Oficina de Enlace de la OSCE en Asia Central

It is well-known that most instances of many NP-complete problem, such as SAT, are easy to solve, and that computationally hard cases are rare. One model of understanding where the hard problems are is the phase transition.

Almost all NP-complete problems exhibit a phase transition, in which the probability (with respect to a uniform distribution) of an accepting instance abruptly changes from 0 to 1 when a constrainedness parameter, also called an order parameter, is adjusted [TCK91]. Intuitively, the problem goes from being underconstrained, with a high density of solutions, to being overconstrained, and very unlikely to contain a solution, as a certain threshold of constrainedness is passed. The phase transition has been observed to be a significant challenge to many algorithms of interest, in which the complexity often peaks at the critical value for the parameter. In this region, the probability of a solution is low but nonnegligible. For example, it is known that in the problem of SAT for 3-CNF formulas, phase transition occurs when the constrainedness parameter of _mn, where n is the number of clauses and m the number of variables approaches a critical value of about 4.2 [GW96]. This occurs because with fewer than 4.2m clauses, the variables are likely underconstrained and it is easy to find a solution, since there are a lot of solutions. On the other hand, with more than 4.2m clauses, the algorithm is likely to backtrack very early, because a contradiction is reached very easily. At the boundary of roughly 4.2m clauses, there is often a high density of well-separated almost solutions which causes most backtracking algorithms to search very deeply in the search tree, or “thrash” [TCK91]. It is further conjectured that all NP-complete problems have at least one order parameter in which hard-to-solve problems are around a critical value of this parameter. The probability of a solution abruptly changes from almost zero to almost

one at this transition.

In the diagrams below, we show examples of search trees in problem instances that are overconstrained or underconstrained, respectively.

In these cases, the complexity of the search is pruned by either the depth of the search tree in the case of overconstrained instances or the branching factor of each node in the case of underconstrained instances. In contrast, an example of a search tree in an instance near the phase transition would be as follows.

In this case, the search tree is deep due to the abundance of near-solutions and wide due to the relative rarity of both solutions allowing one to end the search or constraints allowing one to prune branches.

In the problem Partition, we are to partition a set of n integers distributed from [1, m]. The constrainedness parameter of interest is observed to be _log(m)n [GW96]. In the diagram below, we show, empirically, the probability of a perfect partition (difference of at most one between two subsets) existing among n uniformly random integers among [1, m] with respect

to _log(m)n , the constrainedness parameter. The data is approximated by running 10000 test

Note the increasing steepness of the transition from almost never having a perfect parti- tion to almost always having a perfect partition as the problem size is scaled, approaching that of a step function. Empirical testing has also demonstrated that most known depth-first search algorithms for NP-complete problems are observed to be increasingly complex in the area of the phase transition. In the following diagram, we are plotting the complexity of invoking the Complete Karmarkar-Karp (CKK) algorithm on the same 10000 instances of Partition of n uniformly random integers among [1, m].

Some NP-complete problems of interest in election systems have also been shown to exhibit phase transitions. In [Wal09], it is shown that manipulation of scoring protocols of three candidates exhibits a phase transition when the constrainedness parameter of

√ n m

reaches a constant critical value, irrespective of the problem size. The probability of a manipulation existing makes an abrupt change from 0 to 1 for large m as one crosses this phase transition. In other words, Θ(√n) manipulators are asymptotically needed for one to affect the outcome of an election with n established voters. However, unlike the case of other NP-complete problems, Walsh further shows empirically that instances in which finding a manipulation, or showing that none exist, using known algorithms, do not directly correspond to the phase transition of this problem. It is found that, invoking the Complete Karmarkar-Karp (CKK) algorithm on these manipulation instances, finding a manipulation or showing that none exist can be done on average with a constant number of branches of the CKK algorithm. This is true even when the parameters of the election are chosen within the phase transition of the manipulation problem. In other words, empirical testing did not demonstrate an increasing complexity peak, as seen in the above diagram, as one crosses the phase transition with respect to the constrainedness parameter of interest in the problem of manipulation.

exponentially rare subset of hard instances contribute to the worst-case hardness of the ma- nipulation problem. The same phase transition parameter of manipulability, in which the probability of manipulation leaps from almost 0 to almost 1 when the

√ n

m reaches a critical

value, is also demonstrated empirically by Connett [Con10] for other election systems, includ- ing those involving multiple rounds. Although the exact constant factor appears different in each election system, in all cases, it requires Θ(√n) manipulators to affect the outcome of an election with n established voters.

Chapter 3

Worst-Case Complexity of

Manipulating k-Approval Elections

In this chapter, we evaluate the worst-case computational complexity of manipulation, bribery, and control, in scoring protocols and families of scoring protocols of the form k- approval, k-veto, and some f (m)-approval elections. These systems are defined in Section 2.1.

3.1

Table of Results

We summarize the worst-case complexity results of this chapter in the tables below, with new results in bold.1

Unweighted Cases

1-app 2-app 3-app k-app, k ≥ 4

Constructive Manipulation P P P P

Constructive Bribery P P NPC NPC

Constructive Control by Adding Voters P P P NPC

Constructive Control by Deleting Voters P P NPC NPC Constructive Control by Adding Candidates NPC NPC NPC NPC Constructive Control by Deleting Candidates NPC NPC NPC NPC

1_{In the table, swb is the complexity of Simple Weighted b-Edge Cover of Multigraphs and sbw that of}

1-veto 2-veto 3-veto k-veto, k ≥ 4

Constructive Manipulation P P P P

Constructive Bribery P P P NPC

Constructive Control by Adding Voters P P NPC NPC

Constructive Control by Deleting Voters P P P NPC

Constructive Control by Adding Candidates NPC NPC NPC NPC Constructive Control by Deleting Candidates NPC NPC NPC NPC Weighted Voter Cases

1-app 2-app 3-app k-app, k ≥ 4

Constructive Manipulation P NPC NPC NPC

Constructive Bribery P NPC NPC NPC

Constructive Control by Adding Voters P ? ? NPC

Constructive Control by Deleting Voters P sbw NPC NPC Constructive Control by Adding Candidates NPC NPC NPC NPC Constructive Control by Deleting Candidates NPC NPC NPC NPC

1-veto 2-veto 3-veto k-veto, k ≥ 4

Constructive Manipulation NPC NPC NPC NPC

Constructive Bribery NPC NPC NPC NPC

Constructive Control by Adding Voters P sbw NPC NPC

Constructive Control by Deleting Voters P ? ? NPC

Constructive Control by Adding Candidates NPC NPC NPC NPC Constructive Control by Deleting Candidates NPC NPC NPC NPC Unweighted $Bribery Cases

1-app 2-app 3-app k-app, k ≥ 4

Constructive $Bribery P ? NPC NPC

1-veto 2-veto 3-veto k-veto, k ≥ 4

3.2

Misuses in Elections with a Fixed Number of Can-