Fair packing of independent sets

03/25/2020 ∙ by Nina Chiarelli, et al. ∙ 0

In this work we add a graph theoretical perspective to a classical problem of fairly allocating indivisible items to several agents. Agents have different profit valuations of items and we allow an incompatibility relation between pairs of items described in terms of a conflict graph. Hence, every feasible allocation of items to the agents corresponds to a partial coloring, that is, a collection of pairwise disjoint independent sets. The sum of profits of vertices/items assigned to one color/agent should be optimized in a maxi-min sense. We derive complexity and algorithmic results for this problem, which is a generalization of the classical Partition and Independent Set problems. In particular, we show that the problem is strongly NP-complete in the classes of bipartite graphs and their line graphs, and solvable in pseudo-polynomial time in the classes of cocomparability graphs and biconvex bipartite graphs.

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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 well-known 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 maxi-min 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 NP-hard 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 NP-hard as mentioned in [4]. In fact, an instance of the strongly NP-complete 3-Partition 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 NP-hard for any constant and strongly NP-hard for being part of the input.

Note that the problem is still only weakly NP-hard for constant even for arbitrary profit functions, since we can construct a pseudo-polynomial 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 Max-Min 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 NP-hard.

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 NP-hard for (see Observation 1.1).

Bipartite permutation graphs

PP

Biconvex bipartite graphs

PP (Thm. 3.2)

Bipartite graphs

sNPc (Thm. 2.3)

Permutation graphs

PP

Interval graphs

PP

Cocomparability graphs

PP (Thm. 3.1)

Comparability graphs

sNPc

Perfect graphs

sNPc

Line graphs of bipartite graphs

sNPc (Thm. 2.4)

Cographs

PP
Figure 1: Relationships between various graph classes and the complexity of the Fair -Division Under Conflicts problem. The arrow from a class to a class means that every graph in is also in . Label ‘PP’ means that the problem is solvable in pseudo-polynomial time for each fixed in the given class, label ‘sNPc’ means that the problem is strongly NP-complete for all fixed , and label ‘?’ means that the complexity is open. For all graph classes in the figure, the problem is solvable in strongly polynomial time for , as it coincides with the weighted independent set problem.

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 NP-complete, the decision version of our Fair -Division Under Conflicts is strongly NP-complete 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 NP-hard (Section 2.2) although the corresponding Independent Set problem is polynomial-time 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 pseudo-polynomial time algorithm. This result is based on an insightful pseudo-polynomial 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 NP-hard even for for general graphs, while Observation 1.1 shows the weak NP-hardness of the problem for constant in the absence of conflicts. In what follows, we show that Fair -Division Under Conflicts is strongly NP-hard also for all , for various well-known 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 non-sustainable 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) NP-complete. Then, for every , the decision version of Fair -Division Under Conflicts with conflict graphs from is (strongly) NP-complete.

Proof

Let be a sustainable class of graphs for which the decision version of Fair -Division Under Conflicts is (strongly) NP-complete 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) NP-complete.∎

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 NP-complete. Then, for every , the decision version of Fair -Division Under Conflicts with conflict graphs from is strongly NP-complete.

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 NP-complete 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 NP-complete. By using a similar argument, we even get a strong inapproximability result for general graphs.

Theorem 2.2

For every and every , it is NP-hard 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 polynomial-time 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 polynomial-time 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 NP-hard 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 NP-hardness 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 NP-complete 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 NP-complete 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 NP-complete 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 NP-complete in the class of line graphs of bipartite graphs.∎

3 Pseudo-polynomial algorithms for special graph classes

As shown in Theorem 2.3, for each , Fair -Division Under Conflicts is strongly NP-complete in the class of bipartite graphs. This rules out the existence of a pseudo-polynomial time algorithm for the problem in the class of bipartite graphs, unless . In this section we show that for every there is a pseudo-polynomial 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 well-known 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 NP-complete in the class of comparability graphs. For cocomparability graphs, we prove that the problem is solvable in pseudo-polynomial 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

  1. 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 vertex-disjoint 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 component-wise 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.

  1. [label=(),itemindent=*]

  2. If appears at least twice as a coordinate of , then we set

    (2)

    Note that since appears at least twice as a coordinate of , there is no partial -coloring of the subgraph of induced by such that equality (1) holds for all . Thus, equation (2) is correct.

  3. If does not appear as any coordinate of , then we set

    (3)

    Since does not appear as any coordinate of , every partial -coloring of the subgraph of induced by such that equality (1) holds for all is a partial -coloring of the subgraph of induced by and vice versa. This implies relation (3).

  4. 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 NP-hard 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:

  1. [label=.,itemindent=*]

  2. For all , with there is .

  3. For all , with there is .

  4. 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 pseudo-polynomial running time will be prohibitive in practice. The high-level idea of the algorithm is illustrated in Algorithm 1.

  apply Lemma 3 for getting the cocomparability graph and vertices ,
  let