G be a connected graph with δ ( G ) = 1 and S be a multiset of positive integers.Then, G is S -fair only if G is a star.Proof. Assume that G is S -fair. Let f be a function in M ( G, S ) and k be the S -fairness constant.Let v ∈ V ( G ) be a vertex of degree 1, and let u be the neighbor of v . As (cid:80) f ( N G ( v )) = k , we11et that f ( u ) = k . Assume for contradiction that G is not a star. Then, | V ( G ) | ≥
4, otherwise G is a star. Let w be a neighbor of u other than v such that deg G ( w ) ≥
2. (If no such w exists,then G is a star.) As (cid:80) f ( N G ( w )) = k and f ( u ) = k , we get that (cid:80) f ( N G ( w ) \ { u } ) = 0. Since N G ( w ) \ { u } (cid:54) = ∅ and labels are positive, this is a contradiction. Lemma 4.2.
Let G be a cycle graph on n vertices and S be a multiset of positive integers. Let k be the required S -fairness constant. Then: • If n mod 4 = 0 , then G is S -fair if and only if S contains labels a, b, k − a, k − b with α S ( a ) = α S ( b ) = α S ( k − a ) = α S ( k − b ) = n/ , for some a, b ∈ N such that a, b < k . • If n mod 4 (cid:54) = 0 , then G is S -fair if and only if S contains only one label, k/ , with α S ( k/
2) = n .Proof. Assume first that G is S -fair. Denote V ( G ) = { v , v , . . . , v n } in the cyclic order, andlet f ∈ M ( G, S ). Then, for every v ∈ V ( G ) , (cid:80) f ( N ( v )) = k . As G is a cycle, for every i ∈ [ n ] , N ( v i ) = { v ( i −
1) mod n , v ( i +1) mod n } . Thus, we have that for every i ∈ [ n ] , f ( v ( i −
1) mod n )+ f ( v ( i +1) mod n ) = k . If we expand these equations, we get the following: f ( v ) + f ( v ) = f ( v ) + f ( v ) , . . . , f ( v n − ) + f ( v n − ) = f ( v n − ) + f ( v ) . (1) f ( v ) + f ( v ) = f ( v ) + f ( v ) , . . . , f ( v n − ) + f ( v n ) = f ( v n ) + f ( v ) . (2) f ( v ) + f ( v ) = f ( v ) + f ( v ) , . . . , f ( v n − ) + f ( v ) = f ( v ) + f ( v ) . (3) f ( v ) + f ( v ) = f ( v ) + f ( v ) , . . . , f ( v n ) + f ( v ) = f ( v ) + f ( v ) . (4)From these equations, we get the following relations: f ( v ) = f ( v ) = f ( v ) = . . . = f ( v n − ) (5) f ( v ) = f ( v ) = f ( v ) = . . . = f ( v n − ) (6) f ( v ) = f ( v ) = f ( v ) = . . . = f ( v n − ) (7) f ( v ) = f ( v ) = f ( v ) = . . . = f ( v n ) (8)Notice that Equations 5, 6, 7 and 8 contain all the vertices v i , i ∈ [ n ], such that i mod 4 = 1, i mod 4 = 2 , i mod 4 = 3 and i mod 4 = 0, respectively. Now, we consider the following cases: • If n mod 4 = 1, then ( n −
3) mod 4 = 2 , ( n −
2) mod 4 = 3 , ( n −
1) mod 4 = 0 and n mod 4 = 1. By the observation that Equation 6 contains all the vertices v i , i ∈ [ n ],such that i mod 4 = 2, f ( v n − ) = f ( v ). Similarly, f ( v n − ) = f ( v ) , f ( v n − ) = f ( v ) and f ( v n ) = f ( v ). So, by Equations 5, 6, 7 and 8, we get that all the vertices have the samelabel. Thus, S contains only one label, k/
2, with α S ( k/
2) = n . • If n mod 4 = 2, then ( n −
3) mod 4 = 3 , ( n −
2) mod 4 = 0 , ( n −
1) mod 4 = 1 and n mod 4 = 2. By the observation that Equation 6 contains all the vertices v i , i ∈ [ n ], suchthat i mod 4 = 2, f ( v n ) = f ( v ). Similarly, f ( v n − ) = f ( v ). So, by Equations 5, 6, 7and 8, we get that f ( v ) = f ( v ) and f ( v ) = f ( v ). Denote f ( v ) = a and f ( v ) = b . As f ( v ) + f ( v ) = k and f ( v ) + f ( v ) = k , we get that a = b = k/
2. Thus, S contains onlyone label, k/
2, with α S ( k/
2) = n . • If n mod 4 = 3, then ( n −
3) mod 4 = 0 , ( n −
2) mod 4 = 1 , ( n −
1) mod 4 = 2 and n mod 4 = 3. By the observation that Equation 6 contains all the vertices v i , i ∈ [ n ],such that i mod 4 = 2, f ( v n − ) = f ( v ). Similarly, f ( v n − ) = f ( v ) , f ( v n − ) = f ( v ) and f ( v n ) = f ( v ). So, by Equations 5, 6, 7 and 8, we get that all the vertices have the samelabel. Thus, S contains only one label, k/
2, with α S ( k/
2) = n .12 If n mod 4 = 0, then ( n −
3) mod 4 = 1 , ( n −
2) mod 4 = 2 , ( n −
1) mod 4 = 3 and n mod 4 = 0, so we do not get any new relation. Denote f ( v ) = a and f ( v ) = b . As f ( v ) + f ( v ) = k and f ( v ) + f ( v ) = k , we get f ( v ) = k − a and f ( v ) = k − b .By Equations 5, 6, 7 and 8, S contains 4 labels a, b, k − a, k − b with α S ( a ) = α S ( b ) = α S ( k − a ) = α S ( k − b ) = n/ n mod 4 = 0 and S contains 4 labels a, b, k − a, k − b with α S ( a ) = α S ( b ) = α S ( k − a ) = α S ( k − b ) = n/
4, for any a, b ∈ N such that a, b < k . Denote V ( G ) = { v , v , . . . , v n } in the cyclic order. Let f : V ( G ) → S be a bijective function defined as follows.For every i ∈ [ n/ , let f ( v i − ) = a, f ( v i − ) = b, f ( v i − ) = k − a and f ( v i ) = k − b . It iseasy to see that f ∈ M ( G, S ), so G is S -fair.Finally, assume that n mod 4 (cid:54) = 0 and S contains only one label, k/
2, with α S ( k/
2) = n .Let f : V ( G ) → S be a bijective function defined as follows. For every v ∈ V ( G ) , f ( v ) = k/ f ∈ M ( G, S ), so G is S -fair.We first prove that the Fair-NET problem is
FPT parameterized by fvs + α + ∆. Thefollowing two lemmas will be helpful in proving it. Lemma 4.3.
There exists an O ( | V ( G ) | ) -time algorithm that, given (i) a graph G , (ii) a multisetof positive integers S , (iii) an induced subgraph F of G such that its a forest, and (iv) a bijection f (cid:48) from N G ( F ) ∪ leaves( F ) to a subset S (cid:48) of S , returns another bijection f (cid:48)(cid:48) from N G ( F ) ∪ V ( F ) toa set S (cid:48)(cid:48) such that S (cid:48) ⊆ S (cid:48)(cid:48) and f (cid:48) = f (cid:48)(cid:48) | V (cid:48) . Moreover, if G is S -fair and f (cid:48) = f | N G ( F ) ∪ leaves( F ) for some f ∈ M ( G, S ) , then f (cid:48)(cid:48) = f | N G ( F ) ∪ V ( F ) .Proof. Let (
G, k ) be an instance of
Fair-NET . Let k be the required S -fairness constant. Let F = { T , T , . . . , T t } be the set of connected components of F . For every tree T ∈ F , wedo the following. Let r be an arbitrarily chosen non-leaf vertex of T . Then, consider T asa rooted tree with r as the root vertex. Let d be the depth of the tree T . We partition thevertex set V ( T ) = V ∪ V . . . ∪ V d , such that V i contains all the vertices of T at depth i .Note that V = { r } and V d ⊆ leaves( T ). Now, consider a vertex v (cid:54) = r in T . We partition N G ( v ) = { p v } ∪ (cid:0) children T ( v ) ∩ leaves( T ) (cid:1) ∪ (cid:0) children T ( v ) \ leaves( T ) (cid:1) ∪ (cid:0) N G ( v ) \ V ( T ) (cid:1) , where p v is the parent of v in T , children T ( v ) ∩ leaves( T ) is the set of children of v in T which areleaves of T , children T ( v ) \ leaves( T ) is the set of children of v in T which are non-leaf verticesof T and N G ( v ) \ T is the set of neighbors of v not in T .Given f (cid:48) , we define another function f T on V (cid:48) = V ( T ) \ leaves( T ) recursively as follows.(i) Base Case:
For all v ∈ V (cid:48) such that v has a leaf child, let w be an arbitrarily chosen leafchild of v . Then, f T ( v ) = k − (cid:80) f (cid:48) ( N G ( w ) \ V ( T )).(i) Recursive Step:
For all v ∈ V (cid:48) such that v doesn’t have any leaf child, let w be an arbitrar-ily chosen child of v . Then, f T ( v ) = k − (cid:80) f T (children T ( w ) \ leaves( T )) − (cid:80) f (cid:48) (children T ( w ) ∩ leaves( T )) − (cid:80) f (cid:48) ( N G ( w ) \ T ).Note that, if v ∈ V i , then children T ( v ) ∈ V i +1 . So, we compute f T by processing vertices of T in the order V d − , . . . , V . We now define f (cid:48)(cid:48) from N G ( F ) ∪ V ( F ) to S (cid:48)(cid:48) = S (cid:48) ∪ f T ∪ f T . . . f T t as follows.(i) For every v ∈ N G ( F ) ∪ L, f (cid:48)(cid:48) ( v ) = f (cid:48) ( v ).(i) For every i ∈ [ t ] and v ∈ V ( T i ) \ leaves( T i ) , f (cid:48)(cid:48) ( v ) = f T i ( v ). S (cid:48)(cid:48) may not be a subset of S . f (cid:48) = f (cid:48)(cid:48) | V (cid:48) and therefore S (cid:48) ⊆ S (cid:48)(cid:48) . The above recursive procedure visits every vertex of G at most once, so it runs in time O ( | V ( G ) | ).Now, suppose that G is an S -fair graph and f (cid:48) = f | N G ( F ) ∪ leaves( F ) for some f ∈ M ( G, S ).As f ∈ M ( G, S ), for every T ∈ F and v ∈ V ( T ) , (cid:80) f ( N ( v )) = k . Consider a tree T ∈ F .Let v be a non-leaf vertex of T . Then, for all w ∈ children T ( v ) , (cid:80) f ( N G ( w )) = k ⇒ f ( v ) + (cid:80) f (children T ( w ) \ leaves( T )) − (cid:80) f (children T ( w ) ∩ leaves( T )) − (cid:80) f ( N G ( w ) \ V ( T )) = k .As f (cid:48) = f | N G ( F ) ∪ leaves( F ) , for all v ∈ V ( T ) \ leaves( T ) and w ∈ children T ( v ) , f ( v ) = k − (cid:80) f (children T ( w ) \ leaves( T )) − (cid:80) f (cid:48) (children T ( w ) ∩ leaves( T )) − (cid:80) f (cid:48) ( N G ( w ) \ V ( T )). If w isa leaf node, then children T ( w ) = ∅ . So, we can write f | N G ( F ) ∪ V ( F ) as follows.(i) For every v ∈ N G ( F ) ∪ L, f ( v ) = f (cid:48) ( v ).(i) For every i ∈ [ t ] and v ∈ V ( T i ) \ leaves( T i ), – if v has a leaf child w , then f ( v ) = k − (cid:80) f (cid:48) ( N G ( w ) \ V ( T )). – else, let w be a child of v , then f ( v ) = k − (cid:80) f (children T ( w ) \ leaves( T )) − (cid:80) f (cid:48) (children T ( w ) ∩ leaves( T )) − (cid:80) f (cid:48) ( N G ( w ) \ V ( T )).As f (cid:48)(cid:48) and f | N G ( F ) ∪ V ( F ) have same base case and recursive step, f (cid:48)(cid:48) = f | N G ( F ) ∪ V ( F ) . This alsoimplies that S (cid:48)(cid:48) ⊆ S .The following corollary directly follows from the above lemma. Corollary 4.1.
Let G be a graph and S be multiset of positive integers. Let F be an inducedsubgraph of G that is a forest. Then, in time O ( α ( S ) | N G ( F ) | + | leaves( F ) | · | V ( G ) | ) , we can computea superset of the set G of functions from N G ( F ) ∪ V ( F ) to S such that for every f ∈ M ( G, S ) ,there exists a g ∈ G such that g = f | N G ( F ) ∪ V ( F ) .Proof. As the number of different functions f (cid:48) from N G ( F ) ∪ leaves( F ) to S is O ( α ( S ) | N G ( F ) | + | leaves( F ) | ),we can compute a superset of G by repeatedly applying Lemma 4.3 for each such f (cid:48) and adding f (cid:48)(cid:48) to G if S (cid:48)(cid:48) ⊆ S . Lemma 4.4.
The
Fair-NET problem is
FPT parameterized by α + ∆ for disjoint union ofstars.Proof. The
FPT algorithm is based on ILP. We first give the algorithm and then prove its cor-rectness.
Algorithm:
Let (
G, k ) be an instance of
Fair-NET for disjoint union of stars. Thus, G = K ,n + K ,n + . . . + K ,n t for some t, n , n , . . . , n t ∈ N . Note that ∆ = ∆( G ) = max( n , n , . . . , n t ).Denote V ( K ,n i ) = { v i } ∪ B i where v i is the highest degree vertex in K ,n i and B i is the set ofall other vertices in K ,n i , for every i ∈ [ t ]. Let k be the required S -fairness constant of G . ByObservations 2.5 and 2.6, G is S -fair if and only if there exists a bijection f : V ( G ) → S such that for all i ∈ [ t ] , f ( v i ) = (cid:80) f ( B i ) = k . So, G is S -fair only if α S ( k ) ≥ t . Let S (cid:48) = S \ { s = k, s = k, . . . , s t = k } and let (cid:98) α = α ( S (cid:48) ). Let (cid:96) , (cid:96) , . . . , (cid:96) (cid:98) α be the uniquelabels in S (cid:48) . Let D = { B , B , . . . , B t } . We partition D into D , D , . . . , D ∆ such that for every i ∈ [∆], every set B ∈ D i is of size i . As the number of unique labels are (cid:98) α , for every i ∈ [∆],any set in D i can have at most (cid:98) α i different label assignments. For every i ∈ [∆] , let L i be theset of feasible label assignments for D i , i.e., the label assignments for any set in D i for which thesum of the labels is k . For every i ∈ [∆], every label assignment la ∈ L i is a set { t la , t la , . . . , t (cid:98) αla } ,where, for every j ∈ [ (cid:98) α ], t jla denoted the number of times label (cid:96) j is used in the label assignment la . For every i ∈ [∆] and la ∈ L i , we have a variable n i,la . For any i ∈ [∆], la ∈ L i and afunction f ∈ M ( G, S ), n i,la represents the number of times label assignment la is used in D i for f . Then, the algorithm works as follows. 14 If α S ( k ) < t , then return False . • Solve the following ILP to find n i,la , for every i ∈ [∆] and la ∈ L i . ∀ i ∈ [∆] , (cid:88) la ∈L i n i,la = |D i | . (9) ∀ j ∈ [ (cid:98) α ] , (cid:88) i ∈ [∆] (cid:88) la ∈L i n i,la · t jla = α S (cid:48) ( (cid:96) j ) . (10) ∀ i ∈ [∆] , ∀ la ∈ L i ; n i,la ≥ . (11) • If the ILP returns a feasible solution, then return
True ; otherwise, return
False . Correctness:
Equation 9 ensures that for every i ∈ [∆], the number of label assignments usedin D i is equal to the number of sets D i has. Equation 10 ensures that for every j ∈ [ (cid:98) α ], thenumber of times label (cid:96) j is used is equal to the number of times it appears in S (cid:48) . For every i ∈ [∆] and for every B ∈ D i , all the vertices in B have the same neighborhood which is just asingle vertex. Thus by Observation 2.2, it is sufficient to know the label assignment for B ; wecan arbitrarily assign labels to the vertices in B once we have decided which labels to use for B .Keeping this interpretation in mind, we now prove the correctness, i.e. the algorithm returns True if and only if G is S -fair.Assume first that the algorithm returns True . It means the ILP assigned non-negativeinteger values for the variables n i,la , i ∈ [∆] and la ∈ L i such that Equations 9 and 10 aresatisfied. As for every i ∈ [∆], L i is the set of feasible label assignments, this implies that wegot a label assignment for every B ∈ D such that sum of the labels of the vertices in B is k .Thus, G is S -fair.Conversely, let G be S -fair. Then, by Observations 2.5 and 2.6, there exists a bijection f : V ( G ) → S such that for all i ∈ [ t ] , f ( v i ) = (cid:80) f ( B i ) = k . So, α S ( k ) ≥ t . Moreover, every B ∈ D has a feasible label assignment so the ILP admits a feasible solution. Thus, the algorithmwill return True . As the number of variables n i,la is O (∆ · (cid:98) α ∆ ), by Theorem 2.1, the Fair-NET problem is
FPT parameterized by α + ∆ for disjoint union of stars. Theorem 4.1.
The
Fair-NET problem is
FPT parameterized by fvs + α + ∆ .Proof. Let (
G, k ) be an instance of
Fair-NET . Let k be the required S -fairness constant. Then,we compute a minimum feedback vertex set F V S of G in time O (5 | F V S | · | F V S | · | V ( G ) | ), usingthe algorithms given by Chen et al. [9]. Let fvs = | F V S | and F = V ( G ) \ F V S . Then bydefinition of feedback vertex set, G [ F ] is a forest. Let F be the set of connected components of G [ F ]. So, F is a collection of trees.Let T be a tree in F . Let v be a leaf vertex of T such that v is not connected to any vertexin F V S , then deg G ( v ) = 1. If deg G ( v ) = 1, then by Lemma 4.1, either G is a star and G = T or G is a disjoint union of T and G [ V ( G ) \ V ( T )]. By this argument, for any tree T ∈ F , eitherall the leaves of T have at least one neighbor in F V S or none of the leaves are connected toany vertex in
F V S . So, we can partition G = G + G , where G is a connected graph whereall the leaves of the forest G [ F ] have at least one neighbor in F V S and G is a disjoint unionof stars. Note that, G is an induced subgraph of G [ F ]. By Observation 2.5, G is S -fair if anyonly if G is S -fair and G is S \ S -fair, for some S ⊆ S .Let L be the set of leaves of G [ F ]. As ∆( G ) ≤ ∆ , | L | ≤ ∆ · fvs . By Corollary 4.1, we cancompute in time O ( α (∆+1) fvs · | V ( G ) | ), a superset H of the set G of functions from V ( G ) to S such that for every f ∈ M ( G, S ), there exists a g ∈ G such that g = f | V ( G ) . We can compute G from H by going over every set h ∈ H and checking whether G is fair under h . If G (cid:54) = ∅ , then15 is g ( V ( G ))-fair for every function g ∈ G . Then, for every function g ∈ G , check whether G is (cid:0) S \ g ( V ( G )) (cid:1) -fair, using Lemma 4.4. If for some function g ∈ G , G is (cid:0) S \ g ( V ( G )) (cid:1) -fair,then by Observation 2.5, G is S -fair. Also, by Corollary 4.1 and Lemma 4.4, the Fair-NET problem is
FPT parameterized by fvs + α + ∆.We now prove that the Fair-NET problem is
FPT parameterized by vc + α . Theorem 4.2.
The
Fair-NET problem is
FPT parameterized by vc + α .Proof. The
FPT algorithm is based on ILP. We first give the algorithm and then prove itscorrectness.
Algorithm:
Let (
G, k ) be an instance of
Fair-NET . Let k be the required S -fair sum. Let S (cid:48) be the set of unique labels in S . Note that | S (cid:48) | = α ( S ). Let V C ⊆ V ( G ) be a vertex cover of G of size vc . Let I = V ( G ) \ V C . By the definition of vertex cover, I is an independent set of G , i.e., no two vertices in I have an edge between them. We partition I into I , I , . . . , I m suchthat for every i ∈ [ m ], I i is an inclusion-wise maximal set of vertices in I which have the sameneighborhood in G . As I is a independent set, m ≤ vc . For every v ∈ V C , we define a binaryindicator set { t v , t v , . . . , t vm } , where t vj = 1 if v is adjacent to the vertices in I j , otherwise t vj = 0,for every j ∈ [ m ].By Observation 2.2, if G is S -fair and if we know the labels of V C under some function f ∈ M ( G, S ), then for every i ∈ [ m ], it is sufficient to know the number of times every labelis used in I i under f to get a bijective function f (cid:48) : V ( G ) → S such that f (cid:48) ∈ M ( G, S ).Keeping this insight in mind, let n i,(cid:96) be a variable whose value (to be computed below) will beinterpreted as the number of times label (cid:96) is used in I i for some function f : V ( G ) → S , forevery (cid:96) ∈ S (cid:48) , i ∈ [ m ]. Then, the algorithm works as follows. • Construct the set G containing all possible functions g : V C → S . Note that |G| ≤ α vc . • For every g ∈ G and (cid:96) ∈ S (cid:48) , let α g ( (cid:96) ) denote the number of times (cid:96) appears in g ( V C ). • For every g ∈ G : – Solve the following ILP to find an assignment to the variables n i,(cid:96) , for every i ∈ [ m ] , (cid:96) ∈ S (cid:48) . ∀ v ∈ V C, (cid:88) i ∈ [ m ] t vi (cid:88) (cid:96) ∈ S (cid:48) n i,(cid:96) · (cid:96) = k (12) ∀ i ∈ [ m ] , (cid:88) (cid:96) ∈ S (cid:48) n i,(cid:96) = | I i | (13) ∀ (cid:96) ∈ S (cid:48) , (cid:88) i ∈ [ m ] n i,(cid:96) = α S ( (cid:96) ) − α g ( (cid:96) ) (14) ∀ i ∈ [ m ] , ∀ (cid:96) ∈ S (cid:48) ; n i,(cid:96) ≥ . (15) – If the ILP returns a feasible solution, then return
True if the following statementholds. ∀ i ∈ [ m ] , (cid:88) v ∈ V C t vi · f ( v ) = k (16) • Return
False . 16 orrectness:
Equation 12 ensures that for every vertex in
V C , the neighborhood sum is k .Equation 13 ensures that for every i ∈ [ m ], the total number of labels used in I i is equal to thesize of I i . Equation 14 ensures that for every unique label (cid:96) ∈ S (cid:48) , the total number of times it isused is equal to the number of times it appear in S . Finally, Equation 16 ensures that for everyvertex in IS , the neighborhood sum is k . As for every i ∈ [ m ], all the vertices in I i have thesame neighborhood, we only check the sum once per I i . Keeping this interpretation in mind,we now prove the correctness, i.e. the algorithm returns True if and only if G is S -fair.Assume first that the algorithm returns True . It means the ILP assigned non-negativeinteger values to n i,(cid:96) , for every i ∈ [ m ] and (cid:96) ∈ S (cid:48) , such that Equations 12, 13 and 14 aresatisfied as well as that Equation 16 returned True . As explained above, that implies that forevery vertex in G , the neighborhood sum is the same and equals to k . Thus, G is S -fair.Conversely, let G be S -fair. Then, there exists a bijection f : V ( G ) → S such that for every v ∈ V ( G ) , (cid:80) f ( N ( v )) = k . As G is the set of all possible functions from V C to S , f | V C ∈ G ,and hence there exists an iteration where the algorithm examines g = f | V C . For g = f | V C , theILP admits a feasible solution. Moreover, Equation 16 then holds. Thus, the algorithm willreturn
True . As |G| ≤ α vc and the number of variables n i,(cid:96) is at most 2 vc · α , by Theorem 2.1,the Fair-NET problem is
FPT parameterized by vc + α .We now prove that the Fair-NET problem is
FPT parameterized by fvs for regular graphs.
Theorem 4.3.
The
Fair-NET problem is
FPT parameterized by fvs for regular graphs.Proof.
Let r ∈ N . Let ( G, k ) be an instance of
Fair-NET for r -regular graphs. Let k be therequired S -fairness constant. Then, we compute a minimum feedback vertex set F V S of G intime O (5 | F V S | · | F V S | · | V ( G ) | ), using the algorithms given by Chen et al. [9]. Let fvs = | F V S | and F = V ( G ) \ F V S . Then, G [ F ] is a forest. We distinguish G into the following three casesbased on the value of r . Case [ r = 1 ]: In this case, G is a collection of edges, i.e., G = tP for some t ∈ N .Then, by Observation 2.5 and by the definition of S -fair labeling, G is S -fair if and only if S = (cid:85) i ∈ [ t ] { a i , k − a i } where for all i ∈ [ t ] , a i ∈ { , , . . . , k } . So, we can solve Fair-NET problem in O ( | S | log S ) = O ( | V ( G ) | log | V ( G ) | ) time by sorting S and checking whether S satisfies the above property. Case [ r = 2 ]: In this case, G is a collection of cycles, i.e. G = C n + C n + . . . + C n t for some t ∈ N . By the definition of feedback vertex set and the minimality of F V S , every cycle containsexactly one vertex from
F V S . So, t = fvs . Also, by Observation 2.5 and by Lemma 4.2, forevery f ∈ M ( G, S ), every cycle is assigned at most 4 distinct labels, by f . So, α ( S ) ≤ fvs .As ∆( G ) = 2 and α ≤ fvs , by Theorem 4.1, the Fair-NET problem is
FPT parameterized by fvs in this case.
Case [ r ≥ ]: As G [ F ] is a forest, | E ( F ) | ≤ | V ( F ) | − ≤ | V ( G ) | −
1. Also, G is a r -regulargraph so | E ( G ) | = r ·| V ( G ) | /
2. This implies that, at least ( r/ − ·| V ( G ) | +1 edges are incidentto vertices of F V S . As every vertex of
F V S is incident to r edges, ( r/ − ·| V ( G ) | +1 ≤ r · fvs .Since r ≥
3, we get that | V ( G ) | = O ( f vs ). As the Fair-NET problem can always be solvedin time O ( | V ( G ) | !) using brute-force approach by going over all the permutations of labels, the Fair-NET problem is
FPT parameterized by fvs in this case as well.Finally, we give a simple lemma proving that the
Fair-NET problem is
FPT parameterizedby vc + ∆. Lemma 4.5.
The
Fair-NET problem is
FPT parameterized by vc + ∆ .Proof. Let G be a graph of maximum degree ∆ and let V C ⊆ V ( G ) be a vertex cover of G of size vc . Then, by the definition of vertex cover, it is easy to see that | V ( G ) | ≤ vc · ∆. As17he Fair-NET problem can always be solved in time O ( | V ( G ) | !) using brute-force approachby going over all the permutations of labels, the Fair-NET problem is
FPT parameterized by vc + ∆. In this paper, we initiated a systematic algorithmic study of
Fair-NET and presented a com-prehensive picture of the parameterized complexity of the problem. We showed NP -hardnessresults on special graph classes, which implied that the problem is para-NP -hard with respect toseveral combinations of structural graph parameters. We also showed that the problem is FPT for some combinations of structural graph parameters.While our work is comprehensive, we stress that it also opens a whole new world of re-search questions within computational social choice. For illustration, let us mention a few suchquestions:(1) Establishing the parameterized complexity of
Fair-NET with respect to tw + ∆.(2) Establishing the parameterized complexity of Fair-NET with respect to tw + ∆ + α . Byemploying the standard dynamic programming technique over tree decomposition, we canshow that the problem is FPT with respect to tw + ∆, when α is a constant. But, theparameterized complexity when α is not a constant is still open.(3) Establishing the classical complexity of Fair-NET where S = [ n ], ,where n denotes thenumber of vertices in the input graph.(4) Studying the scenario where there is no input infrastructure graph and the objective is toconstruct one which is S -fair.(5) Analysis of related labellings such as { } - Fair-NET [6] and vertex-bimagic labeling [4].(6) Introducing additional fairness notions for non-eliminating tournaments, perhaps by refin-ing/extending/modifying the notion of S -fairness.(7) Introducing manipulation and bribery to Fair-NET .18 eferences [1]
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