A Unifying Framework for Manipulation Problems

01/29/2018 ∙ by Dušan Knop, et al. ∙ Technion University of Bergen University of Bonn 0

Manipulation models for electoral systems are a core research theme in social choice theory; they include bribery (unweighted, weighted, swap, shift, ...), control (by adding or deleting voters or candidates), lobbying in referenda and others. We develop a unifying framework for manipulation models with few types of people, one of the most commonly studied scenarios. A critical insight of our framework is to separate the descriptive complexity of the voting rule R from the number of types of people. This allows us to finally settle the computational complexity of R-Swap Bribery, one of the most fundamental manipulation problems. In particular, we prove that R-Swap Bribery is fixed-parameter tractable when R is Dodgson's rule and Young's rule, when parameterized by the number of candidates. This way, we resolve a long-standing open question from 2007 which was explicitly asked by Faliszewski et al. [JAIR 40, 2011]. Our algorithms reveal that the true hardness of bribery problems often stems from the complexity of the voting rules. On one hand, we give a fixed-parameter algorithm parameterized by number of types of people for complex voting rules. Thus, we reveal that R-Swap Bribery with Dodgson's rule is much harder than with Condorcet's rule, which can be expressed by a conjunction of linear inequalities, while Dodson's rule requires quantifier alternation and a bounded number of disjunctions of linear systems. On the other hand, we give an algorithm for quantifier-free voting rules which is parameterized only by the number of conjunctions of the voting rule and runs in time polynomial in the number of types of people. This way, our framework explains why Shift Bribery is polynomial-time solvable for the plurality voting rule, making explicit that the rule is simple in that it can be expressed with a single linear inequality, and that the number of voter types is polynomial.

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1 Introduction

Problems of manipulation, bribery and control constitute a fundamental part of computational social choice. Many such problems are known to be NP-hard (or worse). However, their input can naturally be partitioned into several parts, like the number of voters, the number of candidates, and others. This motivates the study of such problems by the powerful tools of parameterized complexity. One of the most fundamental parameters is the number of candidates , which in many real-life scenarios can be expected to be reasonably small.

A by-now classical example in this direction is the -Swap Bribery problem, which takes as input an election consisting of a set of candidates and a set of voters with their individual preference lists (for ), which are total orders over . Additionally, for each voter and each pair of consecutive candidates , there is some cost of swapping the order of and in . The objective is to find a minimum-cost set of swaps of consecutive candidates in the preference lists in order to make a designated candidate the winner of the thus-perturbed election under a fixed voting rule .

This problem was introduced by Elkind et al. [EFS09] and has since been studied for many classical voting rules  [DS12, FRRS14, KKM17c, SFE17]. In particular, its computational complexity has been thoroughly analyzed with respect to the number of candidates . The observation that is often small motivated the search for fixed-parameter algorithms for -Swap Bribery parameterized by , which are algorithms that run in time for some computable function , here denotes the size of the input election; if such an algorithm exists, we then say that the problem is fixed-parameter tractable with respect to the parameter .

Despite the problem’s importance, for a long time, only the “uniform cost” case of -Swap Bribery was known to be fixed-parameter tractable for various voting rules, parameterized by the number of candidates; here, uniform cost refers to the special case that all voters have the same cost function, that is, for all . This is a fundamental result due to Dorn and Schlotter [DS12], who showed that -Swap Bribery with uniform cost can be solved in time for all voting rules that are “linearly describable”. Many classical voting rules are indeed linearly describable, like any scoring protocol, Copeland, Maximin, or Bucklin.

Recently, Knop et al. [KKM17c] gave the first fixed-parameter algorithms for -Swap Bribery for general cost functions for most voting rules studied in the literature (scoring protocol, Copeland, Maximin, Bucklin etc.), thereby removing the uniform cost assumption. This way, they resolved a long-standing open problem. Moreover, their algorithm runs in time for many rules , and thus improves the double-exponential run time by Dorn and Schlotter. Their key idea was to reduce the problem to so-called -fold integer programming, which allowed them to solve the problem efficiently for bounded number of candidates despite their integer program having an unbounded number of variables. Their approach also solved -Swap Bribery for  being the Kemeny rule, even for general cost functions, though the Kemeny rule is not known to be linearly describable (cf. [FHH11, p. 338]). However, this does not apply for Dodgson’s and Young’s rules.

1.1 The challenge

Even so, there are some notable voting rules for which the complexity of -Swap Bribery remained open even in the uniform cost case. This includes the Dodgson rule and the Young rule. Those rules are based on the notion of Condorcet winner, which is a candidate who beats any other candidate in a head-to-head contest. The Condorcet voting rule is very natural and dates back to the 18th century; however, clearly there exist elections without a Condorcet winner. In such a situation one proclaims those candidates as winners who are “closest” to being a Condorcet winner; different notions of closeness then yield different voting rules:

  • [leftmargin=*]

  • Closeness measured as the of number of swaps in voter’s preference orders defines the Dodgson rule.

  • Closeness measured as the number of voter deletions defines the Young rule.

Thus, a candidate is a Dodgson winner if s/he can be made a Condorcet winner by a minimum number of swaps in the voter’s preference orders over all candidates; analogously for the Young rule and voter deletions.

Kemeny rule111Faliszewski [FHH09] give a fixed-parameter algorithm for the -Bribery problem when Kemeny; -Bribery is, in a sense, simpler than -Swap Bribery because in Bribery the cost of bribing a voter does not depend on how we bribe, while in Swap Bribery the cost is the sum of costs for each performed swap.,

When considering -Swap Bribery, the Dodgson rule and the Young rule are much more complicated to handle than other rules; the reasons are several. First, for many voting rules , the winner of an election can be found in polynomial time, and solving this winner determination problem is certainly a necessary subtask when solving -Swap Bribery. However, for , already winner determination is NP-hard, and so even verifying a solution (that is indeed the winner of the perturbed election) is intractable. However, for , already winner determination is NP-hard, and so even verifying that  is indeed the winner of the perturbed election is intractable. In fact, winner determination for these voting rules is complete for parallel access to NP [HSV05, HHR97, RSV03], denoted -complete222The class

contains all problems solvable in polynomial time by a deterministic Turing machine which has access to an

NP oracle, but must ask all of its oracle queries at once (i.e., the queries can not depend on each other).. Second, for more than 25 years the winner determination problem for the Dodgson rule and Young rule was only known to be solvable by an ILP-based algorithm [BITT89] with doubly-exponential dependence in ; a single-exponential algorithm is only known since recently [KKM17c]. Even though winner determination for these rules turns out to be fixed-parameter tractable with parameter , there is a sharp difference: for the Kemeny rule, a simple procedure enumerating all possible preference orders suffices to determine the winner, while for Dodgson and Young, only a double-exponential ILP-based algorithm was known for a long time [BITT89] and a single-exponential algorithm is only known recently [KKM17c]. This provides a sharp contrast to the Kemeny rule, for which simply enumerating all possible preference orders suffices to determine the winner. Faliszewski et al. [FHH09] describe these difficulties:

  • It is interesting to consider which features of Kemeny elections allow us to employ the above [ILP-based] attack, given that the same approach does not seem to work for either Dodgson or Young elections. One of the reasons is that the universal quantification implicit in Dodgson and Young elections is over an exponentially large search space, but the quantification in Kemeny is, in the case of a fixed candidate set, over a fixed number of options.

Thus, it is not clear how to solve -Swap Bribery even for uniform cost with any fixed-parameter algorithm for being the Dodgson rule or the Young rule. These complications led Faliszewski et al. [FHH11] to explicitly ask for the complexity of -Swap Bribery parameterized by the number of candidates under these rules.

1.2 Our contributions

We start by making a key observation about the majority of fixed-parameter algorithms for -Swap Bribery when is small. A typical such result is an algorithm for -Swap Bribery for being Condorcet’s voting rule. That algorithm uses two key ingredients:

  1. [leftmargin=*]

  2. There are at most preference orders of , and hence each voter falls into one of types; thus, an input election is expressible as a society , where is the number of voters of type .

  3. Expressing that a candidate is a Condorcet winner is possible using a conjunction of linear inequalities in terms of .

As those key properties hold almost universally for voting rules , one might be tempted to think that if there are many types of voters, the -Swap Bribery problem must be hard, and if there are few types of voters, the problem must be easy. However, two points arise as counter-evidence. First, very recently, Knop et al. [KKM17c] showed that even if there are many types of voters who differ by their cost functions, the -Swap Bribery problem remains fixed-parameter tractable for a wide variety of voting rules . Second, as already mentioned, it was open since 2007 whether -Swap Bribery with Dodgson’s and Young’s voting rule are fixed-parameter tractable for few candidates, even for uniform cost functions.

From voters and candidates to societies. Here, we take a novel perspective. We observe that the two key ingredients (1) and (2) apply much more widely than for -Swap Bribery; namely, they are also present in many other manipulation, bribery and control problems. We therefore abstract away the specifics of such problems and introduce general notions of “society”, “moves in societies”, and “winning conditions”. Let be the number of types of people (e.g., voters in an election or a referendum). A society is simply a non-negative

-dimensional integer vector encoding the numbers of people of each type. A

move is a -dimensional integer vector whose elements sum up to zero; it encodes how many people move from one type to another. A change is a -dimensional vector (typically associated with a move) encoding the effect of a move on a society, such that is again a society. Finally, a winning condition is a predicate encoding some desirable property of a society, such as that a preferred candidate has won or that a preferred agenda was selected in a referendum. Specifically, we study winning conditions which are describable by formulas in Presburger Arithmetic (PA). PA is a logical language whose atomic formulas are linear inequalities over the integers, which are then joined with logical connectives and quantifiers. Thus, winning conditions describable by PA formulas widely generalize the class of linearly describable voting rules by Dorn and Schlotter [DS12].

Our main technical contribution informally reads as follows:

Theorem 1 (informal).

Deciding satisfiability of PA formulas with two quantifiers is fixed-parameter tractable with respect to the dimension and length of formula, provided its coefficients and constants are given in unary.

The importance of Theorem 1 arises from its applicability to the following general manipulation problem that we introduce here. This general manipulation problem, which we call Minimum Move, captures that many manipulation problems can be cast as finding a minimum move with respect to some objective function; in particular, it encompasses the well-studied -Swap Bribery problem. We study Minimum Move for linear objective functions and winning conditions  expressible with PA formulas of the form “”. For all such , with the help of Theorem 1, we show that Minimum Move is fixed-parameter tractable for combined parameter the descriptive complexity (length) of the winning condition  and the number of “types of people”, that is, it is fixed-parameter tractable for parameter lengthe of plus . As an important special case, we obtain the first fixed-parameter algorithm for -Swap Bribery for  the Dodgson rule and the Young Rule with uniform costs. To this end, we model the winning condition of the Dodgson rule and Young rules as a PA formula. For intuition, consider the Young rule: a candidate is a Young winner (with score ) if there exists a set of at most voters such that is a Condorcet winner of the election , and for all sets of at most voters any other candidate is not a Condorcet winner of the election . This formula has one quantifier alternation, and its length (for a fixed score ) is bounded by some function of ; finally we have to take a disjunction of such formulas over all possible scores . For a candidate set , the number of types of people is bounded by . Consequently, we finally settle the long-standing open question about the complexity of -Swap Bribery for , that was explicitly raised by Faliszewski [FHH11]:

Theorem 2.

-Swap Bribery with uniform cost is fixed-parameter tractable parameterized by the number of candidates for  being the Dodgson rule or the Young rule; it can be solved in time for some computable function .

Beyond this fundamental problem, we show that a host of other well-studied manipulation problems are captured by our fixed-parameter algorithm for Minimum Move:

Corollary 3.

For , the following problems are fixed-parameter tractable for uniform costs when parameterized by the number of candidates: -$Bribery, -CCDV/CCAV, -Possible Winner, and -Extension Bribery.

Let us turn our attention to the parameter “number of types of people” . Our main contribution here is the following:

Theorem 4 (informal).

For any quantifier-free winning condition , Minimum Move can be solved in time polynomial in the number of types and exponential only in the number of linear inequalities of .

Note that in many models of bribery and control, the number of potential types of people (i.e., types that can occur in any feasible solution) is polynomial in the number of people on input. For example, in Shift Bribery, every voter can be bribed to change their preferences order to one of orders; thus the number of potential types is . Similarly, in CCAV / CCDV (constructive control by adding or deleting voters), every voter has an active/latent bit; thus the number of potential types is . Similar arguments also work for Support Bribery where we change voters’ approval counts, and with a more intricate argumentation also for some voting rules and Bribery and $Bribery. In this sense, the fact that we need to consider potential voter types in -Swap Bribery almost seems like an anomaly, rather than a rule. In summary, the complexity of Minimum Move depends primarily on the descriptive complexity of the winning condition , because in many cases the number of types of people is polynomially bounded.

Another consequence of Theorem 1 are the first fixed-parameter algorithms for two important manipulation problems beyond -Swap Bribery. The Resilient Budget problem asks, for a given society whether allocating budget is sufficient in order to repel any adversary move of cost at most with a counter-move of cost at most (so that the winning condition is still satisfied). Similarly, Robust Move asks for a move of cost at most which causes the winning condition to be satisfied even after any adversary move of cost at most . For formal definitions and results, cf. Sect. 4.2.

1.3 Interpretation of results

Intuitively, the results obtained with Theorem 1 can be interpreted as follows. Dodgson-Swap Bribery is fixed-parameter tractable parameterized by ; however, this comes with at least two limitations as compared to prior work for simpler voting rules . First, our methods do not extend beyond the uniform cost scenario, and this remains a major open problem. Second, our result requires the input election to be given in unary, while prior work allows it to be given in binary (this is sometimes called the succinct case [FHH09]). This is easily explained by the different descriptive complexities of the respective voting rules: for example, while Condorcet’s voting rule can be formulated as a quantifier-free PA formula, formulating Dodgson’s rule requires a long disjunction of formulas which use two quantifiers and a bounded number of disjunctions.

Theorem 4 lets us discuss more specifically the complexity of various voting rules. For example, the Plurality voting rule can be expressed with a single linear inequality encoding that a preferred candidate obtained more points than the remaining candidates altogether. Thus, all problems which can be modeled as Minimum Move are polynomial-time solvable with the Plurality voting rule. This interprets the result of Elkind et al. [EFS09, Theorem 4.1] that Plurality-Shift Bribery is polynomial-time solvable: the number of potential voter types is polynomial, and Plurality has a simple description. Continuing, we may compare =Borda with =Copeland. The winning condition for =Borda can be described with inequalities, while =Copeland requires inequalities. Thus Borda-Swap Bribery is solvable in time , while Copeland-Swap Bribery requires time . Finally, all descriptions of Kemeny’s voting rule we are aware of require inequalities, and thus result in Kemeny-Swap Bribery being solvable in time . We do not claim these complexities to be best possible, but conjecture the existence of lower bounds separating the various voting rules; in particular, we believe that Kemeny-Swap Bribery requires double-exponential time.

Finally, our work provides a natural next step in unifying the many different models that have been proposed for voting, bribing and manipulation problems. In this direction, Faliszewski et al. [FHH11] study what happens when multiple bribery and manipulation actions can occur in an election; e.g., CCAV asks for constructive control by adding voters while CCDV by deleting voters; similarly for CCAC and CCDC for adding/deleting candidates. Faliszewski et al. unify those various (up to that point separately studied) attacks. Similarly, Knop et al. [KKM17c] formulate the -Multi Bribery problem, which also incorporates swaps and perturbing approval counts. The problem we put forward in this paper, Minimum Move, in some sense generalizes and simplifies all those “meta”-problems.

1.4 Related work

We have reviewed most of the relevant computational social choice work already. However, there seems to be some confusion in the literature that deserves clarification. The paper of Faliszewski et al. [FHH09] pioneering the concept of bribery in elections indeed considers the voting rules Kemeny, Dodgson and Young, and provides a fixed-parameter algorithm for Kemeny-Bribery. There are three features of their paper that we wish to discuss.

First, turning their attention to Dodgson-Bribery, they write:

  • Applying the integer programming attack for the case of bribery within Dodgson-like election systems […] is more complicated. These systems involve a more intricate interaction between bribing the voters and then changing their preferences. For Dodgson elections, after the bribery, we still need to worry about the adjacent switches within voters’ preference lists that make a particular candidate a Condorcet winner. […] This interaction seems to be too complicated to be captured by an integer linear program, but building on the flavor of the Bartholdi et al. 

    [BITT89] ILP attack we can achieve the following: Instead of making a winner, we can attempt to make have at most a given Dodgson or Young score.

They call this problem DodgsonScore-Bribery and provide positive results for it. Notice, however, that finding a bribery which makes  have a certain Dodgson score does not prevent another candidate to have a lower score and winning the bribed election. Thus, solving DodgsonScore-Bribery can be very far from the desired result.

Second, the authors then observe that a brute force approach enumerating all briberies solves the Dodgson-Bribery problem in polynomial time for constantly many candidates; however, theirs is not a fixed-parameter algorithm for parameter .

Third, they then introduce another voting system called Dodgson, which is similar to Dodgson, and provide a fixed-parameter algorithm for winner determination. However, as in the case of Dodgson-Bribery, they do not provide a fixed-parameter algorithm for Dodgson-Bribery.

The issue is then that a subsequent paper of Falisezwski et al. [FHH11] claims that the Dodgson rule is “integer-linear-program implementable” and that this implies a certain election control problem generalizing Bribery to be fixed-parameter tractable [FHH11, Theorem 6.2]. We believe the authors do not sufficiently differentiate between determining the winner with one ILP, as is the case for most simple voting rules, and with multiple ILPs, as is the case for Dodgson. Thus, we believe there is no evidence that the Dodgson rule is “integer-linear-program implementable”. Yet, this may be possible and this question still deserves attention. Whatever the reason, we are convinced that their [FHH11, Theorem 6.1] does not hold for =Dodgson. Hence, we believe that ours are the first fixed-parameter algorithms for any Bribery-like problem for  {Dodgson, Young}.

2 Preliminaries

Let be integers. We define and . Throughout, we reserve bold face letters (e.g. ) for vectors. For a vector its -th coordinate is .

Next, we provide notions and notations for -Swap Bribery.

Elections. An election  consists of a set of candidates and a set  of voters, who indicate their preferences over the candidates in , represented via a preference order which is a total order over . We often identify a voter with their preference order . Denote by the rank of candidate  in ; ’s most preferred candidate has rank 1 and their least preferred candidate has rank . For distinct candidates , write if voter  prefers  over .

Swaps. Let be an election and let be a voter. For candidates , a swap means to exchange the positions of and in ; denote the perturbed order by . A swap  is admissible in if . A set of swaps is admissible in if they can be applied sequentially in , one after the other, in some order, such that each one of them is admissible. Note that the perturbed vote, denoted by , is independent from the order in which the swaps of are applied. We extend this notation for applying swaps in several votes and denote it . We specify ’s cost of swaps by a function .

Voting rules. A voting rule  is a function that maps an election to a subset , called the winners. A candidate  is a Condorcet winner if any other  satisfies ; then we say that beats in a head-to-head contest. The Young score of is the size of the smallest subset such that is a Condorcet winner in . Analogously, the Dodgson score of is the size of the smallest admissible set of swaps such that is a Condorcet winner in . Then, is a Young (Dodgson) winner if it has minimum Young (Dodgson) score.

We aim to solve the following problem:

-Swap Bribery Input: An election , a designated candidate and swap costs for . Find: A set of admissible swaps of minimum cost so that wins the election under the rule .

3 Moves in Societies and Presburger Arithmetic

Let be the number of types of people.

Definition 5.

A society is a non-negative -dimensional integer vector .

In most problems, we are interested in modifying a society by moving people between types.

Definition 6.

A move is a vector .

Intuitively, is the number of people of type turning type .

Definition 7.

A change is a vector whose elements sum up to . We say that is the change associated with a move if , and we write . A change is feasible with respect to society if , i.e., if applying the change to results in a society.

One more useful notion is that of a move costs vector:

Definition 8.

A move costs vector is a vector in which satisfies the triangle inequality, i.e., for all distinct .

Definition 9 (-move).

Let and be a move costs vector. A move is a -move if .

Finally, we want to check that the society (e.g., resulting from applying some moves) satisfies a certain desired condition. This condition depends on the problem we are modeling: in variants of bribery, it says that a preferred candidate is elected as a winner or to be a part of a committee under a given voting rule; in the context of lobbying, it says that a preferred agenda was selected. To allow large expressibility, we make a very broad definition:

Definition 10.

A winning condition of width is a predicate with free variables.

3.1 Presburger Arithmetic

For two formulas and , we write to denote their equivalence.

Definition 11 (Presburger Arithmetic).

Let

be the set of quantifier-free Presburger Arithmetic (PA) formulas with free variables which are a disjunction of at most conjunctions of linear inequalities , each of length at most , where and for each inequality. Then, let

be the set of PA formulas equivalent to some DNF formula from . Finally, let

be the set of PA formulas with quantifier depth , free variables , and dimension ; here, for each and with . The length of is the number of symbols it contains, which is polynomially bounded in . By and we denote the sets of PA formulas whose leading quantifier is or , respectively.

Example.

A simple example of PA is the following formula.

Here , and .

Figure 1: Visualization of PA formula in DNF.

We study winning conditions expressible in PA, and state our complexity results with respect to the descriptive complexity of , which is its number of variables, quantifiers, logical connectives, and unary encoding length of coefficients and constants.

Vocabulary. We express relevant definitions by simple PA formulas over integral variables and with integer coefficients and constants:

  • ,

  • ,

  • is a linear map ; thus if we let , then

  • , and,

  • -.

We note that, for elections, our definition of winning condition generalizes the notion of “linearly-definable voting rules” by Dorn and Schlotter [DS12]. Precisely, those rules belong to with ; we will show that Dodgson and Young are in with . Thus, our winning conditions capture an extensive set of voting rules.

3.2 Modeling Problems as Minimum Move

We model moves in societies by the following general problem:

Minimum Move Input: A society , an objective function , a winning condition . Find: A move minimizing s.t. .

It models many well-studied problems:

Multi bribery. Knop et al. [KKM17c] introduce a generalization of various bribery problems called -Multi Bribery. Informally, we are given an election where each voter further has an approval count, and is either active or latent the status of which can be changed at certain cost; likewise, there are costs for perturbing their preference order or approval count. This problem generalizes Bribery, $Bribery, Swap Bribery, Shift Bribery, Support Bribery, Extension Bribery, Possible Winner, Constructive Control by Adding/Deleting Voters and other problems.

Notice that there are at most possible preference orders, at most possible approval counts, and states “active” or “latent”. Thus, there are at most potential types of voters, and we can express the input election as a society . A move costs vector  describing the costs of moving a voter from one type to another is obtained by calculating (possibly using a shortest path algorithm) the least costs based on the given cost functions. Let be a PA formula which is satisfied if the preferred candidate wins under the voting rule in a society . Then, a bribery of minimum cost in a -Multi Bribery instance can be modeled as solving Minimum Move with . This modeling, combined with Theorem 2 and Corollary 17, yields Corollary 3.

Multiwinner elections. Bredereck et al. [BFN16] study the complexity of Shift Bribery in committee elections, that is, in elections with multiple winners. The modeling is exactly the same as above, except for the winning condition which will be a long disjunction over all committees which include the preferred candidate.

Lobbying in referenda. Bredereck et al. [BCH14] study the complexity of Lobbying in referenda. There, voters cast ballots with their “yes”/“no” answers to issues. The task is to push an agenda, i.e., a certain outcome. Again, voters fall into groups according to their ballots, the costs of changing their opinions forms a move costs vector, and a winning condition expresses that the selected agenda succeeded.

4 Sentences With Two Quantifiers

We shall now introduce the building blocks of our proof of Theorem 1. Woods [Woo15] gives an algorithm that efficiently converts any quantifier-free PA formula into an equivalent DNF formula of bounded length:

Lemma 12 (Woods [Woo15, Proposition 5.1]).

Let  be a quantifier-free PA formula with containing inequalities, whose coefficients and right-hand sides are bounded in absolute value by and , respectively. Then can be converted into an equivalent DNF formula with at most disjunctions, each containing at most conjunctions with the same bound on and .

It is often useful for the quantifiers of a PA formula to range over integer points of polyhedra, e.g. (we do not write for brevity, as we assume everything to be integer); again, our definition is not restrictive by the fact that we can always rearrange:

Parametric ILP. A special case of PA are parametric ILPs, which can be viewed333Parametric ILPs are typically viewed as ILPs with a varying right hand side, that is, deciding the sentence ; it is known that our formulation is equivalent, as shown by Crampton et al. [CGKW17], who call it ILP Resiliency. as deciding the sentence

where and are integer matrices. A consequence of an algorithm of Eisenbrand and Shmonin [ES08] is the following:

Corollary 13 ([Es08, Theorem 4.2], [Cgkw17, Corollary 1]).

Any parametric ILP whose entries of , and are given in unary, is fixed-parameter tractable when parameterized by and .

ILP and disjunctions. We shall use a folklore result about implementing disjunctions in ILP when the domains of variables can be bounded. For that, we need another definition.

Definition 14 (-bounded, -small PA formula).

Let be a PA formula, and let . Then we say that is -bounded if

i.e., the set of feasible solutions does not change by restricting all quantifiers and free variables to the corresponding box of size .

Moreover, we say that any is -small if it is -bounded, that is, its coefficients and constants are bounded by .

A special case are ILPs which are -small; they correspond to PA formulas with and ; for such formulas we show:

Lemma 15 (ILP disjunctions [folklore]).

Let for be -small ILPs with for each . Then, a -small system with can be constructed in time such that

Proof.

Let , let for

be binary variables, and consider the following system:

Assume it has an integer solution . Then there is an index such that and thus holds; thus, the system has an integer solution. In the other direction, assume that the system has a solution ; then let . We shall prove that holds for all . Since is -small, each of its row sums has terms which are a multiple of two numbers, each bounded by , and thus is at most . Moreover, since , we have and thus the right hand side is and every assignment of feasible for satisfies it. Clearly, the new system has variables, inequalities, is -bounded and and thus it is -small, and can be constructed in the claimed time. ∎

We now prove Theorem 1; we restate it in formal terms here:

Theorem 16 (formal version of Theorem 1).

Let be a -small sentence (i.e., without free variables and thus ). Then can be decided in time for some computable function .

Proof.

Let . Clearly, to decide we can instead decide . Consider the formula : by Lemma 12, there exists an equivalent -small DNF formula such that the number of its disjunctions and conjunctions is a function of just the original and .

Thus, from now on focus on the case . Our next task is to construct an instance of Parametric ILP equivalent to deciding . To do that, replace with , where is of dimension and the system of linear inequalities has bounded length, coefficients and right sides. Now we use Lemma 15. Assume that is a disjunction of linear systems, each of at most conjunctions, and by the assumptions of the theorem we know that is -small. Plugging into Lemma 15, we have , , and , and we obtain a formula equivalent to . Thus, we are left with deciding with the following parameters:

  • [leftmargin=*]

  • The coefficients and right-hand sides are bounded by some computable function .

  • The dimensions of are bounded by .

Thus, we are in the setting of Corollary 13 and we can decide the above sentence by a fixed-parameter algorithm. ∎

Corollary 17.

Let be a disjunction of many -small formulas. Then Minimum Move with objective and winning condition can be solved in time for some computable function .

Proof.

Let , and let , for each . Now we need to perform binary search with parameter on the sentence

This sentence is obviously equivalent to

which can be decided by application of Theorem 16. This is possible in the claimed time, as, for each , has quantifier depth , is -small for and has dimension . A minimum move can then be constructed by coordinate-wise binary search; cf. [KQR10, Thm. 4.1]. ∎

4.1 Application: Swap Bribery for the Dodgson Rule and Young Rule

We now prove Theorem 2 by giving our fixed-parameter algorithm for -Swap Bribery with  the Dodgson Rule or the Young Rule.

Proof of Theorem 2.

Fix an instance of -Swap Bribery. As there are possible total orders on , each voter has one of these orders. Thus we view the election as a society . Let be a move costs vector defined as “swap distance between types  and ” (this is simply the number of inversions between the permutations  and  [EFS09, Proposition 3.2]); observe that a society is in swap distance at most from if and  is a feasible -move for . Moreover, let be defined as for every type and for every two types , where is a type for latent voters; observe that is in voter deletion distance at most from if and is a -move.

Our plan is to express the winning condition for using a PA formula which is a disjunction of polynomially many (in and ) formulas from , and then solve Minimum Move with and using Corollary 17; recall that since the instance is uniform, we have for all , and we let be the move costs vector obtained from . For , we proceed analogously. In the end, we will verify that are bounded by a function of , and that are polynomial in the input size.

Expressing and . The winning condition for candidate in both the Dodgson and Young rule can be viewed as  being closest to being a Condorcet winner with respect to some distance measure. Specifically, for the Dodgson rule, this distance is the number of swaps in the preference orders, and in the Young rule, the distance is the number of voter deletions. For this reason, finding a bribery which makes a winner corresponds to finding a bribery which, for some , makes be in distance or less from being a Condorcet winner, and, simultaneously, making every other candidate be in distance at least from being a Condorcet winner.

Let us fix ; later we will argue that we can go through all relevant choices of . First, we will express the condition that in a society , candidate beats candidate in a head-to-head contest:

where is the preference order shared by all voters of type . Then, it is easy to express that is a winner in under Condorcet’s rule:

Finally, we express that is a Dodgson-winner with score by as follows:

Then,