We study the problem of coalitional manipulation---where @math manipulators try to manipulate an election on @math candidates---under general scoring rules, with a focus on the Borda protocol. We do so both in the weighted and unweighted settings. We focus on minimizing the maximum score obtainable by a non-preferred candidate. In the strongest, most general setting, we provide an algorithm for any scoring rule as described by a vector @math : for some @math , it obtains an additive approximation equal to @math , where @math is the sum of voter weights. For Borda, both the weighted and unweighted variants are known to be @math -hard. For the unweighted case, our simpler algorithm provides a randomized, additive @math approximation; in other words, if there exists a strategy enabling the preferred candidate to win by an @math margin, our method, with high probability, will find a strategy enabling her to win (albeit with a possibly smaller margin). It thus provides a somewhat stronger guarantee compared to the previous methods, which implicitly implied a strategy that provides an @math -additive approximation to the maximum score of a non-preferred candidate. For the weighted case, our generalized algorithm provides an @math -additive approximation, where @math is the sum of voter weights. This is a clear advantage over previous methods: some of them do not generalize to the weighted case, while others---which approximate the number of manipulators---pose restrictions on the weights of extra manipulators added. Our methods are based on carefully rounding an exponentially-large configuration linear program that is solved by using the ellipsoid method with an efficient separation oracle.
In the weighted case, the situation is different: for all positional scoring rules @math , except plurality-like rules, @math -WCM is @math -hard when @math @cite_1 @cite_19 @cite_3 . In particular, this holds for Borda-WCM. However, the computational hardness of Borda-UCM still remained open for quite some time, until finally shown to be @math -hard as well @cite_0 @cite_23 in 2011, even for the case of @math and adding @math manipulators.
@cite_1 @cite_3 @cite_0 @cite_19 @cite_23
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We investigate manipulation of the Borda voting rule, as well as two elimination style voting rules, Nanson's and Baldwin's voting rules, which are based on Borda voting. We argue that these rules have a number of desirable computational properties. For unweighted Borda voting, we prove that it is NP-hard for a coalition of two manipulators to compute a manipulation. This resolves a long-standing open problem in the computational complexity of manipulating common voting rules. We prove that manipulation of Baldwin's and Nanson's rules is computationally more difficult than manipulation of Borda, as it is NP-hard for a single manipulator to compute a manipulation. In addition, for Baldwin's and Nanson's rules with weighted votes, we prove that it is NP-hard for a coalition of manipulators to compute a manipulation with a small number of candidates.Because of these NP-hardness results, we compute manipulations using heuristic algorithms that attempt to minimise the number of manipulators. We propose several new heuristic methods. Experiments show that these methods significantly outperform the previously best known heuristic method for the Borda rule. Our results suggest that, whilst computing a manipulation of the Borda rule is NP-hard, computational complexity may provide only a weak barrier against manipulation in practice. In contrast to the Borda rule, our experiments with Baldwin's and Nanson's rules demonstrate that both of them are often more difficult to manipulate in practice. These results suggest that elimination style voting rules deserve further study. The Borda voting rule is a positional scoring rule where, for m candidates, for every vote the first candidate receives m- 1 points, the second m- 2 points and so on. A Borda winner is a candidate with highest total score. It has been a prominent open problem to determine the computational complexity of UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA: Can one add a certain number of additional votes (called manipulators) to an election such that a distinguished candidate becomes a winner? We settle this open problem by showing NP-hardness even for two manipulators and three input votes. Moreover, we discuss extensions and limitations of this hardness result. Positional scoring rules in voting compute the score of an alternative by summing the scores for the alternative induced by every vote. This summation principle ensures that all votes contribute equally to the score of an alternative. We relax this assumption and, instead, aggregate scores by taking into account the rank of a score in the ordered list of scores obtained from the votes. This defines a new family of voting rules, rank-dependent scoring rules (RDSRs), based on ordered weighted average (OWA) operators, which, include all scoring rules, and many others, most of which of new. We study some properties of these rules, and show, empirically, that certain RDSRs are less manipulable than Borda voting, across a variety of statistical cultures. Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz.