Nadja Betzler

Unweighted coalitional manipulation under the Borda rule Is NP-hard

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.

Parameterized computational complexity of Dodgson and Young elections

Abstract We show that the two NP-complete problems of Dodgson Score and Young Score have differing computational complexities when the winner is close to being a Condorcet winner. On the one hand, we present an efficient fixed-parameter algorithm for determining a Condorcet winner in Dodgson elections by a minimum number of switches in the votes. On the other hand, we prove that the corresponding problem for Young elections, where one has to delete votes instead of performing switches, is W[2]-complete. In addition, we study Dodgson elections that allow ties between the candidates and give fixed-parameter tractability as well as W[2]-completeness results depending on the cost model for switching ties.

Towards a dichotomy for the Possible Winner problem in elections based on scoring rules

To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the Possible Winner problem that asks, given a set of partial votes, whether a distinguished candidate can still become a winner. In this work, we consider the computational complexity of Possible Winner for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, k-approval, and Borda. Generalizing previous NP-hardness results for some special cases, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2,1,...,1,0), while it is solvable in polynomial time for plurality and veto.

Towards a Dichotomy of Finding Possible Winners in Elections Based on Scoring Rules

To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the Possible Winner problem that asks, given a set of partial votes, if a distinguished candidate can still become a winner. In this work, we consider the computational complexity of Possible Winner for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, k-approval, and Borda. Generalizing previous NP-hardness results for some special cases and providing new many-one reductions, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2,1,...,1,0), while it is solvable in polynomial time for plurality and veto.

Experiments on data reduction for optimal domination in networks

We present empirical results on computing optimal dominating sets in networks by means of data reduction through efficient preprocessing rules. Thus, we demonstrate the usefulness of so far only theoretically considered data reduction techniques for practically solving one of the most important network problems in combinatorial optimization.

Separator-based data reduction for signed graph balancing

Polynomial-time data reduction is a classical approach to hard graph problems. Typically, particular small subgraphs are replaced by smaller gadgets. We generalize this approach to handle any small subgraph that has a small separator connecting it to the rest of the graph. The problem we study is the NP-hard Balanced Subgraph problem, which asks for a 2-coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. The data reduction scheme unifies and generalizes a number of previously known data reductions, and can be applied to a large number of graph problems where a coloring or a subset of the vertices is sought. To solve the instances that remain after reduction, we use a fixed-parameter algorithm based on iterative compression with a very effective heuristic speedup. Our implementation can solve biological real-world instances exactly for which previously only approximations were known. In addition, we present experimental results for financial networks and random networks.

Optimal edge deletions for signed graph balancing

The Balanced Subgraph problem (edge deletion variant) asks for a 2-coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. We present an exact algorithm for BALANCED SUBGRAPH based on a combination of data reduction rules and a fixed-parameter algorithm. The data reduction is based on finding small separators and a novel gadget construction scheme. The fixed-parameter algorithm is based on iterative compression with a very effective heuristic speedup. Our implementation can solve biological real-world instances exactly for which previously only approximations [DasGupta et al., WEA 2006] were known.

On Bounded-Degree Vertex Deletion parameterized by treewidth

Given an undirected graph G and an integer d>=0, the NP-hard Bounded-Degree Vertex Deletion problem asks to delete as few vertices as possible from G such that the resulting graph has maximum vertex degree d. Our main result is to prove that Bounded-Degree Vertex Deletion is W[1]-hard with respect to the parameter treewidth. As a side result, we obtain that the NP-hard Vector Dominating Set problem is W[1]-hard with respect to the parameter treewidth. On the positive side, we show that Bounded-Degree Vertex Deletion becomes fixed-parameter tractable when parameterized by the combined parameter treewidth and number of vertices to delete, and when parametrized by the feedback edge set number.

Partial Kernelization for Rank Aggregation: Theory and Experiments

Rank Aggregation is important in many areas ranging from web search over databases to bioinformatics. The underlying decision problem Kemeny Score is NP-complete even in case of four input rankings to be aggregated into a “median ranking”. We study efficient polynomial-time data reduction rules that allow us to find optimal median rankings. On the theoretical side, we improve a result for a “partial problem kernel” from quadratic to linear size. On the practical side, we provide encouraging experimental results with data based on web search and sport competitions, e.g., computing optimal median rankings for real-world instances with more than 100 candidates within milliseconds.

Parameterized Computational Complexity of Dodgson and Young Elections

We show that, other than for standard complexity theory with known NP-completeness results, the computational complexity of the Dodgson and Young election systems is completely different from a parameterized complexity point of view. That is, on the one hand, we present an efficient fixed-parameter algorithm for determining a Condorcet winner in Dodgson elections by a minimum number of switches in the votes. On the other hand, we prove that the corresponding problem for Young elections, where one has to delete votes instead of performing switches, is W[2]-complete. In addition, we study Dodgson elections that allow ties between the candidates and give fixed-parameter tractability as well as W[2]-hardness results depending on the cost model for switching ties.