1 Introduction
Allocating resources to several agents in a satisfactory way is a classical problem in combinatorial optimization. In particular, interesting questions arise if agents have different valuations of resources or if additional constraints are imposed for a feasible allocation. In this work we study the fair allocation of
indivisible goods or items to a set of agents. Each agent has its own additive utility function over the set of items. The goal is to assign every item to exactly one of the agents such that the minimal utility over all agents is as large as possible. Related problems of fair allocation are frequently studied in Computational Social Choice, see, e.g., [9]. In the area of Combinatorial Optimization a similar problem is wellknown as the Santa Claus problem (see [5]), which can be also seen as weight partitioning as well as a scheduling problem.In this paper we look at the problem from a graph theoretical perspective and add a major new aspect to the problem. We allow an incompatibility relation between pairs of items, meaning that incompatible items should not be allocated to the same agent. This can reflect the fact that items rule out their joint usage or simply the fact that certain items are identical (or from a similar type) and it does not make sense for one agent to receive more than one of these items. We will represent such a relation by a conflict graph where vertices correspond to items and edges express incompatibilities. Now, every feasible allocation to one agent must be an independent set in the conflict graph. This means that the overall solution can also be expressed as a partial coloring of the conflict graph , but in addition every vertex/item has a profit value for every color/agent and the sum of profits of vertices/items assigned to one color/agent should be optimized in a maximin sense.
We believe that this problem combines aspects of independent sets, graph coloring, and weight partitioning in an interesting way, offering new perspectives to look at these classical combinatorial optimization problems.
Disjunctive constraints represented by conflict graphs were considered for a wide variety of combinatorial optimization problems. We just mention the knapsack problem ([21, 22]), bin packing ([19]), scheduling (e.g., [8, 14]) and problems on graphs (e.g., [12]).
For a formal definition of our problem we consider a set of items with cardinality and profit functions . The satisfaction level of an ordered partition of (with respect to ) is defined as the minimum of the resulting profits , where . The classical fair division problem can be stated as follows.
Fair Division of Indivisible Goods
Input:  A set of items, profit functions . 

Task:  Compute an ordered partition of with maximum satisfaction level. 
For the special case, where all profit functions are identical, i.e., , the problem can also be represented in a scheduling setting. There are identical machines and jobs, which have to be assigned to the machines by a partitioning. The goal is to maximize the minimal completion time (corresponding to the satisfaction level) over all machines. It was pointed out in [13] that this problem is weakly NPhard even for machines. Indeed, it is easy to see that an algorithm deciding the above scheduling problem for two machines would also decide the classical Partition problem: given integers , can they be partitioned into two subsets with equal sums? For , one can simply add jobs of length one half of the sum of weights in the instance of Partition. If is not fixed, but part of the input, the same scheduling problem is strongly NPhard as mentioned in [4]. In fact, an instance of the strongly NPcomplete 3Partition problem with elements and target bound could be decided by any algorithm for the scheduling problem with jobs, machines and a desired minimal completion time equal to . We conclude for later reference.
Observation 1.1
Fair Division of Indivisible Goods, even with identical profit functions, is weakly NPhard for any constant and strongly NPhard for being part of the input.
Note that the problem is still only weakly NPhard for constant even for arbitrary profit functions, since we can construct a pseudopolynomial algorithm solving the problem with a dimensional dynamic programming array.
The first elaborate treatment of Fair Division of Indivisible Goods was given in [7], where two approximation algorithms with bounded (but not constant) approximation ratio were given. They also mention that the problem cannot be approximated by a factor better than (under ). In [15] further approximation results were derived. In 2006 Bansal and Sviridenko [5] coined the term Santa Claus problem, which corresponds to the variant of the above problem when is not fixed but part of the input. Since then a huge number of approximation results have appeared on this problem of allocating indivisible goods exploring different concepts of objective functions and various approximation measures.
A different specialization is assumed in the widely studied Restricted MaxMin Fair Allocation problem. This is a special case of Fair Division of Indivisible Goods where every item has a fixed valuation and every kid either likes or ignores item , i.e., the profit function . A fairly recent overview of approximation results both for this restricted setting as well as for the general case of the Santa Claus problem can be found in [3].
In this paper we study a generalization of Fair Division of Indivisible Goods, where a conflict graph on the set of items to be divided is introduced. An edge means that items and should not be assigned to the same subset of the partition. The conflict graph immediately gives rise to (partial) colorings of the graph which were studied by Berge [6] and de Werra [24].
Definition 1
A partial coloring of a graph is a sequence of pairwise disjoint independent sets in .
Combining the profit structure with the notion of coloring we define for the profit functions and for each partial coloring a tuple , called the profit profile of . The minimum profit of a profile, i.e., , is the satisfaction level of . Now we can define the problem considered in this paper:
Fair Division Under Conflicts
Input:  A graph , profit functions . 

Task:  Compute a partial coloring of with maximum satisfaction level. 
In the hardness reductions of this paper we will frequently use the decision version of this problem: for a given , does there exists a partial coloring of with satisfaction level at least ?
Note that an optimal partial coloring does not necessarily select all vertices from . Furthermore, note also that for , the problem coincides with the weighted independent set problem. In particular, since the case of unit weights and generalizes the independent set problem, we obtain the following result.
Observation 1.2
Fair Division Under Conflicts is strongly NPhard.
Thus, the addition of the conflict structure gives rise to a much more complicated problem, since Fair Division of Indivisible Goods (which arises naturally as a special case for an edgeless conflict graph ) is trivial for and only weakly NPhard for (see Observation 1.1).
In this contribution we first introduce a general concept of extendable graph families and show that for every such graph class in which Independent Set is NPcomplete, the decision version of our Fair Division Under Conflicts is strongly NPcomplete when the conflict graphs are in (Section 2.1). By a similar reasoning we can also reach a strong inapproximability result for our problem. For bipartite conflict graphs as well as their line graphs Fair Division Under Conflicts can be shown to be strongly NPhard (Section 2.2) although the corresponding Independent Set problem is polynomialtime solvable. On the other hand, for the relevant special case of biconvex bipartite graphs (cf. [17], [18]), Fair Division Under Conflicts can be solved by a pseudopolynomial time algorithm. This result is based on an insightful pseudopolynomial algorithm for the problem on a cocomparability conflict graph (Section 3). See Fig. 1 for a summary of results.
2 Hardness results
Observation 1.2 shows that Fair Division Under Conflicts is strongly NPhard even for for general graphs, while Observation 1.1 shows the weak NPhardness of the problem for constant in the absence of conflicts. In what follows, we show that Fair Division Under Conflicts is strongly NPhard also for all , for various wellknown graph classes.
2.1 General hardness results
We start with the following general property of graph classes. Let us call a class of graphs sustainable if every graph in the class can be enlarged to a graph in the class by adding to it one vertex. More formally, is sustainable if for every graph there exists a graph and a vertex such that . Clearly, any class of graphs closed under adding isolated vertices, or under adding universal vertices is sustainable. This property is shared by many well known graph classes, including planar graphs, bipartite graphs, chordal graphs, perfect graphs, etc. Furthermore, all graph classes defined by a single nontrivial forbidden induced subgraph are sustainable.
Lemma 1
For every graph with at least two vertices, the class of free graphs is sustainable.
Proof
Let be the class of free graphs and let . Since has at least two vertices, it cannot have both a universal and an isolated vertex. If has no universal vertex, then the join of with results in a graph in properly extending . If has no isolated vertex, then the disjoint union of with results in a graph in properly extending .∎
For an example of a nonsustainable graph class closed under vertex deletion, consider the family of all cycles and their induced subgraphs. Then every cycle is in but cannot be extended to a larger graph in . The importance of sustainable graph classes for Fair Division Under Conflicts is evident from the following theorem.
Theorem 2.1
Let be a sustainable class of graphs for which the decision version of Fair Division Under Conflicts is (strongly) NPcomplete. Then, for every , the decision version of Fair Division Under Conflicts with conflict graphs from is (strongly) NPcomplete.
Proof
Let be a sustainable class of graphs for which the decision version of Fair Division Under Conflicts is (strongly) NPcomplete and let . Let be an instance of Fair Division Under Conflicts (decision version) such that . Since is sustainable, there exists a graph such that for some additional vertices . We now define the profit functions . For all , let
and in addition let, for all , let
Observe that has a partial coloring such that for all if and only if has a partial coloring such that for all . Since all the numbers involved in the reduction are polynomially bounded we conclude that Fair Division Under Conflicts with conflict graphs from is also (strongly) NPcomplete.∎
Since the Independent Set problem is a special case of the Fair Division Under Conflicts, Theorem 2.1 immediately implies the following.
Corollary 1
Let be a sustainable class of graphs for which Independent Set is NPcomplete. Then, for every , the decision version of Fair Division Under Conflicts with conflict graphs from is strongly NPcomplete.
It is known (see, e.g., [2]) that for every graph that has a component that is not a path or a subdivision of the claw, Independent Set is NPcomplete on free graphs. Thus, for every such graph , Lemma 1 and Corollary 1 imply that for every , Fair Division Under Conflicts (decision version) with free conflict graphs is strongly NPcomplete. By using a similar argument, we even get a strong inapproximability result for general graphs.
Theorem 2.2
For every and every , it is NPhard to approximate Fair Division Under Conflicts within a factor of , even for unit profit functions.
Proof
Fix an integer . We give a reduction from the Independent Set problem: find a maximum independent set in a given graph . We construct a graph by taking copies of and by adding all possible edges between vertices from different copies. Furthermore we take unit profit functions from to . We claim that the maximum size of an independent set in equals the maximum satisfaction level of a partial coloring in (with respect to the unit profit functions ). Given a maximum independent set in of size one can immediately obtain a partial coloring of with satisfaction level by inserting all vertices of in the th copy of into , for all . On the other hand, given a partial coloring of with satisfaction level , one can simply choose , which is an independent set completely contained in one copy of . Thus, corresponds to an independent set in of size .
Suppose that for some there exists a polynomialtime algorithm that approximates Fair Division Under Conflicts within a factor of on input instances with unit profit functions. We will show that this implies the existence of a polynomialtime algorithm approximating the Independent Set problem within a factor of where . As shown by Zuckerman [25], this would imply .
Consider an input graph to the Independent Set problem. The algorithm proceeds as follows. If , then the graph is of constant order and the problem can be solved optimally in time. If , then the graph is constructed following the above reduction, a partial coloring is computed using algorithm on equipped with unit profit functions, and a subset of corresponding to is returned. Clearly, the algorithm runs in polynomial time and computes an independent set in . Let denote the maximum satisfaction level of a partial coloring in . By the above claim, the independence number of equals . Thus, to complete the proof, it suffices to show that . By assumption on , we have that . We want to show that , or, equivalently, . After some straightforward algebraic manipulations, this inequality simplifies to the equivalent inequality , which is true by assumption.∎
2.2 Bipartite graphs and their line graphs
In this section we show that for all , Fair Division Under Conflicts is NPhard in two classes of graphs where the Independent Set problem is solvable in polynomial time: the class of bipartite graphs and the class of line graphs of bipartite graphs. Recall that for a given graph , its line graph has a vertex for each edge of , with two distinct vertices adjacent in the line graph if and only if the corresponding edges share an endpoint in .
The proof for bipartite graphs shows strong NPhardness even for the case when all the profit functions are equal.
Theorem 2.3
For each integer , the decision version of Fair Division Under Conflicts is strongly NPcomplete in the class of bipartite graphs.
Proof
We use a reduction from the decision version of the Clique problem: Given a graph and an integer , does contain a clique of size ? Consider an instance of Clique such that . We define an instance of Fair Division Under Conflicts (decision version) consisting of a bipartite conflict graph , profit functions , and a lower bound on the required satisfaction level. The graph has a vertex for each vertex of the graph as well as for each edge of and new vertices . It is defined as follows:
The lower bound on the satisfaction level is defined by setting . For ease of notation we set and we furthermore introduce a second integer such that , where . (Note that .) With this, the profit functions , for all , are defined as
Note that all the profits introduced as well as the number of vertices and edges of are polynomial in . To complete the proof, we show that has a clique of size if and only if has a partial coloring with satisfaction level at least . First assume that has a clique of size . We construct a partial coloring of by setting
Observe that the partial coloring gives rise to the corresponding profit profile with all entries equal to , which establishes one of the two implications.
Suppose now that there exists a partial coloring of for which the profit profile has all entries . Since for each , the total profit of the set is only , the partial coloring must use exactly one of the vertices in each color class. We may assume without loss of generality that for all . Let be the set of uncolored vertices in w.r.t. the partial coloring . Since for each of the profit functions , the difference between the overall sum of the profits of vertices of and is equal to , we clearly have , which implies that . Next, observe that every vertex of belongs to either or to , since otherwise we would have , contrary to the assumption that the satisfaction level of is at least .
Consider the sets and . Then and, since , it follows that contains exactly vertices from (if , then ) and at least vertices from . Let denote the set of all vertices of with a neighbor in . By the construction of and since , it follows that is of cardinality at least . Furthermore, since is independent, we have . Consequently, , hence equalities must hold throughout. In particular, is a clique of size in .∎
Theorem 2.4
For each integer , the decision version of Fair Division Under Conflicts is strongly NPcomplete in the class of line graphs of bipartite graphs.
Proof
Note that it suffices to prove the statement for . For , Theorem 2.1 applies, since the class of line graphs of bipartite graphs is sustainable. Indeed, if is the line graph of a bipartite graph , then the graph obtained from by adding to it an isolated vertex is the line graph of the bipartite graph obtained from by adding to it an isolated edge.
For , we use a reduction from the following problem: Given a bipartite graph and an integer , does contain two disjoint matchings and such that is a perfect matching and ? This problem was shown to be NPcomplete by Pálvölgi (see [20]). Consider an instance of this problem such that and is even. Then we define the following instance of the decision version of Fair Division Under Conflicts with a conflict graph , where is the line graph of . The lower bound on the satisfaction level is defined by setting . The profit functions are defined as for all , and for all . Clearly, all the profits introduced as well as the number of vertices and edges of are polynomial in . Recall that every matching in corresponds to an independent set in .
We now show that the instances of the two decision problems have the same answers. Suppose first that has two disjoint matchings and such that is a perfect matching and . Then the sequence is a partial coloring of such that
Conversely, suppose that has a partial coloring with satisfaction level at least . Then the independent sets and in are disjoint matchings in . Moreover, since
we obtain and . Thus, is a perfect matching in and any set of edges in is a matching in disjoint from . This proves that the decision version of Fair Division Under Conflicts is strongly NPcomplete in the class of line graphs of bipartite graphs.∎
3 Pseudopolynomial algorithms for special graph classes
As shown in Theorem 2.3, for each , Fair Division Under Conflicts is strongly NPcomplete in the class of bipartite graphs. This rules out the existence of a pseudopolynomial time algorithm for the problem in the class of bipartite graphs, unless . In this section we show that for every there is a pseudopolynomial time algorithm for the Fair Division Under Conflicts in a subclass of bipartite graphs, the class of biconvex bipartite graphs. The algorithm reduces the problem to the class of bipartite permutation graphs. To solve the problem in the class of bipartite permutation graphs, we develop a solution in a more general class of graphs, the class of cocomparability graphs. A graph is a comparability graph if it has a transitive orientation, that is, if each of the edges of
can be replaced by exactly one of the ordered pairs
and so that the resulting set of directed edges is transitive (that is, for every three vertices , if and , then ). A graph is a cocomparability graph if its complement is a comparability graph. Comparability graphs and cocomparability graphs are wellknown subclasses of perfect graphs. The class of cocomparability graphs is a common generalization of the classes of interval graphs, permutation graphs, and trapezoid graphs (see, e.g., [10, 16]).Since every bipartite graph is a comparability graph, Theorem 2.3 implies that for each , Fair Division Under Conflicts is strongly NPcomplete in the class of comparability graphs. For cocomparability graphs, we prove that the problem is solvable in pseudopolynomial time. The key result in this direction is the following lemma, which will also be used in our proof of Theorem 3.2.
Lemma 2
For every , given a cocomparability graph and profit functions , the set of all profit profiles of partial colorings of can be computed in time , where .
The proof is based on a directed acyclic graph representing a transitive orientation of the complement of .
Proof
Let be a cocomparability graph. In time , we compute the complement of and a transitive orientation of it [23]. Since is a directed acyclic graph, one can compute in linear time a topological sort of , that is, an ordering of the vertices such that if is an arc of , then (see, e.g., [11]). Note that

a set with is independent in if and only if is a directed path in .
Thus, a partial coloring in corresponds to a collection of vertexdisjoint directed paths in , and vice versa. We process the vertices of in the ordering given by the topological sort of and try all possibilities for the color (if any) of the current vertex in order to extend a partial coloring of the already processed subgraph of with . (In terms of , we choose which of the directed paths will be extended into .) To avoid introducing additional terminology and notation, we present the details of the algorithm in terms of partial colorings of (instead of systems of disjoint paths in ).
For each and each tuple , we compute the set of all tuples such that there exists a partial coloring of the subgraph of induced by (which is empty if ) such that and
(1) 
for all . Note that for each , the possible values of the th coordinate of any member of belong to the set where . Thus, each set has at most elements. Note also that the total number of sets is of the order .
In what follows we explain how to compute the sets . For , the only feasible choice for the tuple is and we set . This is correct since the only partial coloring of the graph with no vertices is the tuple . Suppose that and that the sets are already computed for all . Fix a tuple . To describe how to compute the set , we will use the following notation. Addition and subtraction of tuples is defined componentwise and for all , we denote by the tuple with all coordinates equal to , except that the th coordinate is equal to . We consider three cases. For each of them, we first give a formula for computing the set and then we argue why the formula is correct.

[label=(),itemindent=*]

If appears exactly once as a coordinate of , say , then we set
(4) where denotes the set of all vertices such that is an arc of . (Note that for all , since is a topological sort of .)
Let and consider a partial coloring of the subgraph of induced by such that and equality (1) holds for all . Then . In particular, . Let and let
Note that if then . Indeed, digraph is an orientation of the complement of , in which vertices and are adjacent (recall that they belong to the independent set in ). This implies that either or is an arc of , but since and is a topological sort of , the pair must be an arc of . Let be the tuple obtained from by replacing with , and let be the tuple obtained from by replacing with . Then is a partial coloring of the subgraph of induced by such that equality obtained from (1) by replacing with and with holds for each . Furthermore, . This shows that if , then the tuple belongs to the union
For the converse direction, let , let be the tuple obtained from by replacing with , and let . Then, there exists a partial coloring of the subgraph of induced by such that for each , we have and equality obtained from (1) by replacing with and with holds. Let be the tuple obtained from by replacing with . To show that is a partial coloring of the subgraph of induced by , it suffices to verify that is an independent set in . If , then is independent. Suppose that . Then, by (), corresponds to a directed path in ending in . Extending this path with vertex results in a directed path in with vertex set , which shows, again by (), that is independent in . Clearly, we have that , and hence is a partial coloring of the subgraph of induced by equality (1) holds for each . Furthermore, . This shows that if , then the tuple belongs to . Therefore, equation (4) is correct.
Finally, the set of all profit profiles of partial colorings of equals to the union, over all , of the sets .
The algorithm can be easily modified so that for each profit profile also a corresponding partial coloring is computed. We would just need to store, for each , each , and each tuple , one partial coloring of the subgraph of induced by such that and equality (1) holds for all .
It remains to estimate the time complexity of the algorithm. For each
and each of the tuples , we can decide which of the three cases (i)–(iii) occurs in time . Step (2) takes constant time, step (3) takes time , and step (4) can be implemented in time . Altogether, this results in running time for each fixed and each tuple . Consequently, the total running time of the algorithm is . ∎Lemma 2 implies the following.
Theorem 3.1
For every , Fair Division Under Conflicts is solvable in time for cocomparability conflict graphs , where .
Proof
By Lemma 2, we can compute the set of all profit profiles of partial colorings of in the stated running time. For each profit profile in , we can determine the satisfaction level of the corresponding partial coloring of . Taking the maximum satisfaction level over all profiles gives the optimal value of Fair Division Under Conflicts for (.∎
Recall from Theorem 2.3 that Fair Division Under Conflicts is strongly NPhard for bipartite conflict graphs. Thus, we consider in the following the more restricted case of biconvex bipartite conflict graphs. Recall that a bipartite graph is biconvex if it has a biconvex ordering, that is, an ordering of and such that for every vertex (resp. ) the neighborhood (resp. ) is a consecutive interval in the ordering of (resp. ordering of ).
It is known that a connected biconvex bipartite graph can always be ordered in such a way that the first and last vertices on one side have a special structure. Fix a biconvex ordering of , say and . Define (resp. ) as the vertex in (resp. ) whose neighborhood is not properly contained in any other neighborhood set (see [1, Def. 8]). In case of ties, is the smallest such index (and the largest). We always assume that , otherwise the ordering in could be mirrored. Under these assumptions, the neighborhoods of vertices appearing in the ordering before and after are nested.
Lemma 3 (Abbas and Stewart [1])
Let be a connected biconvex graph. Then there exists a biconvex ordering of the vertices of such that:

[label=.,itemindent=*]

For all , with there is .

For all , with there is .

The subgraph of induced by vertex set is a bipartite permutation graph.
Property (iii) can be put in context with Theorem 3.1. Indeed, it is known that permutation graphs are a subclass of cocomparability graphs (see, e.g., [10]). This gives rise to the following result that Fair Division Under Conflicts on biconvex bipartite graphs is indeed easier (from the complexity point of view) than on general bipartite graphs. It should be pointed out that the contribution of Theorem 3.2 is the identification of the complexity status of the problem, but not a practically relevant algorithm, since the pseudopolynomial running time will be prohibitive in practice. The highlevel idea of the algorithm is illustrated in Algorithm 1.
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