Further parameters measuring incompleteness

In Subsection 8.3.1, we have seen that for a broad class of scoring rules there is no hope for fixed-parameter tractability with respect to the “number of undetermined pairs per vote”. In contrast, the fixed-parameter tractability results with respect to the parameter “total number of undetermined pairs” show that being not too far from complete information makes the Possible Winner problem provably easier (8.3.2).

However, this result suffers from the fact that the considered parameter may assume quite large values for many instances. This motivates the study of further parameter-izations measuring the amount of incompleteness. In the following, we suggest some parameterizations and ideas that might serve as a basis for future research.

• Instead of measuring the amount of incompleteness per vote, it also seems plau-sible to measure the amount of incompleteness per candidate. This directly leads to the parameterizations “maximum/average number of undetermined pairs in which a candidate is involved”.

• The parameterizations suggested in the previous subsection measure the amount of incompleteness. An “opposite” perspective is to measure the amount of com-pleteness within the partial votes, for example, by parameterizations based on the “number of determined pairs”. This directly leads to the “dual parameter-izations” with respect to the parameterizations from the previous subsections.

For example, the number of determined pairs per vote is the total number of pairs minus the number of undetermined pairs per vote. Such parameterizations can be considered as extending the special case of Manipulation where the number of determined pairs in every non-linear vote is zero.

• The development of a refined search tree in Section 8.3.2 led to the identification of the special case that all undetermined pairs of candidates are isolated. In this case, Possible Winner becomes solvable in polynomial time for a class of scoring rules including Borda and k-approval. The concept of isolated pairs can be considered as a measure of “local disturbance”. Along these lines, it might be interesting to investigate instances allowing for isolated triples and,

more generally, isolated tuples of bounded size. However, since the size of an isolated tuple is smaller than two times the “number of undetermined pairs per vote”, the hardness results from Subsection 8.3.1 can be transferred.

• An interesting parameterization concerns the “number of possible extensions per vote”.⁵ In general, the computation of this parameter, that is, counting the number of linear extensions for one partial vote is computationally hard [47].

Fortunately, the set E(v) of extensions of a partial order v, can be generated in time constant in|E(v)| [180]. Note that it is easy to see that this parameteriza-tion is fpt-equivalent to the parameter “number of undetermined pairs per vote”.

However, it might allow for a different view that is helpful to design algorithms.

Concluding, we think that the development and investigation of further parameteriza-tions measuring the amount of (in)completeness is an important challenge for future research. Clearly, this is a general conceptual task also of interest for voting rules other than scoring rules.

The discussion above dealt with the identification of new single parameterizations that might lead to tractability for meaningful cases. In the following chapter, we present an alternative way to obtain fixed-parameter tractable cases by investigating combined parameters (where known single parameters lead to W[1]-hardness). More specifically, the usefulness of combined parameters still capturing meaningful scenarios will be exhibited using Possible Winner for k-approval voting as example.

5There are some recent considerations of the counting variant of Possible Winner [7].

Chapter 9

Combined parameters for k-approval

Under the k-approval rule every voter can assign one point to exactly k alternatives and an alternative with most points in total wins. In the case that every voter pro-vides complete information, the winner can be easily determined. However, there are settings in which the voters may only provide partial information on their preferences.

This directly leads to the central combinatorial problem considered in this part: the Possible Winner problem, which asks whether a specific alternative can still become a winner. In Chapter 7, we provided NP-completeness for every k ∈ {2, . . . , m − 2}

with m denoting the number of alternatives if the number of votes is unbounded. In Chapter 8, we showed that Possible Winner for k-approval is also NP-complete if there are only two partial votes (and k is part of the input).

These hardness results motivate a multivariate complexity analysis with respect to the combined parameter “number of votes” and “number of candidates to which a voter gives one/zero points” for k-approval. Can we efficiently solve Possible Winner when these parameters are both small? This setting might look restrictive on a first glance but it naturally reflects scenarios in which one is interested in finding a small group of winners (or losers). For example, a small committee awards a small number of grants or picks out a limited number of students for graduate school. Another example might appear in a human resource department where few people select few employees out of a large pool of job applicants. As concrete example one might look at the decision about the Nobel prize for peace in 2009, where a committee consisting of five people had to select up to three winners out of about 200 candidates. At a certain point, a committee member might have already known that he (or she) prefers Obama and Bono to Berlusconi, but might have not decided on the order of Obama and Bono yet. This immediately leads the way to the question whether, given a set of “partial preferences”, a certain candidate may still win and hence motivates the study of the Possible Winner problem for k-approval (see Section 7.2 for a formal definition).

In the above described scenarios, the only “unbounded” part of the input is the number of candidates. Hence, directly related questions are whether we can ignore or delete candidates which are not relevant for the decision process and how to identify such candidates. In this context, parameterized algorithmics provides the concept of

kernelization by means of polynomial-time data reduction rules that “preprocess” an instance such that the size of the “reduced” instance only depends on the parameter.

Basic definitions are provided in Section 1.3.1. Although kernelization has been ap-plied successfully in many areas (see [34, 126] for surveys), it seems hardly explored for problems in the voting context. In fact, we are only aware of recent results for Dodg-son Score [106, 108] (see Chapter 6 for more details) and Swap Bribery [74] as well as some “partial kernelization” results for Kemeny Score, provided in Chapter 4.

In this chapter, we use kernelization to show the fixed-parameter tractability of Possible Winner for k-approval in two “symmetric” scenarios.

1. We consider the combined parameter “number of incomplete votes” t and “num-ber of candidates to which every voter gives zero points” k^′ := m− k for m can-didates. Making use of a simple observation we show that Possible Winner admits a polynomial-size problem kernel with respect to (t, k^′) and provide two algorithms: a simple search tree where the exponential part of the running time is bounded by 2^O(k′)for constant t and a dynamic programming algorithm where the exponential part of the running time is bounded by 2^O(t) for constant k^′. The bound on the dynamic programming table is based the same idea as for the dynamic programming algorithm for Dodgson Score (see Section 6.1). This indicates that this approach may become of general interest.

2. We consider the combined parameter t and k, where k denotes the “number of candidates to which a voter assigns one point”. We observe that here one cannot argue symmetrically to the first scenario. Using other arguments, we provide a superexponential-size problem kernel showing the fixed-parameter tractability of Possible Winner with respect to (t, k). For the special case of 2-approval, we give a polynomial-size kernel with O(t²) candidates by applying an additional reduction rule based on maximum matching techniques. Using a methodology due to Bodlaender et al. [35], our main technical result of this chapter shows that Possible Winner is very unlikely to admit a polynomial-size problem kernel with respect to (t, k).

As in Chapters 7 and 8, all results are given for the unique winner case, that is, looking for a single candidate with maximum score, but they directly transfer to the cowinner case. Note that although the unique-winner and cowinner are used in the definition of Possible Winner and Manipulation in general [65, 131, 194], for k-approval and some of our introductory examples, this seems not to model all situations directly. In particular, using k-approval voting, one often is interested if a distinguished candidate can be part of a winning set of size k. Hence, we discuss other problem variants asking for a set of winners in Chapter 10. However, we stress that in many cases it might also be of interest who is a unique possible winner (the scenario considered in this work).

For example, when voting for a board with k members, a unique winner might become the head of the board or get some special award.

Some of the reduction rules given in this chapter will not directly decrease the instance size by removing candidates or votes but instead only decrease the number of possible extensions of a vote, for example, by “fixing” candidates. To fix a candidate at a certain position means to specify its relation to all other candidates. Clearly, a candidate may not be fixed at every position in a specific partial vote. For every

9.1 Fixed number of zero-positions 147 candidate c^′∈ C and a partial vote v ∈ V , let

L(v, c^′) :={c^′′∈ C | c^′′≻ c^′ in v} and R(v, c^′) :={c^′′∈ C | c^′≻ c^′′ in v}.

Then, fixing a candidate c^′∈ C as good as possible means to add L(v, c^′)≻ c^′≻ C \ (L(v, c^′)∪ {c^′})

to v. Analogously, fixing a candidate as bad as possible is realized by adding C \ (R(v, c^′)∪ {c^′}) ≻ c^′ ≻ R(v, c^′) to v. Fixing a subset of candidates as good/bad as possible means that the single candidates are fixed as good/bad as possible processing them in an arbitrary order. If a candidate c^′ ∈ C is fixed in all partial votes, this implies that also its score s(c^′) is fixed and hence s(c^′) is well-defined. Furthermore, we say that a candidate c^′ may shift a candidate c^′′ to the left (right) in a partial vote v if c^′′ ≻ c^′ (c^′ ≻ c^′′) in v, that is, setting c^′ to a one-position (zero-position) implies to set c^′′ to a one-position (zero-position) as well.

As discussed in the previous chapters, the votes of an input instance of Possible Winner can be partitioned into a (possibly empty) set of linear votes, called V^l, and a set of proper (nonlinear) partial votes, called V^p. We state all our results for the parameter t :=|V^p|. All positive results also hold for the parameter number of total votes n :=|V^l| + |V^p|. However, this means that we have to “reduce” the number of linear votes such that it is bounded by the considered parameter. To this end, in some of our reduction rules, we replace the set of linear votes by an equivalent set, that is, the maximum partial scores remain unchanged, by using Lemma 7.1 (see Section 7.3.1).

To apply Lemma 7.1, for some instances, it might be necessary to add an additional dummy candidate to achieve the Property 1. In all considered cases, this can be done in a straightforward way without changing the parameter values of an instance and thus will not be further discussed. Note that we only state polynomial-size and not provide explicit bounds on the number of linear votes in a reduced instance. This is clearly sufficient to state a polynomial kernel. A further refinement seems not to be of interest from practical point of view since in our case it always make sense to store the maximum partial scores itself instead of “encoding” them into a new set of linear votes of bounded size.

9.1 Fixed number of zero-positions

In this section, we investigate Possible Winner under (m− k^′)-approval with k^′ <

m, that is, k^′ denotes the number of zero-positions. We give a polynomial kernel with respect to (t, k^′) for Possible Winner where t is the number of partial votes.

In addition, we provide two algorithms; a simple branching algorithm with running time 2^O(k′)· poly(n, m) for constant t and a dynamic programming algorithm with running time 2^O(t)· poly(n, m) for constant k^′.