Fair allocation of indivisible goods under conflict constraints
Nina Chiarelli, Matjaž Krnc, Martin Milanič, Ulrich Pferschy, Nevena Pivač, Joachim Schauer
FFair allocation of indivisible goodsunder conflict constraints
Nina Chiarelli · Matjaˇz Krnc · MartinMilaniˇc · Ulrich Pferschy · NevenaPivaˇc · Joachim SchauerAbstract
We consider the fair allocation of indivisible items to several agentsand add a graph theoretical perspective to this classical problem. Therebywe introduce an incompatibility relation between pairs of items described interms of a conflict graph. Every subset of items assigned to one agent has toform an independent set in this graph. Thus, the allocation of items to theagents corresponds to a partial coloring of the conflict graph. Every agenthas its own profit valuation for every item. Aiming at a fair allocation, ourgoal is the maximization of the lowest total profit of items allocated to anyone of the agents. The resulting optimization problem contains, as specialcases, both
Partition and
Independent Set . In our contribution we derivecomplexity and algorithmic results depending on the properties of the givengraph. We can show that the problem is strongly NP-hard for bipartite graphsand their line graphs, and solvable in pseudo-polynomial time for the classes ofchordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphsof bounded treewidth. Each of the pseudo-polynomial algorithms can also beturned into a fully polynomial approximation scheme (FPTAS).
Keywords
Fair division · Conflict graph · Partial coloring.
A preliminary version containing some of the results presented here appeared in [17].N. ChiarelliUniversity of Primorska, FAMNIT and IAM, Koper, Slovenia, E-mail:[email protected]. KrncUniversity of Primorska, FAMNIT, Koper, Slovenia, E-mail: [email protected]. MilaniˇcUniversity of Primorska, FAMNIT and IAM, Koper, Slovenia E-mail: [email protected]. PferschyUniversity of Graz, Austria, E-mail: [email protected]. PivaˇcUniversity of Primorska, FAMNIT and IAM, Koper, Slovenia E-mail:[email protected]. SchauerFH JOANNEUM, Austria, E-mail: [email protected] a r X i v : . [ c s . D M ] J a n N. Chiarelli et al.
Mathematics Subject Classification (2010) · · · · · Allocating resources to several agents in a satisfactory way is a classical prob-lem in combinatorial optimization. In particular, interesting questions arise ifagents have different valuations of resources or if additional constraints areimposed for a feasible allocation. In this work we study the fair allocationof n indivisible goods or items to a set of k agents. Each agent has its ownadditive utility function over the set of items. The goal is to assign every itemto exactly one of the agents such that the minimal utility over all agents isas large as possible. Related problems of fair allocation are frequently studiedin Computational Social Choice, see, e.g., [15]. In the area of CombinatorialOptimization a similar problem is well-known as the Santa Claus problem(see [9]), which can also be seen as weight partitioning as well as a schedulingproblem.In this paper we look at the problem from a graph theoretical perspectiveand add a major new aspect to the problem. We allow an incompatibilityrelation between pairs of items, meaning that incompatible items should notbe allocated to the same agent. This can reflect the fact that items rule outtheir joint usage or simply the fact that certain items are identical (or from asimilar type) and it does not make sense for one agent to receive more thanone of these items. We will represent such a relation by a conflict graph wherevertices correspond to items and edges express incompatibilities. Now, everyfeasible allocation to one agent must be an independent set in the conflictgraph. This means that the overall solution can also be expressed as a partial k -coloring of the conflict graph G , but in addition every vertex/item has aprofit value for every color/agent and the sum of profits of vertices/itemsassigned to one color/agent should be optimized in a maxi-min sense.We believe that this problem combines aspects of independent sets, graphcoloring, and weight partitioning in an interesting way, offering new perspec-tives to look at these classical combinatorial optimization problems.For a formal definition of our problem we consider a set V of items withcardinality | V | = n and k profit functions p , . . . , p k : V → Z + . An ordered k -partition of V is a sequence ( X , . . . , X k ) of k pairwise disjoint subsets of V such that (cid:83) ki =1 X i = V . The satisfaction level of an ordered k -partition( X , . . . , X k ) of V (with respect to p , . . . , p k ) is defined as the minimum of theresulting profits p j ( X j ) := (cid:80) v ∈ X j p j ( v ), where j ∈ { , . . . , k } . The classicalfair division problem can be stated as follows. Fair k -Division of Indivisible Goods Input:
A set V of n items, k profit functions p , . . . , p k : V → Z + . Task:
Compute an ordered k -partition of V with maximum satisfactionlevel. air allocation of indivisible goods under conflict constraints 3 A discussion of the concept of fairness arising from this optimization ofabsolute profit values is beyond the scope of this paper. For further discussion,the reader could start with Balinski [6].This assignment of n indivisible items to k entities can be compared to theapportionment problem considered by Balinski and Ramires [8], see also [7],where n items (usually identical seats of a parliament) are assigned to k states.In the former, fairness is expressed by the minimum value of the sum of profits,while in the latter fairness means that the apportioned value should be as closeas possible to the valuation of the state (i.e., size of population).For the special case, where all k profit functions are identical, i.e., p = p = . . . = p k , the problem can also be represented in a scheduling setting. There are k identical machines and n jobs, which have to be assigned to the machinesby a k -partitioning. The goal is to maximize the minimal completion time(corresponding to the satisfaction level) over all k machines. It was pointed outin [23] that this problem is weakly NP-hard even for k = 2 machines. Indeed,it is easy to see that an algorithm deciding the above scheduling problemfor two machines would also decide the classical Partition problem: given n integers a , . . . , a n , can they be partitioned into two subsets with equalsums? For k ≥
3, one can simply add jobs of length one half of the sumof weights in the instance of
Partition . If k is not fixed, but part of theinput, the same scheduling problem is strongly NP-hard as mentioned in [5].In fact, an instance of the strongly NP-complete problem with3 m elements and target bound B could be decided by any algorithm for thescheduling problem with n = 3 m jobs, k = m machines and a desired minimalcompletion time equal to B . We conclude for later reference. Observation 1
Fair k -Division of Indivisible Goods , even with k iden-tical profit functions, is weakly NP-hard for any constant k ≥ and stronglyNP-hard for k being part of the input. Note that the problem is still only weakly NP-hard for constant k even for ar-bitrary profit functions, since we can construct a pseudo-polynomial algorithmsolving the problem with a k -dimensional dynamic programming array.The first elaborate treatment of Fair k -Division of Indivisible Goods was given in [11], where two approximation algorithms with non-constant ap-proximation ratios were given. The authors also mention that the problemcannot be approximated by a factor better than 1 / (cid:54) = NP). In [26]further approximation results were derived. In 2006 Bansal and Sviridenko [9]coined the term Santa Claus problem, which corresponds to the variant of theabove problem when k is not fixed but part of the input. Since then a hugenumber of approximation results have appeared on this problem of allocat-ing indivisible goods exploring different concepts of objective functions andvarious approximation measures.A different specialization is assumed in the widely studied Restricted Max-Min Fair Allocation problem. This is a special case of
Fair k -Division of In-divisible Goods where every item v i ∈ V has a fixed valuation p ( v i ) and ev-ery kid either likes or ignores item v i , i.e., the profit function p j ( v i ) ∈ { , p ( v i ) } . N. Chiarelli et al.
A fairly recent overview of approximation results both for this restricted set-ting as well as for the general case of the Santa Claus problem can be foundin [3].In this paper we study a generalization of
Fair k -Division of IndivisibleGoods , where a conflict graph G = ( V, E ) on the set V of items to be divided isintroduced. An edge { i, j } ∈ E means that items i and j should not be assignedto the same subset of the partition. Disjunctive constraints represented byconflict graphs were considered for a wide variety of combinatorial optimizationproblems. We just mention the knapsack problem ([33,34]), bin packing ([31]),scheduling (e.g., [13, 25]), and problems on graphs (e.g., [21]).The conflict graph immediately gives rise to (partial) colorings of the graphwhich were studied by Berge [10] and de Werra [22]. Definition 1 A partial k -coloring of a graph G is a sequence ( X , . . . , X k ) of k pairwise disjoint independent sets in G .Combining the profit structure with the notion of coloring we define forthe k profit functions p , . . . , p k : V → Z + and for each partial k -coloring c = ( X , . . . , X k ) a k -tuple ( p ( X ) , . . . , p k ( X k )), called the profit profile of c .The minimum profit of a profile, i.e., min kj =1 { p j ( X j ) } , is the satisfaction level of c . Now we can define the problem considered in this paper: Fair k -Division Under Conflicts Input:
A graph G = ( V, E ), k profit functions p , . . . , p k : V → Z + . Task:
Compute a partial k -coloring of G with maximum satisfactionlevel.In the hardness reductions of this paper we will frequently use the decisionversion of this problem: for a given q ∈ Z + , does there exist a partial k -coloringof G with satisfaction level at least q ?Note that an optimal partial k -coloring ( X , . . . , X k ) does not necessarilyselect all vertices from V . Furthermore, note also that for k = 1, the problemcoincides with the Weighted Independent Set problem: given a graph G = ( V, E ) and a weight function on the vertices, find an independent set ofmaximum total weight. In particular, since the case of unit weights and k = 1generalizes the Independent Set problem, we obtain the following result.
Observation 2
Fair -Division Under Conflicts is strongly NP-hard. Thus, the addition of the conflict structure gives rise to a much morecomplicated problem, since
Fair k -Division of Indivisible Goods (whicharises naturally as a special case for an edgeless conflict graph G ) is trivial for k = 1 and only weakly NP-hard for k ≥ Fair k -Division Under Conflicts for different classes of conflictgraphs. We study the boundary between strongly NP-hard cases and thosewhere a pseudo-polynomial algorithm can be derived for constant k . Obser-vation 1 implies that this is the only type of positive result we can achieve. air allocation of indivisible goods under conflict constraints 5 Moreover, considering Observation 2, it only makes sense to consider graphclasses where the independent set problem is polynomially solvable. One suchprominent example is the class of perfect graphs (see [28]). Thus, in this paperwe concentrate (mainly) on various subclasses of perfect graphs as depicted inFigure 1. Additionally, we show how to adapt the algorithm for chordal graphsto obtain a pseudo-polynomial algorithm for graphs of bounded treewidth.
Bipartite permutation graphs PP Biconvex bipartite graphs PP (Thm. 9)Bipartite graphs sNPc (Thm. 5) Permutation graphs PP Interval graphs PP Cocomparability graphs PP (Thm. 8) Chordal graphs PP (Thm. 12)Comparability graphs sNPc Perfect graphs sNPc
Line graphs of bipartite graphs sNPc (Thm. 6)Graphs of bounded treewidth PP (Thm. 13)Forests PP Fig. 1
Relationships between various graph classes and the complexity of
Fair k -DivisionUnder Conflicts (decision version). An arrow from a class G to a class G means thatevery graph in G is also in G . Label ‘ PP ’ means that for each fixed k the problem issolvable in pseudo-polynomial time in the given class, and label ‘ sNPc ’ means that for eachfixed k ≥ k = 1, as it coincides with the weighted independent setproblem. Our contributions and structure of the paper.
We first show that forall k ≥
1, the decision version of our
Fair k -Division Under Conflicts is strongly NP-complete for conflict graphs from any graph class G for which Independent Set is NP-complete, provided a certain mild technical ‘extend-ability’ condition is satisfied (Section 2.1). By a similar reasoning we can alsoreach a strong inapproximability result for our problem. For bipartite conflictgraphs as well as their line graphs
Fair k -Division Under Conflicts canbe shown to be strongly NP-hard (Section 2.2) although the corresponding N. Chiarelli et al.
Independent Set problem is polynomial-time solvable. On the other hand,for the relevant special case of biconvex bipartite graphs (cf. [29], [30]),
Fair k -Division Under Conflicts can be solved by a pseudo-polynomial timealgorithm. This result is based on an insightful pseudo-polynomial algorithmfor the problem on a cocomparability conflict graph (Section 3). Besides theseresults, in Section 3 we present dynamic programming based solutions for theclasses of chordal graphs and graphs of bounded treewidth. See Figure 1 for asummary of results. Preliminary definitions and notation.
All graphs considered in this paperare finite, simple, and undirected. A vertex in a graph G is said to be isolated if it has no neighbors and universal if it is adjacent to all other vertices. A clique in a graph G is a set of pairwise adjacent vertices and an independent set is a set of pairwise nonadjacent vertices. A matching in G is a set of pairwisedisjoint edges, and a matching M is perfect if every vertex of G is an endpointof an edge of M . For a graph G = ( V, E ) and a set X ⊆ V , we denote by G [ X ] the subgraph of G induced by X , that is, the graph with vertex set X inwhich two vertices are adjacent if and only if they are adjacent in G . Giventwo graphs G and H , we say that G is H -free if no induced subgraph of G isisomorphic to H . Observation 2 shows that
Fair k -Division Under Conflicts is strongly NP-hard even for k = 1 for general graphs, while Observation 1 shows the weakNP-hardness of the problem for constant k ≥ Fair k -Division Under Conflicts is stronglyNP-hard also for all k ≥
2, for various well-known graph classes.2.1 General hardness resultsWe start with the following general property of graph classes. Let us call agraph class G sustainable if every graph in the class can be enlarged to a graphin the class by adding to it one vertex. More formally, G is sustainable if forevery graph G ∈ G there exists a graph G (cid:48) ∈ G and a vertex v ∈ V ( G (cid:48) ) suchthat G (cid:48) − v = G . Clearly, any class of graphs closed under adding isolatedvertices, or under adding universal vertices is sustainable. This property isshared by many well known graph classes, including planar graphs, bipartitegraphs, chordal graphs, perfect graphs, etc. Furthermore, all graph classesdefined by a single nontrivial forbidden induced subgraph are sustainable. Lemma 1
For every graph H with at least two vertices, the class of H -freegraphs is sustainable.Proof Let G be the class of H -free graphs and let G ∈ G . Since H has atleast two vertices, it cannot have both a universal and an isolated vertex. If air allocation of indivisible goods under conflict constraints 7 H has no universal vertex, then the graph obtained from G by adding to ita universal vertex results in a graph in G properly extending G . If H has noisolated vertex, then the disjoint union of G with the one-vertex graph resultsin a graph in G properly extending G . (cid:117)(cid:116) For an example of a graph class G closed under vertex deletion that is notsustainable, consider the family of all cycles and their induced subgraphs. Thenevery cycle is in G but cannot be extended to a larger graph in G . The impor-tance of sustainable graph classes for Fair k -Division Under Conflicts isevident from the following theorem. Theorem 3
Let G be a sustainable class of graphs and let k be a positiveinteger such that the decision version of Fair k -Division Under Conflicts is (strongly) NP-complete. Then, for every (cid:96) ≥ k , the decision version of Fair (cid:96) -Division Under Conflicts with conflict graphs from G is (strongly) NP-complete.Proof Let G be a sustainable class of graphs for which the decision version of Fair k -Division Under Conflicts is (strongly) NP-complete and let (cid:96) > k .Let ( G, p , . . . , p k , q ) be an instance of Fair k -Division Under Conflicts (decision version) such that G ∈ G . Since G is sustainable, there exists a graph G (cid:48) ∈ G such that G (cid:48) − { x , . . . , x (cid:96) − k } = G for some (cid:96) − k additional vertices x , . . . , x (cid:96) − k . We now define the profit functions p (cid:48) , . . . , p (cid:48) (cid:96) : V ( G (cid:48) ) → Z + . Forall j = 1 , . . . , k , let p (cid:48) j ( v ) = (cid:40) p j ( v ) if v ∈ V ( G ) , v ∈ { x j | ≤ j ≤ (cid:96) − k } . and in addition let, for all j = k + 1 , . . . , (cid:96) , let p j ( v ) = (cid:40) q if v = x j − k , v ∈ V ( G (cid:48) ) \ { x j − k } . Observe that G (cid:48) has a partial k -coloring ( X (cid:48) , . . . , X (cid:48) k ) such that p (cid:48) j ( X (cid:48) j ) ≥ q for all j = 1 , . . . , (cid:96) if and only if G has a partial k -coloring ( X , . . . , X k )such that p j ( X j ) ≥ q for all j = 1 , . . . , k . Since all the numbers involvedin the reduction are polynomially bounded we conclude that Fair (cid:96) -DivisionUnder Conflicts with conflict graphs from G is also (strongly) NP-complete. (cid:117)(cid:116) In the
Independent Set problem, we are given a graph G and the task isto compute an independent set in G of maximum size. Since this is a specialcase of the Fair -Division Under Conflicts , Theorem 3 immediatelyimplies the following. Corollary 1
Let G be a sustainable class of graphs for which the decisionversion of Independent Set is NP-complete. Then, for every k ≥ , thedecision version of Fair k -Division Under Conflicts with conflict graphsfrom G is strongly NP-complete. N. Chiarelli et al.
It is known (see, e.g., [2]) that for every graph H that has a componentthat is not a path or a subdivision of the claw (the complete bipartite graph K , ), the decision version of Independent Set is NP-complete on H -freegraphs. Thus, for every such graph H , Lemma 1 and Corollary 1 imply thatfor every k ≥ Fair k -Division Under Conflicts (decision version) with H -free conflict graphs is strongly NP-complete. By using a similar argument,we also get a strong inapproximability result for general graphs. Theorem 4
For every k ≥ and every ε > , it is NP-hard to approximate Fair k -Division Under Conflicts within a factor of | V ( G ) | − ε , even forunit profit functions.Proof Fix an integer k ≥
1. We give a reduction from the
Independent Set problem. We construct a graph G (cid:48) by taking k copies of G and by adding allpossible edges between vertices from different copies. Furthermore we take k “unit” profit functions p , . . . , p k from V ( G (cid:48) ) to { } . We claim that the max-imum size of an independent set in G equals the maximum satisfaction levelof a partial k -coloring in G (cid:48) (with respect to the profit functions p , . . . , p k ).Given a maximum independent set I in G of size q one can immediately obtaina partial k -coloring ( X , . . . , X k ) of G (cid:48) with satisfaction level q by inserting allvertices of I in the j -th copy of G into X j , for all j = 1 , . . . , k . On the otherhand, given a partial k -coloring ( X , . . . , X k ) of G (cid:48) with satisfaction level q ,one can simply choose X , which is an independent set completely containedin one copy of G . Thus, X corresponds to an independent set in G of size q .Suppose that for some ε ∈ (0 ,
1) there exists a polynomial-time algorithm A that approximates Fair k -Division Under Conflicts within a factor of | V ( G ) | − ε on input instances with unit profit functions. We will show thatthis implies the existence of a polynomial-time algorithm A (cid:48) approximatingthe Independent Set problem within a factor of | V ( G ) | − ε (cid:48) where ε (cid:48) = ε/ G to the Independent Set problem. The al-gorithm A (cid:48) proceeds as follows. If | V ( G ) | < k − ε ) /ε , then the graph is ofconstant order and the problem can be solved optimally in O (1) time. If | V ( G ) | ≥ k − ε ) /ε , then the graph G (cid:48) is constructed following the abovereduction, a partial k -coloring ( X , . . . , X k ) is computed using algorithm A on G (cid:48) equipped with k unit profit functions, and a subset of V ( G ) corre-sponding to X is returned. Clearly, the algorithm runs in polynomial timeand computes an independent set in G . Let q denote the maximum satis-faction level of a partial k -coloring in G (cid:48) . By the above claim, the inde-pendence number of G equals q . Thus, to complete the proof, it sufficesto show that | X | ≥ q/ ( | V ( G ) | − ε (cid:48) ). By assumption on A , we have that | X | ≥ q/ ( | V ( G (cid:48) ) | − ε ). We want to show that q/ | V ( G (cid:48) ) | − ε ≥ q/ | V ( G ) | − ε (cid:48) ,or, equivalently, 1 /k − ε | V ( G ) | − ε ≥ / | V ( G ) | − ε/ . After some straightfor-ward algebraic manipulations, this inequality simplifies to the equivalent in-equality | V ( G ) | ≥ k − ε ) /ε , which is true by assumption. (cid:117)(cid:116) air allocation of indivisible goods under conflict constraints 9 k ≥ Fair k -Division Under Conflicts is NP-hard in two classes of graphs where the Independent Set problemis solvable in polynomial time: the classes of bipartite graphs and their linegraphs. Recall that for a given graph G , its line graph has a vertex for eachedge of G , with two distinct vertices adjacent in the line graph if and only ifthe corresponding edges share an endpoint in G .The proof for bipartite graphs shows strong NP-hardness even for the casewhen all the profit functions are equal. Theorem 5
For each integer k ≥ , the decision version of Fair k -DivisionUnder Conflicts is strongly NP-complete in the class of bipartite graphs.Proof We use a reduction from the decision version of the
Clique problem:Given a graph G and an integer (cid:96) , does G contain a clique of size (cid:96) ? Consideran instance ( G, (cid:96) ) of
Clique such that 2 ≤ (cid:96) < n := | V ( G ) | . We define aninstance of Fair k -Division Under Conflicts (decision version) consistingof a bipartite conflict graph G (cid:48) , profit functions p , . . . , p k , and a lower bound q on the required satisfaction level. The graph G (cid:48) = ( A ∪ B, E (cid:48) ) has a vertexfor each vertex of the graph G as well as for each edge of G and k new vertices x , . . . , x k . It is defined as follows: A = V ( G ) ∪ { x } , B = E ( G ) ∪ { x i | ≤ i ≤ k } ,E (cid:48) = { ve | v ∈ V ( G ) is an endpoint of e ∈ E ( G ) } ∪ { vx i | v ∈ V ( G ) , ≤ i ≤ k } . The lower bound q on the satisfaction level is defined by setting q = n + (cid:0) (cid:96) (cid:1) n +( n − (cid:96) ). For ease of notation we set N = n and we furthermore introducea second integer N such that q = N + (cid:16) m − (cid:0) (cid:96) (cid:1)(cid:17) n , where m = | E ( G ) | .(Note that N ≥ n .) With this, the profit functions p i : V ( G (cid:48) ) → Z + , for all i ∈ { , . . . , k } , are defined as p i ( v ) =
1; if v ∈ V ( G ); n ; if v ∈ E ( G ); N ; if v = x ; N ; if v = x ; q ; if v = x j for some j ∈ { , . . . , k } . Note that all the profits introduced as well as the number of vertices and edgesof G (cid:48) are polynomial in n . To complete the proof, we show that G has a cliqueof size (cid:96) if and only if G (cid:48) has a partial k -coloring with satisfaction level atleast q . First assume that G has a clique C of size (cid:96) . We construct a partial k -coloring c = ( X , . . . , X k ) of G (cid:48) by setting X = { x } ∪ { e ∈ E ( G ) | e ⊆ C } ∪ ( V ( G ) \ C ) ,X = { x } ∪ ( E ( G ) \ X ) ,X j = { x j } for 3 ≤ j ≤ k. Observe that the partial k -coloring c gives rise to the corresponding profitprofile with all entries equal to q , which establishes one of the two implications.Suppose now that there exists a partial k -coloring c = ( X , . . . , X k ) of G (cid:48) for which the profit profile has all entries ≥ q . Since for each i ∈ { , . . . , k } , thetotal profit of the set V ( G ) ∪ E ( G ) is only mn + n < n , the partial coloring c must use exactly one of the k vertices x , . . . , x k in each color class. We mayassume without loss of generality that x i ∈ X i for all i ∈ { , . . . , k } . Let U bethe set of uncolored vertices in G (cid:48) w.r.t. the partial coloring c . Since for each ofthe profit functions p i , the difference between the overall sum of the profits ofvertices of G (cid:48) and k · q is equal to (cid:96) , we clearly have (cid:80) v ∈ U p i ( v ) ≤ (cid:96) < n , whichimplies that U ⊆ V ( G ). Next, observe that every vertex of E ( G ) belongs toeither X or to X , since otherwise we would have p ( X ) + p ( X ) < q ,contrary to the assumption that the satisfaction level of c is at least q .Consider the sets W = X ∩ V ( G ) and F = X ∩ E ( G ). Then X = { x } ∪ W ∪ F and, since (cid:80) v ∈ X p ( v ) ≥ q = N + (cid:0) (cid:96) (cid:1) n + ( n − (cid:96) ), it follows that X contains exactly (cid:0) (cid:96) (cid:1) vertices from E ( G ) (if | F | > (cid:0) (cid:96) (cid:1) , then p ( X ) < q )and at least n − (cid:96) vertices from V ( G ). Let C denote the set of all vertices of G (cid:48) with a neighbor in F . By the construction of G (cid:48) and since | F | = (cid:0) (cid:96) (cid:1) , it followsthat C is of cardinality at least (cid:96) . Furthermore, since X is independent, wehave C ∩ W = ∅ . Consequently, n = | V ( G ) | ≥ | C | + | W | ≥ (cid:96) + ( n − (cid:96) ) = n ,hence equalities must hold throughout. In particular, C is a clique of size (cid:96) in G . (cid:117)(cid:116) Theorem 6
For each integer k ≥ , the decision version of Fair k -DivisionUnder Conflicts is strongly NP-complete in the class of line graphs of bi-partite graphs.Proof Note that it suffices to prove the statement for k = 2. For k >
2, Theo-rem 3 applies, since the class of line graphs of bipartite graphs is sustainable.Indeed, if G (cid:48) is the line graph of a bipartite graph G , then the graph obtainedfrom G (cid:48) by adding to it an isolated vertex is the line graph of the bipartitegraph obtained from G by adding to it an isolated edge.For k = 2, we use a reduction from the following problem: Given a bipartitegraph G and an integer Q , does G contain two disjoint matchings M and M such that M is a perfect matching and | M | ≥ Q ? This problem was shownto be NP-complete by P´alv¨olgi (see [32]). Consider an instance ( G, Q ) of thisproblem such that 1 ≤ Q ≤ n/ n = | V ( G ) | is even. Then we definethe following instance of the decision version of Fair -Division UnderConflicts with a conflict graph G (cid:48) , where G (cid:48) is the line graph of G . Thelower bound q on the satisfaction level is defined by setting q = n · Q/
2. Theprofit functions p , p : V ( G (cid:48) ) → Z + are defined as p ( v ) = Q for all v ∈ V ( G (cid:48) ),and p ( v ) = n/ v ∈ V ( G (cid:48) ). Clearly, all the profits introduced as well asthe number of vertices and edges of G (cid:48) are polynomial in n . Recall that everymatching in G corresponds to an independent set in G (cid:48) .We now show that the instances of the two decision problems have thesame answers. Suppose first that G has two disjoint matchings M and M air allocation of indivisible goods under conflict constraints 11 such that M is a perfect matching and | M | ≥ Q . Then the sequence ( M , M )is a partial 2-coloring of G (cid:48) such that p ( M ) = Q | M | = Q · n/ q and p ( M ) = ( n/ · | M | ≥ ( n/ Q = q. Conversely, suppose that G (cid:48) has a partial 2-coloring ( X , X ) with satisfactionlevel at least q . Then the independent sets X and X in G (cid:48) are disjointmatchings in G . Moreover, since p ( X ) = Q | X | ≥ q = Q · n/ p ( X ) = ( n/ · | X | ≥ q = Q · n/ , we obtain | X | ≥ n/ | X | ≥ Q . Thus, X is a perfect matching in G andany set of Q edges in X is a matching in G disjoint from X . This provesthat the decision version of Fair -Division Under Conflicts is stronglyNP-complete in the class of line graphs of bipartite graphs. (cid:117)(cid:116) In this section we turn our attention to classes of graphs for which the
Fair k -Division Under Conflicts is solvable in pseudo-polynomial time. As shownin Theorem 5, for each k ≥ Fair k -Division Under Conflicts is stronglyNP-complete in the class of bipartite graphs, and this rules out the existenceof a pseudo-polynomial time algorithm for the problem in the class of bipartitegraphs, unless P = NP. We show that for every k there is a pseudo-polynomialtime algorithm for the Fair k -Division Under Conflicts in a subclass ofbipartite graphs, the class of biconvex bipartite graphs . The algorithm reducesthe problem to the class of bipartite permutation graphs. To solve the prob-lem in the class of bipartite permutation graphs, we develop a solution in amore general class of graphs, the class of cocomparability graphs (containingpermutation graphs). Further, using a dynamic programming approach, weshow that for every k there is a pseudo-polynomial time algorithm for Fair k -Division Under Conflicts in the classes of chordal graphs and graphsof bounded treewidth. It will be shown in Section 4 that all these pseudo-polynomial dynamic programming algorithms allow the construction of a fullypolynomial time approximation scheme (FPTAS).Let us first fix some notation. Given a graph G and k profit functions p , . . . , p k : V → Z + , we denote by n the number of vertices in G , n = | V ( G ) | .All pseudo-polynomial results in this section depend on an upper bound onthe maximum reachable profit value Q = max ≤ j ≤ k p j ( V ). Given an integer k >
0, the addition and subtraction of k -tuples is defined component-wise,and for all (cid:96) ∈ { , . . . , k } , we denote by e (cid:96) ( x ) the k -tuple with all coordinatesequal to 0, except that the (cid:96) -th coordinate is equal to x .3.1 Cocomparability graphsA graph G = ( V, E ) is a comparability graph if it has a transitive orientation,that is, if each of the edges { u, v } of G can be replaced by exactly one of the ordered pairs ( u, v ) and ( v, u ) so that the resulting set A of directed edges istransitive (that is, for every three vertices x, y, z ∈ V , if ( x, y ) ∈ A and ( y, z ) ∈ A , then ( x, z ) ∈ A ). A graph G is a cocomparability graph if its complement isa comparability graph. Comparability graphs and cocomparability graphs arewell-known subclasses of perfect graphs. The class of cocomparability graphs isa common generalization of the classes of interval graphs, permutation graphs,and trapezoid graphs (see, e.g., [16, 27]).Since every bipartite graph is a comparability graph, Theorem 5 impliesthat for each k ≥ Fair k -Division Under Conflicts is strongly NP-complete in the class of comparability graphs. For cocomparability graphs, weprove that the problem is solvable in pseudo-polynomial time. The key resultin this direction is the following lemma. Lemma 7
For every k ≥ , given a cocomparability graph G = ( V, E ) and k profit functions p , . . . , p k : V → Z + , the set of all profit profiles of par-tial k -colorings of G can be computed in time O ( n k +2 ( Q + 1) k ) , where Q =max ≤ j ≤ k p j ( V ) .Proof Let G be a cocomparability graph. In time O ( n ), we compute thecomplement of G and a transitive orientation D of it [35]. Since D is a directedacyclic graph, one can compute in linear time a topological sort of D , that is,an ordering v , . . . , v n of the vertices such that if ( v i , v j ) is an arc of D , then i < j (see, e.g., [18]). Note that( ∗ ) a set X = { v i , . . . , v i p } ⊆ V with i < . . . < i p is independent in G if andonly if ( v i , . . . , v i p ) is a directed path in D .Thus, a partial k -coloring in G corresponds to a collection of k vertex-disjointdirected paths in D , and vice versa. We process the vertices of G in the orderinggiven by the topological sort of D and try all possibilities for the color (if any)of the current vertex v j in order to extend a partial k -coloring of the alreadyprocessed subgraph of G with v j . (In terms of D , we choose which of the k directed paths will be extended into v j .) To avoid introducing additionalterminology and notation, we present the details of the algorithm in terms ofpartial k -colorings of G instead of systems of disjoint paths in D .For each j ∈ { , , . . . , n } and each k -tuple ( i , . . . , i k ) ∈ { , , . . . , j } k ,we compute the set P j ( i , . . . , i k ) of all k -tuples ( q , . . . , q k ) ∈ Z k + such thatthere exists a partial k -coloring ( X , . . . , X k ) of the subgraph of G induced by { v , . . . , v j } (which is empty if j = 0) such that q (cid:96) = p (cid:96) ( X (cid:96) ) and i (cid:96) = (cid:26) max { r : v r ∈ X (cid:96) } , if X (cid:96) (cid:54) = ∅ ;0 , if X (cid:96) = ∅ (1)for all (cid:96) ∈ { , . . . , k } . Note that for each (cid:96) ∈ { , . . . , k } , the possible val-ues of the (cid:96) -th coordinate of any member of P j ( i , . . . , i k ) belong to the set { , , . . . , Q } where Q = max ≤ j ≤ k p j ( V ). Thus, each set P j ( i , . . . , i k ) has atmost ( Q + 1) k elements. Note also that the total number of sets P j ( i , . . . , i k )is of the order O ( n k +1 ). air allocation of indivisible goods under conflict constraints 13 In what follows we explain how to compute the sets P j ( i , . . . , i k ). For j = 0, the only feasible choice for the k -tuple ( i , . . . , i k ) is (0 , . . . ,
0) and weset P (0 , . . . ,
0) = { } k = { (0 , . . . , } . This is correct since the only partial k -coloring of the graph with no vertices is the k -tuple ( ∅ , . . . , ∅ ). Suppose that j > P j − ( i , . . . , i k ) are already computed for all ( i , . . . , i k ) ∈{ , , . . . , j − } k . Fix a k -tuple ( i , . . . , i k ) ∈ { , , . . . , j } k . To describe how tocompute the set P j ( i , . . . , i k ), we will use the following notation. We considerthree cases. For each of them, we first give a formula for computing the set P j ( i , . . . , i k ) and then we argue why the formula is correct.(i) If j appears at least twice as a coordinate of ( i , . . . , i k ), then we set P j ( i , . . . , i k ) = ∅ . (2)Note that since j appears at least twice as a coordinate of ( i , . . . , i k ),there is no partial k -coloring ( X , . . . , X k ) of the subgraph of G inducedby { v , . . . , v j } such that equality (1) holds for all (cid:96) ∈ { , . . . , k } . Thus,equation (2) is correct.(ii) If j does not appear as any coordinate of ( i , . . . , i k ), then we set P j ( i , . . . , i k ) = P j − ( i , . . . , i k ) . (3)Since j does not appear as any coordinate of ( i , . . . , i k ), every partial k -coloring of the subgraph of G induced by { v , . . . , v j − } such that equal-ity (1) holds for all (cid:96) ∈ { , . . . , k } is a partial k -coloring of the subgraph of G induced by { v , . . . , v j } and vice versa. This implies relation (3).(iii) If j appears exactly once as a coordinate of ( i , . . . , i k ), say i s = j , thenwe set P j ( i , . . . , i k ) = (cid:91) { j (cid:48) : j (cid:48) =0 or vj (cid:48)∈ N − D ( vj ) } { q + e s ( p s ( v j )) | q ∈ P j − ( i , . . . , i s − , j (cid:48) , i s +1 , . . . , i k ) } , (4)where N − D ( v j ) denotes the set of all vertices v j (cid:48) such that ( v j (cid:48) , v j ) is an arcof D . (Note that j (cid:48) < j for all v j (cid:48) ∈ N − D ( v j ), since v , . . . , v n is a topologicalsort of D .)Let q = ( q , . . . , q k ) ∈ P j ( i , . . . , i k ) and consider a partial k -coloring( X , . . . , X k ) of the subgraph of G induced by { v , . . . , v j } such that p (cid:96) ( X (cid:96) ) = q (cid:96) and equality (1) holds for all (cid:96) ∈ { , . . . , k } . Then max { q : v q ∈ X s } = i s = j . In particular, v j ∈ X s . Let X (cid:48) s = X s \ { v j } and let j (cid:48) = (cid:26) max { r : v r ∈ X (cid:48) s } , if X (cid:48) s (cid:54) = ∅ ;0 , if X (cid:48) s = ∅ . Note that if X (cid:48) s (cid:54) = ∅ then v j (cid:48) ∈ N − D ( v j ). Indeed, digraph D is an orientationof the complement of G , in which vertices v j (cid:48) and v j are adjacent (recallthat they belong to the independent set X s in G ). This implies that either( v j , v j (cid:48) ) or ( v j (cid:48) , v j ) is an arc of D , but since j (cid:48) < j and v , . . . , v n is atopological sort of D , the pair ( v j (cid:48) , v j ) must be an arc of D . Let ( i (cid:48) , . . . , i (cid:48) k ) be the k -tuple obtained from ( i , . . . , i k ) by replacing i s with j (cid:48) , and let( X (cid:48) , . . . , X (cid:48) k ) be the k -tuple obtained from ( X , . . . , X k ) by replacing X s with X (cid:48) s . Then ( X (cid:48) , . . . , X (cid:48) k ) is a partial k -coloring of the subgraph of G induced by { v , . . . , v j − } such that equality obtained from (1) by replacing X (cid:96) with X (cid:48) (cid:96) and i (cid:96) with i (cid:48) (cid:96) holds for each (cid:96) ∈ { , . . . , k } . Furthermore,( p ( X ) , . . . , p k ( X k )) = ( p ( X (cid:48) ) , . . . , p k ( X (cid:48) k )) + e s ( p s ( v j )). This shows thatif q = ( q , . . . , q k ) ∈ P j ( i , . . . , i k ), then the k -tuple q belongs to the union (cid:91) { j (cid:48) : j (cid:48) =0 or v j (cid:48) ∈ N − D ( v j ) } { q + e s ( p s ( v j )) | q ∈ P j − ( i , . . . , i s − , j (cid:48) , i s +1 , . . . , i k ) } . For the converse direction, let j (cid:48) ∈ { }∪{ ≤ j (cid:48) ≤ j − | v j (cid:48) ∈ N − D ( v j ) } , let( i (cid:48) , . . . , i (cid:48) k ) be the k -tuple obtained from ( i , . . . , i k ) by replacing i s with j (cid:48) ,and let q = ( q , . . . , q k ) ∈ P j − ( i (cid:48) , . . . , i (cid:48) k ). Then, there exists a partial k -coloring ( X (cid:48) , . . . , X (cid:48) k ) of the subgraph of G induced by { v , . . . , v j − } suchthat for each (cid:96) ∈ { , . . . , k } , we have p (cid:96) ( X (cid:48) (cid:96) ) = q (cid:96) and equality obtainedfrom (1) by replacing X (cid:96) with X (cid:48) (cid:96) and i (cid:96) with i (cid:48) (cid:96) holds. Let ( X , . . . , X k ) bethe k -tuple obtained from ( X (cid:48) , . . . , X (cid:48) k ) by replacing X (cid:48) s with X (cid:48) s ∪ { v j } . Toshow that ( X , . . . , X k ) is a partial k -coloring of the subgraph of G inducedby { v , . . . , v j } , it suffices to verify that X s = X (cid:48) s ∪ { v j } is an independentset in G . If X (cid:48) s = ∅ , then X s = { v j } is independent. Suppose that X (cid:48) s (cid:54) = ∅ . Then, by ( ∗ ), X (cid:48) s corresponds to a directed path in D ending in v j (cid:48) .Extending this path with vertex v j ∈ N + D ( v j (cid:48) ) results in a directed path in D with vertex set X s , which shows, again by ( ∗ ), that X s is independent in G . Clearly, we have that max { r : v r ∈ X s } = j , and hence ( X , . . . , X k ) is apartial k -coloring of the subgraph of G induced by { v , . . . , v j } equality (1)holds for each (cid:96) ∈ { , . . . , k } . Furthermore, ( p ( X ) , . . . , p k ( X k )) = q + e s ( p s ( v j )). This shows that if q ∈ P j − ( i (cid:48) , . . . , i (cid:48) k ), then the k -tuple q + e s ( p s ( v j )) belongs to P j ( i , . . . , i k ). Therefore, equation (4) is correct.Finally, the set of all profit profiles of partial k -colorings of G equals to theunion, over all ( i , . . . , i k ) ∈ { , , . . . , n } k , of the sets P n ( i , . . . , i k ).The algorithm can be easily modified so that for each profit profile alsoa corresponding partial k -coloring is computed. We would just need to store,for each j ∈ { , , . . . , n } , each ( i , . . . , i k ) ∈ { , , . . . , j } k , and each k -tuple( q , . . . , q k ) ∈ P j ( i , . . . , i k ), one partial k -coloring ( X , . . . , X k ) of the sub-graph of G induced by { v , . . . , v i } such that p (cid:96) ( X (cid:96) ) = q (cid:96) and equality (1)holds for all (cid:96) ∈ { , . . . , k } .It remains to estimate the time complexity of the algorithm. For each j ∈{ , . . . , n } and each of the O ( n k ) k -tuples ( i , . . . , i k ) ∈ { , , . . . , j } k , we candecide which of the three cases (i)–(iii) occurs in time O ( k ). Step (2) takes con-stant time, step (3) takes time O (( Q + 1) k ), and step (4) can be implementedin time O ( n ( Q +1) k ). Altogether, this results in running time O ( n ( Q +1) k ) foreach fixed j ∈ { , . . . , n } and each k -tuple ( i , . . . , i k ) ∈ { , , . . . , j } k . Conse-quently, the total running time of the algorithm is O ( n k +2 ( Q + 1) k ). (cid:117)(cid:116) Lemma 7 implies the following. air allocation of indivisible goods under conflict constraints 15
Theorem 8
For every k ≥ , Fair k -Division Under Conflicts is solv-able in time O ( n k +2 ( Q + 1) k ) for cocomparability conflict graphs G , where Q = max ≤ j ≤ k p j ( V ( G )) .Proof By Lemma 7, we can compute the set Π of all profit profiles of partial k -colorings of G in the stated running time. For each profit profile in Π , wecan determine the satisfaction level of the corresponding partial k -coloring of G . Taking the maximum satisfaction level over all profiles gives the optimalvalue of Fair k -Division Under Conflicts for ( G, p , . . . , p k ). (cid:117)(cid:116) Fair k -Division Under Conflicts is stronglyNP-hard for bipartite conflict graphs. Thus, we consider in the following themore restricted case of biconvex bipartite conflict graphs. Recall that a bipar-tite graph G = ( A ∪ B, E ) is biconvex if it has a biconvex ordering , that is,an ordering of A and B such that for every vertex a ∈ A (resp. b ∈ B ) theneighborhood N ( a ) (resp. N ( b )) is an interval of consecutive vertices in theordering of B (resp. ordering of A ).It is known that a connected biconvex bipartite graph G can always beordered in such a way that the first and last vertices on one side have aspecial structure. Fix a biconvex ordering of G , say A = ( a , . . . , a s ) and B = ( b , . . . , b t ). Define a L (resp. a R ) as the vertex in N ( b ) (resp. N ( b t ))whose neighborhood is not properly contained in any other neighborhood set(see [1, Def. 8]). In case of ties, a L is the smallest such index (and a R thelargest). We always assume that a L ≤ a R , otherwise the ordering in A couldbe mirrored. Under these assumptions, the neighborhoods of vertices appearingin the ordering before a L and after a R are nested. Lemma 2 (Abbas and Stewart [1])
Let G = ( A ∪ B, E ) be a connectedbiconvex graph. Then there exists a biconvex ordering of the vertices of G suchthat:i. For all a i , a j with a ≤ a i < a j ≤ a L we have N ( a i ) ⊆ N ( a j ) .ii. For all a i , a j with a R ≤ a i < a j ≤ a s we have N ( a j ) ⊆ N ( a i ) .iii. The subgraph G (cid:48) of G induced by vertex set { a L , . . . , a R }∪ B is a bipartitepermutation graph. Property (iii) can be put in context with Theorem 8. Indeed, it is knownthat permutation graphs are a subclass of cocomparability graphs (see,e.g., [16]). This gives rise to the following result that
Fair k -Division UnderConflicts on biconvex bipartite graphs is indeed easier (from the complexitypoint of view) than on general bipartite graphs. It should be pointed out thatthe contribution of Theorem 9 is the identification of the complexity statusof the problem, but not a practically relevant algorithm, since the pseudo-polynomial running time will be prohibitive in practice. The high-level idea ofthe algorithm is illustrated in Algorithm 1. Algorithm 1
Algorithmic Idea for a Connected Biconvex Graph G apply Lemma 2 for getting the cocomparability graph G (cid:48) and vertices a L , a R let A L := { a , . . . , a L − } and A R := { a R +1 , . . . , a s } for all j ∈ { , . . . , k } do guess a j ∈ A L with largest index (resp. smallest index a j ∈ A R ) included in X j end for each such guess can be represented by a 2 k -tuple σ = ( a , . . . , a k , a , . . . , a k ) for each guess σ dofor all j ∈ { , . . . , k } do exclude all vertices v of the neighborhood N ( a j ) ⊆ B (and N ( a j ) ⊆ B )from insertion into X j by setting their profit p j ( v ) := 0 end for apply Lemma 7 to the cocomparability graph G (cid:48) and the modified profit functions toobtain the set Π σ of all profit profiles ( q , . . . , q k ) of partial k -colorings of G (cid:48) withrespect to the modified profits increase each profit profile by setting q j := q j + p j ( a j ) + p j ( a j ) augment these profiles with vertices from A L and A R end for choose the best solution over all guesses σ Theorem 9
For every k ≥ , Fair k -Division Under Conflicts is solv-able in time O ( n k +2 ( Q + 1) k ) for connected biconvex bipartite conflict graphs G , where Q = max ≤ j ≤ k p j ( V ( G )) .Proof Assuming at first that G is connected, Lemma 2 is applied for obtainingfrom G the cocomparability graph G (cid:48) . However, we have to consider also thevertex sets A L := { a , . . . , a L − } and A R := { a R +1 , . . . , a s } . This is done byconsidering assignments of vertices in A L ∪ A R to the k subsets of a partial k -coloring of G in an efficient way as follows.For every j ∈ { , . . . , k } , we guess, by going through all possibilities, thelargest index vertex a j ∈ A L (resp. smallest index a j ∈ A R ) inserted in X j .One can add an artificial vertex a (resp. a s +1 ) to represent the case that novertex from A L (resp. A R ) is inserted in X j . Thus, every guess is representedby a 2 k -tuple σ = ( a , . . . , a k , a , . . . , a k ). The total number of such guesses(i.e., iterations) is bounded by ( n + 1) k for each of A L and A R , i.e., O ( n k )selections to be considered in total.For each such guess σ we perform the following computations. For every j ∈ { , . . . , k } the vertices in the neighborhood N ( a j ) ⊆ B (and N ( a j ) ⊆ B )of the chosen index must be excluded from insertion into the correspondingset X j . This can be easily realized by setting to 0 the profits p j of all verticesin N ( a j ) (resp. N ( a j )). With these slight modifications of the profits we canapply Lemma 7 for the cocomparability graph G (cid:48) and the modified profitfunctions p σj to obtain the set Π σ of all (pseudo-polynomially many) profitprofiles ( q , . . . , q k ) of partial k -colorings of G (cid:48) with respect to p σ . Every entry q j of a profit profile in Π σ is increased by p j ( a j ) + p j ( a j ), to account forinclusion of the vertices selected by the guess σ .In every guess there are the two vertices a j and a j permanently assignedto X j for every j and their neighborhoods N ( a j ) and N ( a j ) are excluded from air allocation of indivisible goods under conflict constraints 17 X j . Now it follows from properties (i) and (ii) of Lemma 2 that for each vertex a (cid:48) ∈ A L with a (cid:48) < a j (resp. a (cid:48) ∈ A R with a (cid:48) > a j ) the neighborhood N ( a (cid:48) ) isa subset of N ( a j ) (resp. N ( a j )). Thus, these vertices a (cid:48) could also be insertedin X j without any violation of the conflict structure. Therefore, we can startfrom the set Π σ of profit profiles computed for ( G (cid:48) , p σ ) and consider iteratively(in arbitrary order) the addition of a vertex a (cid:48) ∈ A L to one of the color classes X j , as it is usually done in dynamic programming. Each a (cid:48) is considered as anaddition to every profit profile ( q , . . . , q k ) ∈ Π σ and for every index j with a (cid:48) < a j yielding new profit profiles ( q , . . . , q j − , q j + p j ( a (cid:48) ) , q j +1 , . . . , q k ) to beadded to Π σ . An analogous procedure is performed for all vertices a (cid:48) ∈ A R where the addition is restricted to indices j with a (cid:48) > a j .For every guess σ , the running time is dominated by the effort of computingthe O (( Q + 1) k ) profit profiles of ( G (cid:48) , p σ ) according to Lemma 7, since addingany of the O ( n ) vertices a (cid:48) requires only k operations for each profit profile.In this way, we construct the set Π σ of all profit profiles of partial k -colorings of G for each guess σ . It remains to identify the optimal solutionin the set Π := (cid:83) σ Π σ similarly as in the proof of Theorem 8. Going overall O ( n k ) guesses σ , the total running time can be given from Lemma 7 as O ( n k +2 ( Q + 1) k ). (cid:117)(cid:116) For disconnected conflict graphs, we can easily paste together the profitprofiles of all connected components. Note that this construction applies forgeneral graphs.
Lemma 10
Given a conflict graph G consisting of c > connected compo-nents G (cid:96) , (cid:96) = 1 , . . . , c , each of them with a set of profit profiles Π (cid:96) , where thesize of each Π (cid:96) is of order O (( Q + 1) k ) with Q = max ≤ j ≤ k p j ( V ( G )) , Fair k -Division Under Conflicts can be solved for G in time O (( c − Q +1) k ) .Proof We maintain a set of profit profiles Π , initialized by Π := Π , anditeratively merge each of the profit profiles Π , . . . , Π m with Π . To merge aset of profit profiles Π (cid:96) , we consider every pair of profiles from Π and Π (cid:96) andperform a vector addition to obtain a (possibly) new profit profile which isadded to Π . At most ( Q + 1) k such pairs may exist. In each of the c − Π remains bounded bythe trivial upper bound ( Q + 1) k . Finally, the best objective function valueis determined by evaluating all profit profiles. The total running time of thisprocedure is of order O (( c − Q + 1) k ). (cid:117)(cid:116) Running Algorithm 1 for all c components of a graph with n vertices canbe done in time O ( n k +2 ( Q + 1) k ). Applying Lemma 10 on the resulting profitprofiles, we obtain the following corollary. Note that the computational com-plexity does not depend on the size of the components. Corollary 11
For every k ≥ , Fair k -Division Under Conflicts is solvable in time O ( n k +2 ( Q + 1) k + ( c − Q + 1) k ) for biconvex bipar-tite conflict graphs G consisting of c connected components, where Q =max ≤ j ≤ k p j ( V ( G )) . Fair k -Division Under Conflicts on chordal graphs. Recall that a graphis chordal if all its induced cycles are of length three. First we state someknown results on chordal graphs and their tree decompositions.A tree decomposition of a graph G is a pair T = ( T, { B t } t ∈ V ( T ) ) where T is a tree whose every node t is assigned a vertex subset B t ⊆ V ( G ) called abag such that the following conditions are satisfied: every vertex of G is in atleast one bag, for every edge { u, v } ∈ E ( G ) there exists a node t ∈ V ( T ) suchthat B t contains both u and v , and for every vertex u ∈ V ( G ) the subgraphof T induced by the set { t ∈ V ( T ) : u ∈ B t } is connected (that is, a tree).A tree decomposition ( T, { B t } t ∈ V ( T ) ) is rooted if we distinguish one vertex r of T which will be the root of T . This introduces natural parent-child andancestor-descendant relations in the tree T . Following [20], we will say thata tree decomposition ( T, { B t } t ∈ V ( T ) ) is nice if it is rooted and the followingconditions are satisfied: – If t ∈ V ( T ) is the root or a leaf of T , then B t = ∅ ; – Every non-leaf node t of T is one of the following three types: – Introduce node: a node t with exactly one child t (cid:48) such that B t = B t (cid:48) ∪ { v } for some vertex v ∈ V ( G ) \ B t (cid:48) ; – Forget node: a node t with exactly one child t (cid:48) such that B t = B t (cid:48) \{ v } for some vertex v ∈ B t (cid:48) ; – Join node: a node t with exactly two children t and t such that B t = B t = B t .The width of a tree decomposition ( T, { B t } t ∈ V ( T ) ) of a graph G is defined asmax t ∈ V ( T ) | B t | −
1. Lemma 7.4 from [20] shows that every tree decompositionof width at most (cid:96) can be transformed in polynomial time into a nice treedecomposition of width at most (cid:96) . The proof actually shows the followingstatement, which will be useful for our purpose.
Lemma 3
Given a tree decomposition T = ( T, { B t } t ∈ V ( T ) ) of an n -vertexgraph G , one can in time O ( n · max { n, | V ( T ) |} ) compute a nice tree decom-position T (cid:48) of G that has at most O ( n ) nodes and such that every bag of T (cid:48) is a subset of a bag of T . Let us now apply these concepts to chordal graphs. A clique tree of a graph G is a tree decomposition ( T, { B t } t ∈ V ( T ) ) such that the bags are exactly themaximal cliques of G . It is well known (see, e.g., [12]) that a graph is chordalif and only if it has a clique tree, and in such a case a clique tree can beconstructed in linear time (see, e.g., [36]). Furthermore, every chordal graph G has at most | V ( G ) | maximal cliques (see, e.g., [12]). Lemma 4
Given an n -vertex chordal graph G , we can compute in linear timea tree decomposition ( T, { B t } t ∈ V ( T ) ) of G with O ( n ) bags, all of which arecliques. air allocation of indivisible goods under conflict constraints 19 Combining Lemmas 3 and 4 yields the following.
Lemma 5
Given an n -vertex chordal graph G , we can compute in time O ( n ) a nice tree decomposition ( T, { B t } t ∈ V ( T ) ) of G with O ( n ) bags, all of whichare cliques. We will also need the following technical lemma about tree decompositions(see, e.g., [20]).
Lemma 6
Let ( T, { B t } t ∈ V ( T ) ) be a tree decomposition of a graph G and let { a, b } be an edge of T . The forest T − { a, b } obtained from T by deletingedge { a, b } consists of two connected components T a (containing a ) and T b (containing b ). Let A = (cid:16)(cid:83) t ∈ V ( T a ) B t (cid:17) \ ( B a ∩ B b ) and B = (cid:16)(cid:83) t ∈ V ( T b ) X t (cid:17) \ ( B a ∩ B b ) . Then no vertex in A is adjacent to a vertex in B . Before we proceed to the main result for chordal graphs, we need to in-troduce an auxiliary definition. Let G = ( V, E ) be a graph, let U ⊆ V , let c = ( X , . . . , X k ) be a partial k -coloring of G [ X ], and let c (cid:48) = ( Y , . . . , Y k ) bea partial k -coloring of G . We say that c (cid:48) agrees with c on U if X j ∩ U = Y j forall j ∈ { , . . . , k } . Theorem 12
For every k ≥ , Fair k -Division Under Conflicts issolvable in time O ( n k +2 ( Q + 1) k ) for a chordal conflict graph G , where Q = max ≤ j ≤ k p j ( V ( G )) .Proof Fix k ≥ G be a chordal graph equipped with profit functions p , . . . , p k : V ( G ) → Z + . We will show that we can compute the set Π ofall profit profiles of partial k -colorings of G in the stated running time. Themaximum satisfaction level over all profit profiles will then give the optimalvalue of Fair k -Division Under Conflicts for ( G, p , . . . , p k ).We first apply Lemma 5 and compute in time O ( n ) a nice tree decompo-sition ( T, { B t } t ∈ V ( T ) ) of G with O ( n ) bags, all of which are cliques. Recallthat by definition T is a rooted tree decomposition of G . Let r be the root of T . For every node t ∈ V ( T ), we denote by V t the union of all bags B t (cid:48) suchthat t (cid:48) ∈ V ( T ) is a (not necessarily proper) descendant of t in T .We traverse tree T bottom-up and use a dynamic programming approachto compute, for every node t ∈ V ( T ) and every partial k -coloring c of G [ B t ],the family P ( t, c ) of all profit profiles of partial k -colorings of G [ V t ] that agreewith c on B t .Since ( T, { B t } t ∈ V ( T ) ) is a nice tree decomposition, we have B r = ∅ ; inparticular, the trivial partial k -coloring ∅ k consisting of k empty sets is theonly partial k -coloring of G [ B r ]. Thus, since V r = V ( G ) and every partial k -coloring of G agrees with the trivial partial k -coloring of G [ B r ] on B r , theset P ( r, ∅ k ) is the set of all profit profiles of partial k -colorings of G , which iswhat we want to compute.We consider various cases depending on the type of a node t ∈ V ( T ) in thenice tree decomposition. For each of them we give a formula for computingthe set P ( t, c ) from the already computed sets of the form P ( t (cid:48) , c (cid:48) ) where t (cid:48) isa child of t in T , and argue why the formula is correct. t is a leaf node. By the definition of a nice tree decomposition it follows that B t = ∅ .Thus, the only partial k -coloring of G [ B t ] is the trivial one, ∅ k . Clearly, P ( t, ∅ k ) = { (0 , . . . , } .2. t is an introduce node. By definition, t has exactly one child t (cid:48) and B t = B t (cid:48) ∪ { v } holds forsome vertex v ∈ V \ X t (cid:48) . Clearly, V t = V t (cid:48) ∪ { v } , and this is a disjointunion. (If v ∈ V t (cid:48) , then the subtree of T consisting of all bags B τ such that v ∈ B τ is not connected; a contradiction.) Consider an arbitrary partial k -coloring c = ( X , . . . , X k ) of G [ B t ]. We want to compute P ( t, c ) usingthe set P ( t (cid:48) , c (cid:48) ), where c (cid:48) = ( X \ { v } , . . . , X k \ { v } ). (Note that c (cid:48) is apartial k -coloring of G [ B t (cid:48) ].) We claim that the following equality holds: P ( t, c ) = (cid:26) { q + e j ( p j ( v )) | q ∈ P ( t (cid:48) , c (cid:48) ) } , if v ∈ X j for some j ∈ { , . . . , k } ; P ( t (cid:48) , c (cid:48) ) , otherwise . To show the recurrence, note first that if for all j ∈ { , . . . , k } we have v / ∈ X j , then c (cid:48) = c and thus P ( t, c ) = P ( t (cid:48) , c (cid:48) ) in this case. If, however, v ∈ X j for some j ∈ { , . . . , k } , then there can only be one such j , andthus c (cid:48) = ( X , . . . , X j − , X j \ { v } , X j +1 , . . . , X k ). In this case, we will needthe fact that v is not adjacent to any vertex of V t (cid:48) \ B t (cid:48) . Indeed, applyingLemma 6 to a = t and b = t (cid:48) shows that no vertex of V ( G ) \ V t (cid:48) is adjacentto any vertex of V t (cid:48) \ B t (cid:48) , hence the statement follows since v ∈ V ( G ) \ V t (cid:48) .The fact that all neighbors of v in the set V t (cid:48) are contained in B t (cid:48) impliesthat for every partial k -coloring of G [ V t (cid:48) ] that agrees with c (cid:48) on B t (cid:48) , adding v to the j -th color class will result in a partial k -coloring of G [ V t ] that agreeswith c on B t . Thus, there is a bijective correspondence between the set ofpartial k -colorings of G [ V t ] that agree with c on B t and those of G [ V t (cid:48) ] thatagree with c (cid:48) on B t (cid:48) , given by removing v from the j -th color class. Thisimplies the claimed equality P ( t, c ) = { q + e j ( p j ( v )) | q ∈ P ( t (cid:48) , c (cid:48) ) } .3. t is a forget node. By definition, t has exactly one child t (cid:48) in T and B t = B t (cid:48) \ { v } holdsfor some vertex v ∈ V \ B t . Thus, V t = V t (cid:48) . Consider an arbitrary partial k -coloring c = ( X , . . . , X k ) of G [ B t ]. We claim that the following equalityholds: P ( t, c ) = P ( t (cid:48) , c ) ∪ (cid:91) j : X j = ∅ P ( t (cid:48) , ( X , . . . , X j − , { v } , X j +1 . . . , X k )) . Consider an arbitrary partial k -coloring ( Y , . . . , Y k ) of G [ V t ] that agreeswith c on B t . If v (cid:54)∈ Y j for all j ∈ { , . . . , k } , then ( Y , . . . , Y k ) agrees with c on B t (cid:48) . Suppose now that v ∈ Y j for some j ∈ { , . . . , k } . Then, j is unique.Furthermore, since B t (cid:48) is a clique in G and hence in G [ V t (cid:48) ], the fact that v ∈ Y j implies that Y j ∩ B t (cid:48) = { v } , and consequently X j = Y j ∩ B t = ∅ .In this case, the partial k -coloring ( Y , . . . , Y k ) agrees with the partial k -coloring ( X , . . . , X j − , { v } , X j +1 , . . . , X k ) of G [ V t (cid:48) ] on B t (cid:48) . Thus, everypartial k -coloring of G [ V t ] that agrees with c on B t either agrees with c air allocation of indivisible goods under conflict constraints 21 on B t (cid:48) or agrees with ( X , . . . , X j − , { v } , X j +1 . . . , X k ) on B t (cid:48) for some j ∈ { , . . . , k } such that X j = ∅ . Similar arguments can be used to showthe converse inclusion, that is, any partial k -coloring of G [ V t (cid:48) ] that satisfiesone of the above conditions is a partial k -coloring of G [ V t ] that agrees with c on B t . This implies the claimed equality.4. t is a join node. By definition, t has exactly two children t and t in T and it holds that B t = B t = B t . We claim that V t ∩ V t = B t . It is clear that B t ⊆ V t ∩ V t . Assume for contradiction that there is a vertex v ∈ V ( G ) suchthat v ∈ ( V t ∩ V t ) \ B t . Then there are nodes t (cid:48) and t (cid:48) of T such that v ∈ B t (cid:48) , v ∈ B t (cid:48) , and t (cid:48) and t (cid:48) are (possibly not proper) descendants of t and t , respectively. It follows that the subgraph of T consisting of allbags containing v is not connected; a contradiction. Thus B t = V t ∩ V t , asclaimed. Furthermore, applying Lemma 6 to a = t and b = t we can showthat no vertex of V t \ B t is adjacent in G to any vertex of V ( G ) \ V t . Since V t \ B t ⊆ V ( G ) \ V t , this implies that no vertex in V t \ B t is adjacent in G to any vertex of V t \ B t .Consider now an arbitrary partial k -coloring c = ( X , . . . , X k ) of G [ B t ](observe that c is also a partial k -coloring of G [ B t ] and G [ B t ]). In thiscase, we have the following recurrence relation: P ( t, c ) = { q + q − ( p ( X ) , . . . , p k ( X k )) | q ∈ P ( t , c ) , q ∈ P ( t , c ) } . It is clear that for any partial k -coloring ( X (cid:48) , . . . , X (cid:48) k ) of G [ V t ] that agreeswith c on B t , the k -tuples ( X (cid:48) ∩ V t , . . . , X (cid:48) k ∩ V t ) and ( X (cid:48) ∩ V t , . . . , X (cid:48) k ∩ V t )are partial k -colorings of G [ V t ] and G [ V t ] that agree with c on B t and B t , respectively. The fact that no vertex in V t \ B t is adjacent in G to anyvertex in V t \ B t implies that the other direction is also true: given partial k -colorings ( X (cid:48) , . . . , X (cid:48) k ) and ( X (cid:48)(cid:48) , . . . , X (cid:48)(cid:48) k ) of G [ V t ] and G [ V t ] that agreewith c on B t and B t , respectively, we have X (cid:48) j ∩ B t = X (cid:48)(cid:48) j ∩ B t = X j for all j ∈ { , . . . , k } , and thus ( X (cid:48) ∪ X (cid:48)(cid:48) , . . . , X (cid:48) k ∪ X (cid:48)(cid:48) k ) is a partial k -coloring of G [ V t ] that agrees with c on B t . Furthermore, for all j ∈ { , . . . , k } , the factthat V t ∩ V t = B t implies that X (cid:48) j ∩ X (cid:48)(cid:48) j = X j , and hence p j ( X (cid:48) j ∪ X (cid:48)(cid:48) j ) = p j ( X (cid:48) j ) + p j ( X (cid:48)(cid:48) j ) − p j ( X j ). The claimed equality follows.It remains to estimate the time complexity of the algorithm. We computea nice tree decomposition of G in time O ( n ). Each of the O ( n ) bags is aclique, so in total we have O ( n k ) partial k -colorings per bag. Furthermore,note that for each partial coloring ( X , . . . , X k ) of any induced subgraph of G and each j ∈ { , . . . , k } , we have p j ( X j ) ∈ { , , . . . , Q } . Thus, each set P ( t, c )has at most ( Q + 1) k elements. For each of the O ( n k +2 ) pairs ( t, c ) where t isa node of T and c is a partial k -coloring of G [ B t ], we compute the set P ( t, c )using the formula corresponding to the type of node t . The time complexityof this step depends on the type of the node. Case 1 takes constant time. InCase 2, we check in constant time whether v ∈ X j for some j ∈ { , . . . , k } andthen compute the set P ( t, c ) in time O (( Q + 1) k ). In Case 3, we first computein (constant) time O ( k ) the set of indices j ∈ { , . . . , k } such that X j = ∅ . Then, the union given by the formula can be computed in time O (( Q + 1) k ),simply by iterating over all families in the union and keeping track of whichof the O (( Q + 1) k ) profit profiles appear in any of the families. Finally, Case 4can be done in time O (( Q + 1) k ). Altogether, this results in running time O (( Q + 1) k ) for each fixed t ∈ V ( T ) and each partial k -coloring c of B t .Consequently, the total running time of the algorithm is O ( n k +2 ( Q + 1) k ). (cid:117)(cid:116) T, { B t } t ∈ V ( T ) ) of a graph G is defined as max t ∈ V ( T ) | B t | −
1. The treewidth of a graph G is the minimumpossible width of a tree decomposition of G . A graph class G is said to be of bounded treewidth if there exists a nonnegative integer (cid:96) such that each graphin G has treewidth at most (cid:96) . For each fixed treewidth bound (cid:96) , given a graph G of treewidth at most (cid:96) , a tree decomposition of G of width at most (cid:96) canbe computed in linear time [14]. Such a decomposition leads to linear-timealgorithms for many problems that are generally NP-hard (see, e.g., [4, 19]).A similar approach as the one used in the proof of Theorem 12 for solvingthe Fair k -Division Under Conflicts on chordal graphs can be used ongraphs of bounded treewidth. For every k ≥
1, the
Fair k -Division UnderConflicts is solvable in pseudo-polynomial time O ( n ( n + ( Q + 1) k )) forgraphs of bounded treewidth, where Q = max ≤ j ≤ k p j ( V ( G )). (The constanthidden in the O notation depends on the value of k and the bound on thetreewidth.)Fix k, (cid:96) ≥ G, p , . . . , p k ) be the input to Fair k -Division UnderConflicts such that the treewidth of G is at most (cid:96) . In time (cid:96) O ( (cid:96) ) n we cancompute a tree decomposition of G a width at most (cid:96) using the algorithm ofBodlaender [14]. Clearly, the obtained tree decomposition has at most (cid:96) O ( (cid:96) ) n bags. By Lemma 3 it follows that we can compute in time O ( (cid:96) O ( (cid:96) ) n ) anice tree decomposition T = ( T, { B t } t ∈ V ( T ) ) of G of width at most (cid:96) , with O ( n ) bags. Every bag has at most (cid:96) + 1 vertices, so for every bag we have atmost a constant number, ( (cid:96) + 1) k +1 , partial k -colorings, which in total gives O ( n ) pairs ( t, c ) of a node t ∈ V ( T ) and a partial k -coloring c of t . For eachsuch pair ( t, c ), we again compute the family P ( t, c ) of all profit profiles ofpartial k -colorings of G [ V t ] that agree with c on B t . Since T is a nice treedecomposition, every node is of one of the four possible types, and in Cases 1,2, and 4 we have identical equalities as in the corresponding cases in the proofof Theorem 12, while in Case 3 the union over all j such that X j = ∅ of the sets P ( t (cid:48) , ( X , . . . , X j − , { v } , X j +1 . . . , X k )) is replaced by the union over all j suchthat X j ∪ { v } is an independent set in G of the sets P ( t (cid:48) , ( X , . . . , X j − , X j ∪{ v } , X j +1 . . . , X k )). Since we can compute the adjacency matrix of G in time O ( n ), we may assume that adjacency checks can be done in constant time.Thus, the expressions in the formulas corresponding to each of the Cases 2and 3 can be evaluated in time O (( Q + 1) k ), while the corresponding time air allocation of indivisible goods under conflict constraints 23 complexity of Case 4 is O (( Q + 1) k ). Altogether, this gives us the claimedrunning time and yields the following. Theorem 13
For every k ≥ and (cid:96) ≥ , Fair k -Division Under Con-flicts is solvable in time O ( n ( n + ( Q + 1) k )) for a graph G of treewidth atmost (cid:96) , where Q = max ≤ j ≤ k p j ( V ( G )) . All the pseudo-polynomial dynamic programming algorithms presented inthis paper share the following characteristics. Throughout the execution fea-sible states are computed, where every state describes a profit allocationgiven by a feasible solution of
Fair k -Division Under Conflicts . Eachsuch state is represented by a k -dimensional vector ( q , . . . , q k ) ∈ Z k + , whereevery entry q j describes the profit p j ( X j ) assigned to agent j by a par-tial coloring ( X , . . . , X k ). While Pareto-dominated states can be eliminated,the total number of states remains trivially bounded by ( Q + 1) k , where Q = max ≤ j ≤ k p j ( V ( G )). The optimal solution with maximum satisfactionlevel can be determined at the end of such an algorithm by simply goingthrough all generated states and inspecting their satisfaction levels.In a canonical step of our algorithms a vertex v (resp. item) is feasi-bly assigned to an agent j thereby generating a new state ( q , . . . , q j − , q j + p j ( v ) , q j +1 , . . . , q k ) from a previous state ( q , . . . , q k ). The decisions taken bythe algorithms depend only on the graph but not on the profit values of pre-viously generated states. Every vertex is assigned to each agent at most once.Under these preconditions, we can derive a fully polynomial time approxi-mation scheme (FPTAS) for each such dynamic programming algorithm (con-sidering k as a constant). For an optimal satisfaction level z ∗ , an FPTAScomputes for every given ε >
0, an approximate solution with satisfactionlevel z A fulfilling z A ≥ z ∗ / (1 + ε ) with running time polynomial in the size ofthe encoded input and in 1 /ε .The FPTAS is based on the observation that the k profit values of a solutioncan also be seen as k objective function values in a multiobjective optimizationproblem. Thus, the technique for deriving an FPTAS for the multiobjectiveknapsack problem described in [24] can be applied as follows.Denote the upper bound for the profit assigned to agent j by UB j = p j ( V ( G )) and set u j = (cid:100) n log ε UB j (cid:101) , where, as usual, n = | V ( G ) | . Partitionthe profit range for each agent j into u j intervals[1 , (1 + ε ) /n ) , [(1 + ε ) /n , (1 + ε ) /n ) , [(1 + ε ) /n , (1 + ε ) /n ) , . . . [(1 + ε ) ( u j − /n , (1 + ε ) u j /n ] . To obtain an FPTAS from the generic dynamic programming algorithm in-dicated above we restrict the possible profit values q j allocated to agent j tothe lower interval endpoints of these intervals. The FPTAS mimics exactly theoperations of the exact dynamic program, but whenever a vertex v is assigned to j , the resulting profit q j + p j ( v ) is rounded down to the nearest interval end-point. Note that this does not change the steps of the dynamic program sincewe assumed that its decisions do not depend on the profit values of states.Since u j is in O ( n/ε · log ( UB j )), which is polynomial in the length of theencoded input (recall that ε ≤ ln(1 + ε ) < log (1 + ε ) for ε ∈ (0 , O (( n/ε ) k (log Q ) k ).Concerning the loss of accuracy we can proceed similarly to [24] and com-pare an arbitrary state ( q , q , . . . , q k ) of the exact dynamic program to somestate of the FPTAS consisting of lower interval endpoints (˜ q , ˜ q , . . . , ˜ q k ). Forevery state ( q , . . . , q j , . . . , q k ) generated by the exact algorithm after assign-ing i vertices to agent j , we claim that in the FPTAS there exists a state(˜ q , ˜ q , . . . , ˜ q k ) of lower interval endpoints such that q j ≤ (1 + ε ) i/n ˜ q j . (5)This claim can be shown by induction. For i = 1, there was one vertex v assigned to agent j giving profit q j = p j ( v ). In the FPTAS, there will bea state where ˜ q j is the largest lower interval endpoint not exceeding q j . Byconstruction of the intervals, we have (1 + ε ) /n ˜ q j ≥ q j .Assuming the claim to be true for some i −
1, we consider the i -th assign-ment of a vertex v to j . In the exact algorithm, p j ( v ) is added to some value q j for which there exists a lower interval endpoint ˜ q j fulfilling q j ≤ (1+ ε ) ( i − /n ˜ q j .During the FPTAS, p j ( v ) will also be added to ˜ q j and the result will berounded down to a lower interval endpoint ˜ q (cid:48) with (1 + ε ) /n ˜ q (cid:48) ≥ ˜ q j + p j ( v ) ≥ (1 + ε ) − ( i − /n q j + p j ( v ) ≥ (1 + ε ) − ( i − /n ( q j + p j ( v )). Moving terms around,this proves (5) for the new profit q j + p j ( v ).Since there can be at most n vertices assigned to any agent, (5) holds alsofor the satisfaction level of the optimal solution. Summarizing, we conclude: Theorem 14
For all pseudo-polynomial dynamic programming algorithms inthis paper (see Theorem 8, Theorem 10 and Corollary 11, Theorem 12, andTheorem 13) there exists an FPTAS.
To put Theorem 14 in perspective, recall that by Theorem 4 no constant-factor approximation for
Fair k -Division Under Conflicts exists for gen-eral graphs, unless P = NP. In this paper we introduced the
Fair k -Division Under Conflicts andstudied it from a computational complexity point of view, with respect to var-ious restrictions on the conflict graph. In particular, we could show that theproblem is strongly NP-hard on general bipartite conflict graphs, but it canbe solved in pseudo-polynomial time on biconvex bipartite graphs, on chordalgraphs, on cocomparability graphs, and on graphs of bounded treewidth. Thereare other graph classes sandwiched between the two classes of our results, for air allocation of indivisible goods under conflict constraints 25 which the complexity of Fair k -Division Under Conflicts is still open.In particular, we can derive open problems from the following sequence of in-clusions: biconvex bipartite ⊆ convex bipartite ⊆ interval bigraph ⊆ chordalbipartite ⊆ bipartite. We believe that a positive result for convex bipartitegraphs could be within reach. Outside this chain of inclusions, we pose thecomplexity of the problem for planar bipartite conflict graphs as another in-teresting open question. Appendix: A remark on biconvex graphs
Biconvex bipartite graphs were characterized by forbidden induced subgraphs by Tuckerin [37]. The list of forbidden induced subgraphs includes all cycles except the cycle of lengthfour and five additional graphs, including the two graphs F and F depicted in Figure 2. F F Fig. 2
Two forbidden induced subgraphs for biconvex bipartite graphs.
Proposition 1
There exists a disconnected biconvex bipartite graph that is not an inducedsubgraph of any connected biconvex bipartite graph.Proof
Consider the graph G depicted in Figure 3. a b b a a b b a a b b a G Fig. 3
A 12-vertex biconvex bipartite graph and a biconvex labeling of it.As shown by the vertex labeling in the figure, G is a biconvex bipartite graph. Conse-quently, the graph G + K , the disjoint union of G and the complete graph of order two, isalso a biconvex bipartite graph. We will show that G + K is not an induced subgraph ofany connected biconvex bipartite graph.Fix a labeling of G as in Figure 3, take a disjoint copy of K , call it G (cid:48) , and supposefor a contradiction that the disjoint union G + G (cid:48) is an induced subgraph of a connected6 N. Chiarelli et al.biconvex bipartite graph H . Let A and B denote the two parts of a bipartition of H so that { a , . . . , a } ⊆ A (and then { b , . . . , b } ⊆ B ).Since H is connected, it contains a path from V ( G (cid:48) ) to V ( G ). Let P be a shortest suchpath. Since the sets V ( G ) and V ( G (cid:48) ) are disjoint and the are no edges between them, P hasat least three vertices. Let x be the only vertex on P that has a neighbor in G , let y be theneighbor of x on P such that y (cid:54)∈ V ( G ), and let z be defined as follows: z = (cid:26) the neighbor of y on P other than x, if P has at least 4 vertices;the neighbor of y in G (cid:48) , if P has exactly three vertices.Since H is bipartite, it contains no cycle of length three. This implies that vertices x and z are not adjacent to each other.By symmetry of G , we may assume that x ∈ A (and thus y ∈ B and z ∈ A ). Furthermore,by the minimality of P , vertices y and z do not have any neighbors in V ( G ). We make aseries of observations about the neighborhood of x in V ( G ). – Vertex x cannot be adjacent to both b and b , since otherwise H would contain aninduced F with vertex set { x, y, z, b , a , b , a } .By symmetry, we may assume that x is not adjacent to b . – Vertex x is not adjacent to b . Suppose that it is. Then x is not adjacent to b , sinceotherwise the set { x, b , a , b , a , b } would induce a 6-cycle in H . But now, H containsan induced F with vertex set { x, b , a , b , a , b , a } , a contradiction. – Vertex x is adjacent to b . Suppose that this is not the case. Then x is not adjacentto b i for i ∈ { , } , since otherwise H would contain an induced F with vertex set { x, b i , a , b , a , b , a } . Therefore, the only possible neighbor of x in V ( G ) is b . Butnow, H contains an induced F with vertex set { x, b , a , b , a , b , a } , a contradiction. – Vertex x is adjacent to b , since otherwise H would contain an induced F with vertexset { y, x, b , a , b , a , b } .To conclude the proof, we observe that H contains an induced F with vertex set { z, y, x, b , a , b , a , b , a } , a contradiction. (cid:117)(cid:116) References
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