Couples

Here we introduce an extension of the Hospitals/Residents problem called Hospitals/Res-idents with Couples or HRC. In this problem, residents may form couples and thus have joint rankings over the hospitals. This means that instead of ranking the hospitals individually, couples rankpairs of hospitals according to their preferences. This allows them to express intentions such as being applied in the same hospital, or in hospitals that are close to each other. If we allow joint preferences, the notion of stability has to be adopted to fit this context as well. Although we only give the details later, together with the formal definition of the problem in Chapter 6, we show an intuitive example.

We define an instance of HRCthat contains residentss1, s2, c1, c2 and hospitalsh1, h2. Lets1ands2be singles, and let (c1, c2) form a couplec. The capacity of bothh1andh2is 2.

The preference lists of the agents are the following (see Figure 1.3 for an illustration):

L(s1) :h1, h2 L(h1) :s1, c1, s2, c2

L(s2) :h2, h1 L(h2) :c1, s1, c2, s2

L(c) : (h1, h1),(h2, h2),(h1, h2)

Part (a) of 1.3 depicts an assignment M1 that assigns both members of the couples c to h1 and both singles to h2. This assignment is not stable, since the single s1 and the hospital h1 would both benefit from contracting each other, so they form a blocking pair forM1. Part (b) of the figure shows an assignmentM2 where s1and c1 are assigned toh1, and s2 and c2 are assigned to h2. Note that both the couples c and the hospital h2 would benefit from contracting each other. Therefore, they block the assignment, yielding thatM2

is unstable as well. Finally, we illustrate a stable assignmentM3in (c), whereM3assigns the singles toh1, and both members ofctoh2.

The task of the Hospitals/Residents with Couplesproblem is to decide whether a stable assignment exists in an instance where couples are involved. This problem was first introduced by Roth [124] who also discovered that a stable assignment need not exist. Later, Ronn [123] proved that the problem is NP-hard.

In Chapter 6 we will give an example showing that an instance of the Hospitals/Resi-dents with Couplesproblem may admit stable assignments of different sizes. We denote byMaximum Hospitals/Residents with Couplesthe optimization problem where the task is to determine a stable assignment of maximum size for a given instance. Note that this problem is trivially NP-hard, as it contains theHospitals/Residents with Couples problem.

We remark that HRC models a situation that arises in many real world applications

[127, 126]. In the last decade, various approaches have been investigated to deal with its intractability, but most researchers examined different assumptions on the preferences of couples that guarantee some kind of tractability [45, 26, 89, 112]. We examine this problem from different viewpoints in Chapter 6.

CHAPTER 2

Recognizing k -apex graphs

In this chapter, we propose a newly developed FPT algorithm for the following problem:

given a graphG, find a set of k vertices whose deletion makes Gplanar. We parameterize this problem byk, the number of vertices which we are allowed to delete.

Planar graphs are subject of wide research interest in graph theory. There are many generally hard problems which can be solved in polynomial time when considering planar graphs, e.g., Maximum Clique, Maximum Cut, and Subgraph Isomorphism [50, 73].

For problems that remain NP-hard on planar graphs, we often have efficient approximation algorithms. For example, the problems Independent Set, Vertex Cover, and Domi-nating Setadmit an efficient linear-time approximation scheme [10, 94]. The research for efficient algorithms for problems on planar graphs is still very intensive.

Many results on planar graphs can be extended to almost planar graphs, which can be defined in various ways. For example, we can consider possible embeddings of a graph in a surface other than the plane. The genus of a graph is the minimum number of handles that must be added to the plane to embed the graph without any crossings. Although determining the genus of a graph is NP-hard [130], the graphs with bounded genus are subjects of wide research. A similar property of graphs is their crossing number, i.e., the minimum possible number of crossings with which the graph can be drawn in the plane. Determining the crossing number is also NP-hard [63].

Cai [24] introduced another notation to capture the distance of a graphGfrom a graph classF, based on the number of certain elementary modification steps. He defines the distance ofGfromF as the minimum number of modifying steps needed to makeGa member ofF. Here, modification can mean the deletion or addition of edges or vertices. We consider the following question: given a graphGand an integerk, is there a set of at mostkvertices inG, whose deletion makesGplanar?

It was proven by Lewis and Yannakakis [92] that the node-deletion problem is NP-com-plete for every non-trivial hereditary graph property decidable in polynomial time. As pla-narity is such a property, the problem of finding a maximum induced planar subgraph is NP-complete, so we cannot hope to find a polynomial-time algorithm that answers the above question. Therefore, the parameterized complexity framework seems suitable for the analysis of this problem.

The standard parameterized version of our problem is the following:

k-Apex

Input: a graphGand an integerk.

Parameter: the integerk.

Task: decide whether deleting at most k vertices from G can result in a planar graph.

We refer to this parameterized problem as the k-Apex problem, because a set of vertices whose deletion makes the graph planar is sometimes calledapex vertices orapices. We will denote the class of graphs for which the answer of the problem is ‘yes’ by Apex(k). Observe that the parameterkindeed expresses the distance of an input instance from planarity. We note that Cai, who also used the parameterized complexity framework for his examinations, used the notation Planar +kvto denote this class [24].

In the parameterized complexity literature, numerous similar node-deletion problems have been studied. A classical result of this type by Bodlaender [16] and Downey and Fellows [43]

states that the Feedback Vertex Set problem, asking whether a graph can be made acyclic by the deletion of at mostk vertices, is FPT. The parameterized complexity of the directed version of this problem has been a long-standing open question, and it has only been proved recently that it is FPT as well [28]. Fixed-parameter tractability has also been proved for the problem of findingkvertices whose deletion results in a bipartite graph [117], or in a chordal graph [101]. On the negative side, the corresponding node-deletion problem for wheel-free graphs was proved to be W[2]-hard [95].

Considering the graph class Apex(k), we can observe that this family of graphs is closed under taking minors. The celebrated graph minor theorem by Robertson and Seymour states that such families can be characterized by a set of excluded minors [121]. They also showed that for each graph H it can be tested in cubic time whether a graph contains H as a minor [120]. As a consequence, membership for such graph classes can be decided in cubic time. In particular, we know that there exists an algorithm with running timef(k)n³ that can decide whether a graph belongs to Apex(k). However, the proof of the graph minor theorem is non-constructive in the following sense. It proves the existence of an algorithm for the membership test that uses the excluded minor characterization of the given graph class, but does not provide any algorithm for determining this characterization. In 2008, an algorithm was presented by Adler, Grohe, and Kreutzer [5] for constructing the set of excluded minors for a given graph class closed under taking minors, which yields a way to explicitly construct the algorithm whose existence was proved by Robertson and Seymour.

We remark that it follows also from a paper by Fellows and Langston [53] that an algorithm for testing membership in Apex(k) can be constructed explicitly.

Although these results provide a general tool that can be applied to our specific problem, so far no direct FPT algorithm has been proposed for it¹ In this chapter, we present a new algorithm which solves Apex(k) in f(k)n² time. Note that the presented algorithm runs in quadratic time, and hence yields a better running time than any algorithm using the minor testing algorithm that is applied in the above mentioned approaches. Moreover, ifG∈Apex(k) then our algorithm also returns a solution, i.e., a setS∈V(G),|S| ≤ksuch thatG−S is planar.

The presented algorithm is strongly based on the ideas used by Grohe [69] for computing crossing number. Grohe uses the fact that the crossing number of a graph is an upper bound for its genus. Since the genus of a graph in Apex(k) cannot be bounded by a function ofk, we need some other ideas. As in [69], we exploit the fact that in a graph with large treewidth we can always find a large grid minor [122]. Examining the structure of the graph with such a

1Recently, a paper by Ken-ichi Kawarabayashi with titlePlanarity allowing few error vertices in linear timehas been accepted to FOCS 2009 proposing an algorithm for this problem.

2.1. Preliminaries 19

Figure 2.1: The hexagonal gridsH1,H2, and H3.

grid minor, we can reduce our problem to a smaller instance. Applying this reduction several times, we finally get an instance with bounded treewidth. Then we make use of Courcelle’s Theorem [38], which states that every graph property that is expressible in monadic second-order logic can be decided in linear time on graphs of bounded treewidth.

In Section 2.1 we summarize some useful definitions used in this chapter. Section 2.2 outlines the FPT algorithm solving thek-Apexproblem, and Sections 2.3 and 2.4 describe the two phases of the algorithm. The results of this chapter were published in [104].

2.1 Preliminaries

In this section, graphs are assumed to be simple, since both loops and multiple edges are irrelevant in thek-Apexproblem.

A graph H is a minor of a graph G if it can be obtained from a subgraph of G by contracting some of its edges. Here contracting an edge e with endpoints a and b means deletinge, and then identifying verticesaandb.

A graphH is asubdivision of a graphGifGcan be obtained fromH by contracting some of its edges that have at least one endpoint of degree two. Or, equivalently,His a subdivision ofGifH can be obtained fromGby replacing some of its edges with newly introduced paths such that the inner vertices of these paths have degree two in H. We refer to these paths inH corresponding to edges ofGas edge-paths. A graphH is atopological minor ofGifG has a subgraph that is a subdivision ofH. We say thatGandG⁰ aretopologically isomorphic if they both are subdivisions of a graphH.

Theg×ggrid is the graphGg×g whereV(Gg×g) ={vij |1≤i, j≤g} and E(Gg×g) = {vijvi⁰j⁰ | |i−i⁰|+|j−j⁰|= 1}. Instead of giving a formal definition for thehexagonal grid of radiusr, which we will denote by Hr, we refer to the illustration shown in Figure 2.1. A cell of a hexagonal grid is one of its cycles of length 6.

Atree decomposition of a graphGis a pair (T,(Vt)t∈V(T)) whereT is a tree,Vt⊆V(G) for allt∈V(T), and the following are true:

• for allv∈V(G) there exists at∈V(T) such thatv∈Vt,

• for allxy∈E(G) there exists at∈V(T) such thatx, y∈Vt,

• iftlies on the path connectingt⁰ andt⁰⁰ inT, thenVt⊇Vt⁰∩Vt⁰⁰.

Thewidthof such a tree decomposition is the maximum of|Vt|−1 taken over allt∈V(T).

The treewidth of a graph G, denoted by tw(G), is the smallest possible width of a tree decomposition ofG. For an introduction to treewidth see e.g. [18, 41].