hier-5 Pattern-Guidedk-Anonymity
archies [Swe02a],p-sensitivity [TV06],`-diversity [Mac+07], and t-close-ness [LLV07].
Open Questions. Our approximation factor of our greedy heuristic was only a side result. An extensive analysis of the polynomial-time approxima-bility of PATTERN-GUIDEDk-ANONYMITYremains open. Are there provably good approximation algorithms for PATTERN-GUIDEDk-ANONYMITY? Con-cerning exact solutions, are there further polynomial-time solvable special cases beyond PATTERN-GUIDED2-ANONYMITY?
On the experimental side, several issues remain to be attacked. For instance, we used integer linear programming in a fairly straightforward way almost without any tuning tricks (e. g., using the heuristic solution or “standard heuristics” as preprocessing for speeding up integer linear program solving). It also remains to perform tests comparing our heuristic algorithm against methods other than Mondrian, unfortunately, for the others no source code seems freely available.
Part III
Modifying Teams
In this second of two main parts, we develop and discuss two models for team modification tasks—one model is for modifying one team and the other model is for modifying the composition of multiple teams. We show how concepts from voting theory and graph theory help to understand and to perform the tasks.
Effective Training of Team Members
Scenario 1. Assume that a company wants to train their technical service team. A set of important skills has been identified, but not all service team members cover all these skills. Since training a team member is expensive, ensuring that each team member covers all required skills is impossible and is also not really necessary in practice. To increase the likelihood that there is always a team available which covers the required skills, the company defines the following goal: each important skill must be covered by more than half of the team members. Now, the task is to send a small number of team members to advanced training courses in order to reach this goal.
In this task, the goal is that, for each required skill, a majority of team member covers it. This shall be achieved by improving the personal skills of some team members. Majority evaluations are quite common in the area of voting theory as already discussed inChapter 3. Hence, it seems natural to use concepts from voting theory.
For our task, the situation translates into a voting scenario as follows. We have multiple issues (the important skills) and voters (the team members) approving or disapproving each issue. Now, our task is to influence a small number of voters such that each issue is approved by a majority of voters.
This corresponds to the LOBBYINGproblem which is well studied in the con-text of voting theory [Bin+14;Chr+07;Erd+07].8It is known to beNP-hard andW[2]-hard with respect to the number of voters to influence [Chr+07].
This model is also related to our model for “Selecting an Accepted Team”
fromChapter 3. In both models, we have a set of alternatives and a set of voters who each approve or disapprove the alternatives and both models use some kind of majority-wise evaluation. In Chapter 3 we ask for a
8Indeed, constructing a binary matrix with one row for each team member and one column for each important skill such that an entry equals 1 if and only if the corresponding team member covers the corresponding team, the task translates to replacing a minimum number of rows in order to obtain a matrix that has a majority of 1s in each column. This natural matrix modification problem is precisely the LOBBYINGproblem known from voting theory.
set of alternatives which is accepted by all (respectively a majority of the) voters, where acceptance means that the voter approves the majority of the alternatives. In this chapter, however, we ask for a set of voters to influence such that all alternatives are approved by a majority of voters, where influencing means to make them approve every issue.
InChapter 6, we perform a multivariate complexity analysis for LOBBY -INGwith respect to several natural parameters and parameter combinations.
We identify (fixed-parameter) tractable and intractable parameterizations.
Furthermore, we develop and implement a greedy logarithmic-factor approx-imation algorithm and compare it with a known algorithm and an integer linear programming formulation on synthetic and empirical data.
Redistributing Teams of Equal Size
Scenario 2. Assume that a company has a set of important resources which can each serve a limited numbersof people. Hence, each resource is shared by a team of sizes. Due to some technical improvement the resource can in the future serve∆sadditional people. Now, some of the resources can be sold. The corresponding teams have to be dissolved and their team members have to be distributed to the remaining teams. However, there are some conflicts between the teams which excludes an arbitrary redistribution between the teams. Altogether, our input is a set of size-steams, a conflict relation between the teams, and a number∆s. The task is to dissolve some of the teams and to redistribute their members each to the non-conflicting remaining teams such that each new team is of sizes+∆s.
In this task, the teams can be easily represented by vertices in a graph where one puts an edge between two vertices if the corresponding teams are not conflicting. Now, the task is to remove a specified number of vertices from the graph and to distribute their load (team members) to neighboring vertices. If each team is of size one and one has to dissolve half of the teams, then one is searching for a perfect matching in the corresponding graph.
Already this very special case indicates that using connections to known graph-theoretic concepts might be fruitful.
InChapter 7, we develop a “dissolution model” for our task which also has applications in political districting, economization, and distributed systems.
We show relations to established graph-theoretic models such as matchings, flow networks, and star partitions. We analyze the computational
complex-ity of finding “balanced dissolutions” mainly identifying polynomial-time solvable andNP-hard special cases.
A variant of our model also covers a two-party variant of the task where one has two groups of team members and wants to ensure that one group forms the majority in a specified number of new teams. This variant is originally motivated from political districting, but may also be useful for other applications such as distributed systems.
6 Lobbying
Assume that each ofnvoters may or may not approve each ofmissues. If an agent (the lobby) may influence up tokvoters, then the central question of theNP-hard LOBBYINGproblem is whether the lobby can choose the voters to be influenced so that as a result each issue gets a majority of approvals.
This problem can be modeled as a simple matrix modification problem: Can one replacekrows of a binaryn×m-matrix bykall-1 rows such that each column in the resulting matrix has a majority of 1s?
In this chapter, extending on previous work [Chr+07] that showed param-eterized intractability (W[2]-completeness) with respect to the numberk of modified rows, we study how natural parameters such as n, m,k, or the “maximum number of 1s missing for any column to have a majority of 1s” (referred to as “gap value g”) govern the computational complexity of LOBBYING. Among other results, we prove that LOBBYING is fixed-parameter tractable for fixed-parameter mand provide a greedy logarithmic-factor approximation algorithm which solves LOBBYINGeven optimally if m≤4. We also show empirically that this greedy algorithm performs well on general instances. As a further key result, we prove that LOBBYING isLOGSNP-complete for constant valuesg≥1, thus providing, to the best of our knowledge, a first complete problem from voting for this complexity class of limited nondeterminism.
6.1 Motivation and Model
Campaign management comprises all sorts of activities for influencing the outcome of an election, including well-known scenarios such as manipula-tion [Azi+13;BHN09;BNW11;BO91;CSL07;Dav+14;PS06;Sli04;Wal11b]
bribery [DS12;EFS09;EFS12;FHH09;SEF11] and control [BTT92;EFS11;
EPR11]. While these works relate to campaigning in case of classical vot-ing scenarios where one typically wants to make a specific candidate win or to prevent him from winning, Christian et al. [Chr+07] introduced the
6 Lobbying
scenario of lobbying in multiple referenda. Intuitively, the point here is that there arenvoters, each of them providing a yes- or no-answer to each of missues. In other words, we have a referendum onmissues the voters can decide on. Naturally, before the referendum is held, campaigns will be run by various parties and interest groups to influence the outcome of the referendum. Assuming complete knowledge about the current voter opinions and assuming the extreme scenario that an external agent–the
“lobby”–gains complete control over specific voters, Christian et al. [Chr+07]
modeled this very basic scenario as 0/1-matrix modification problem, where the rows represent voters, the columns represent issues to vote on with yes (1) or no (0), and the lobby goal is represented by a 0/1-vector.
LOBBYING
Input: A matrixA∈{0, 1}n×mand an integerk≥0.
Question: Can one modify the entries of at mostkrows inAsuch that in the resulting matrix every column has more 1s than 0s?
In our context,modifyinga row means to simply flip all 0s to 1s. Hence, modifying a minimum number of rows can be interpreted as the lobby influ-encing a minimum number of voters to reach a certain goal. In difference to Christian et al. [Chr+07], we assume that the desired outcome of each column is 1 instead of providing a goal vector of the lobby. That is, 1 corre-sponds to agreement with the lobby goal and 0 correcorre-sponds to disagreement.
Clearly, by appropriately flipping all entries of a column this can be always ensured. Furthermore, we assume that any column with a majority of 1s is removed from the input matrix. The following example, which is an extract from the real-world data fromSubsection 6.3.2, illustrates our model.
Example 6.1. Consider the following four issues and voting behavior of the five faction leaders extracted from the recorded votes of the German parliament. (SeeSubsection 6.3.2for details about the full data set.)
Selected issues:
1. Water access is a human right.
2. Forbid the Nationalist Party.
3. Financial help for Ireland.
4. Financial help for Cyprus.
1 2 3 4
Brüderle No No Yes Yes
Gysi Yes Yes No No
Kauder No No Yes Yes Künast Yes Yes Yes Yes Steinmeier No Yes Yes Yes
6.1 Motivation and Model
Assume that the lobby wants to approve the first two issues and disapprove the last two issues. Then, the matrix translates into the following binary matrix.
0 0 0 0
1 1 1 1
0 0 0 0
1 1 0 0
0 1 0 0
The second column can be removed because the majority of voters already agree with the lobby. Modifying the first and third row yields a solution.
LOBBYINGisNP-complete [Chr+07]. Moreover, in the setting of parame-terized complexity analysis [DF13;FG06;Nie06], Christian et al. [Chr+07]
showed that it isW[2]-complete with respect to the parameter “numberk of rows to modify”, that is, even if only a small number of voters shall be influenced the problem is computationally intractable. In this chapter, we provide a significantly more refined view on the parameterized and mul-tivariate computational complexity [FJR13;Nie10] of LOBBYING. To this end, we identify the following parameters naturally occurring in LOBBYING and analyze their influence on its computational complexity. The studied parameters are:
• n: number of rows;
• m: number of columns;1
• k: number of rows to modify;
• k0:=d(n+1)/2e −k: below-guarantee parameter;2
• g:=maxmj
=1gj: maximum gap value over all columns, where the gap valuegj:=d(n+1)/2e −Pn
i=1Ai,jis the number of missing 1s to make columnjhave more 1s than 0s;
• s: maximum number of 1s per row;
• t: maximum number of 0s per row.
1Christian et al. [Chr+07] argued thatmseldom exceeds 20.
2Clearly, lobbyingd(n+1)/2evoters, that is, modifyingd(n+1)/2erows in the input matrix yields always the desired solution, makingd(n+1)/2ea trivial upper bound fork.
6 Lobbying
maximum gap valueg
maximum number sof 1s per row
(s,k0)
below-guarantee parameterk0
number kof rows
to modify (s,g) numberm
of columns
maximum number tof 0s per row
(s,k) (m,g) number
nof rows (k,k0) (m,k0)
(m,k)
(s,n)
(m,n) (t,n)
W[2]-hard NP-hard
(even for constant parameter values) LOGSNP-hard
FPT
(fixed-parameter tractable) ILP-FPT
trivialFPT
no polynomial kernel
(parameters provide upper bounds on the input size)
Figure 6.1: Parameterized complexity of LOBBYINGand relations between the parameters considered in this chapter. An arc fromxtoy means that there is some function f withx≤f(y). We omit combined parameters where one component is upper-bounded by a function of the other component, for example (n,k) is omit-ted, becausek≤n. ILP-FPTmeans that the fixed-parameter tractability bases on a formulation as integer linear program.
NP-hardness andLOGSNP-hardness holds for constant param-eter values. There is no polynomial-size problem kernel for any (combined) parameter inside the dotted polygon, unless NP⊆coNP/poly.
6.1 Motivation and Model
Parameter Choice. The parametersn,m, andkare naturally occurring parameters as they measure the input size and the solution size, respec-tively. Scenarios with only few voters or issues are clearly interesting and realistic restrictions. Furthermore, also the restriction to instances with small solutions is very natural as a limited budget of the lobby may only allow a (very) small amount of bribery or, more positively, advertisement.
Additionally,k0complements the parameterkand seems promising because it measures the distance from a trivial type of yes-instances. Moreover, observe that the maximum gap value gis a lower bound on the number of rows that have to be modified; it can be precomputed in linear time. Fur-thermore, sets of issues where each single issue needs only a small amount of additional approvals or disapprovals bear good prospects for a lobby with limited budget. Finally,sandtmeasure the density of the input matrix. In our model the density of the matrix can be seen as the degree of agreement between the voters and the lobby. Hence, it is worth to investigate whether high or low density leads to computational tractability.
Parameter Relations. There are various relations between the parame-ters’ values above. For instance, columns containing a majority of 1s can be safely removed (also implying positive gap values for all columns). As this implies that the input matrix has at least as many 0s as 1s, it follows that there has to be at least one row where at least half of the entries are 0s. Hence, t≤m≤2t. In addition, if the input matrix contains a col-umn only consisting of 0s, then one has to modifyd(n+1)/2erows, implying that the corresponding LOBBYING instance is a yes-instance if and only ifk≥ d(n+1)/2e. Also, we assumem≤n·sas otherwise there has to be a column consisting only of 0s which is a trivial input instance since we would only have to check whetherk≥ d(n+1)/2e. The relations between these parameters and the combinations considered in this chapter are illustrated inFigure 6.1.
6.1.1 Related Models
Our central point of reference is the work by Christian et al. [Chr+07] whose key result is the proof ofW[2]-completeness of LOBBYINGwith respect to the numberk. Erdélyi et al. [Erd+07] generalized LOBBYINGto a weighted scenario and showed that an efficient greedy algorithm achieves a loga-rithmic approximation factor (1+ln(m· d(n+1)/2e)) for the weighted case.
6 Lobbying
Moreover, they showed that essentially no better approximation ratio can be proven for their algorithm. Later, models for lobbying in a probabilistic environment were proposed by Binkele-Raible et al. [Bin+14], providing (parameterized) hardness and tractability results; they interpret lobbying as a form of bribery.
As a special case of thecombinatorial reverse auctionproblem [San+02], the optimization version of LOBBYING is also relevant to combinatorial markets in multi-agent systems. In combinatorial reverse auctions, there are different items that a buyer wants to acquire at the lowest cost from the sellers. More precisely, the buyer wants to have an amount of units from each item,U=(u1,u2, . . . ,um). Each sellerisubmits how many goods he or she has for each item, (λi1,λ2i, . . . ,λim), and the pricepifor which he or she would sell. The goal is to minimize the cost while acquiring the required amount of units of each item. It is easy to see how to translate our lobbying problem into a combinatorial reverse auction problem: Each issue jcorresponds to an item j. The buyer’s requirement for each item j is the gap value of the corresponding column,uj:=gj,∀j∈{1, . . . ,m}. Each row i corresponds to a seller with offerλij :=1−Ai,j,∀j∈{1, . . . ,m}and pricepi:=1.
Finally, LOBBYING is also closely related to bribery in judgment ag-gregation[BER11] where the judges submit binary opinions on different propositions and the goal is to bribe as few judges as possible in order to obtain a certain outcome. A LOBBYINGinstance can be formulated as an equivalent instance of the bribery in judgment aggregation problem using a premise-based procedure. More precisely, given a binary matrixA, each propositional variable vj corresponds to a column j, each judge i corre-sponds to a row iwith the judgment set consisting of those variablesvj satisfying Ai,j=1, and the agenda consists of all propositional variables as premises and no conclusions. Now the goal of modifying as few rows as possible in order to have more 1s than 0s in each column is equivalent to bribing as few judges as possible in order to have all possible variables approved by more than half of the judges.
6.1.2 Our Contributions
We initiate a systematic parameterized and multivariate complexity anal-ysis for the LOBBYINGproblem. We contribute to the theoretical as well as to the empirical side. See the preliminaries inChapter 2for definitions
6.1 Motivation and Model
m t s k0 g k n
m ILP-FPT(Thm.6.9, Cor.6.2) FPT(Thm.6.10) t (2m)2.5·2m+o(2m)·m·n (g+1)m·n2·m s s≥3:NP-c,s≤2:P
NP-c (Thm.
6.1)
FPT(Cor.6.3) (Thm.6.1, Thm.6.6) (g+1)4s·s·n2+16g·m
FPT(const.g) FPT
k0 XP,m2g+1·22k0 2n·m
(Thm.6.11) (Prop.6.2) W[2]-h (a)
W[2]-c (a)
g LOGSNP-c
(Thm.6.2) k
n
a[Chr+07]
Table 6.1: Summary of results. For each parameter combination (row by col-umn) the entries indicate whether LOBBYINGis fixed-parameter tractable (FPT), fixed-parameter tractable based on a formulation as integer linear program (ILP-FPT), polynomial-time solvable for constant parameter values (XP),W[2]-hard (W[2]-h), W[2] -complete (W[2]-c),LOGSNP-complete, meaning completeness for a class of limited nondeterminism lying betweenPandNPfor constant parameter values (LOGSNP-c), orNP-complete even for constant parameter values (NP-c). Entries on the main diagonal represent results with respect to single parameters. Furthermore, for (m,n), (t,n), and (s,n), we have problem kernels of polyno-mial size because these parameters naturally bound the input size (seeFigure 6.1). For (m,k0) we are not aware of kernel size bounds. For all other parameter combinations above, under some reasonable complexity-theoretic assumptions, there cannot be a polynomial-size problem kernel (see Section6.2.3).
6 Lobbying
of parameterized complexity classes and the like. Our complexity results are summarized inTable 6.1. Let us sketch a few highlights. In Subsec-tion 6.2.1, we show that LOBBYINGremainsNP-hard fors=3 andk0=1.
Thus, there is no hope for fixed-parameter tractability with respect to the parameterssork0. InSubsection 6.2.2, a special highlight of this chapter is to show that even if the gap parametergis equal to one, LOBBYINGremains intractable in the sense of being complete for the limited nondeterminism classLOGSNP[PY96] (also see the survey by Goldsmith, Levy, and Mund-henk [GLM96]). This is of particular interest for at least two reasons. First, it provides a first natural voting problem complete forLOGSNP. Second, this is one of the so far rare examples where this complexity class is used in the context of parameterized complexity analysis. InSubsection 6.2.3, we reveal limitations of effective polynomial-time preprocessing for LOBBYING, that is, we prove that polynomial-size problem kernels are unlikely to exist.
InSubsection 6.2.4, we show that LOBBYINGbecomes polynomial-time solv-able for matrices with at most two 1s per row. InSubsection 6.2.5, we show that LOBBYING is fixed-parameter tractable for parameterm by means of describing an ILP formulation with at most 2m variables. We further provide an efficient greedy algorithm yielding provably optimal results for input matrices with up to four columns. This algorithm also provides a logarithmic approximation ratio for cases with more than four columns. In Subsection 6.2.6, we develop several fixed-parameter algorithms for various parameter combinations. Finally, inSection 6.3, we experimentally compare our greedy heuristic with the heuristic due to Erdélyi et al. [Erd+07], and an implementation of our ILP formulation providing solutions guaranteed to be optimal. Our empirical results with random data and real-world data indicate that LOBBYINGcan be solved efficiently.