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28
Determining Possible and Necessary Winners under Common Voting Rules Given Partial Orders
"... Usually a voting rule or correspondence requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a profile of part ..."
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Usually a voting rule or correspondence requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a profile of partial orders and a candidate c, two important questions arise: first, is c guaranteed to win, and second, is it still possible for c to win? These are the necessary winner and possible winner problems, respectively. We consider the setting where the number of alternatives is unbounded and the votes are unweighted. We prove that for Copeland, maximin, Bucklin, and ranked pairs, the possible winner problem is NPcomplete; also, we give a sufficient condition on scoring rules for the possible winner problem to be NPcomplete (Borda satisfies this condition). We also prove that for Copeland and ranked pairs, the necessary winner problem is coNPcomplete. All the hardness results hold even when the number of undetermined pairs in each vote is no more than a constant. We also present polynomialtime algorithms for the necessary winner problem for scoring rules, maximin, and Bucklin.
Compilation complexity of common voting rules
, 2010
"... In computational social choice, one important problem is to take the votes of a subelectorate (subset of the voters), and summarize them using a small number of bits. This needs to be done in such a way that, if all that we know is the summary, as well as the votes of voters outside the subelectorat ..."
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Cited by 58 (12 self)
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In computational social choice, one important problem is to take the votes of a subelectorate (subset of the voters), and summarize them using a small number of bits. This needs to be done in such a way that, if all that we know is the summary, as well as the votes of voters outside the subelectorate, we can conclude which of the m alternatives wins. This corresponds to the notion of compilation complexity, the minimum number of bits required to summarize the votes for a particular rule, which was introduced by Chevaleyre et al. [IJCAI09]. We study three different types of compilation complexity. The first, studied by Chevaleyre et al., depends on the size of the subelectorate but not on the size of the complement (the voters outside the subelectorate). The second depends on the size of the complement but not on the size of the subelectorate. The third depends on both. We first investigate the relations among the three types of compilation complexity. Then, we give upper and lower bounds on all three types of compilation complexity for the most prominent voting rules. We show that for lapproval (when l ≤ m/2), Borda, and Bucklin, the bounds for all three types are asymptotically tight, up to a multiplicative constant; for lapproval (when l> m/2), plurality with runoff, all Condorcet consistent rules that are based on unweighted majority graphs (including Copeland and voting trees), and all Condorcet consistent rules that are based on the order of pairwise elections (including ranked pairs and maximin), the bounds for all three types are asymptotically tight up to a multiplicative constant when the sizes of the subelectorate and its complement are both larger than m 1+ǫ for some ǫ> 0.
It Only Takes a Few: On the Hardness of Voting With a Constant Number of Agents
, 2013
"... Many hardness results in computational social choice make use of the fact that every directed graph may be induced by the pairwise majority relation. However, this fact requires that the number of voters is almost linear in the number of alternatives. It is therefore unclear whether existing hardnes ..."
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Many hardness results in computational social choice make use of the fact that every directed graph may be induced by the pairwise majority relation. However, this fact requires that the number of voters is almost linear in the number of alternatives. It is therefore unclear whether existing hardness results remain intact when the number of voters is bounded, as is for example typically the case in search engine aggregation settings. In this paper, we provide sufficient conditions for majority graphs to be obtainable using a constant number of voters and leverage these conditions to show that winner determination for the Banks set, the tournament equilibrium set, Slater’s rule, and ranked pairs remains hard even when there is only a small constant number of voters.
Acyclic sets in kmajority tournaments
 In preparation
"... When Π is a set of k linear orders on a ground set X, and k is odd, the kmajority tournament generated by Π has vertex set X and has an edge from u to v if and only if a majority of the orders in Π rank u before v. Let fk(n) be the minimum, over all kmajority tournaments with n vertices, of the ma ..."
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When Π is a set of k linear orders on a ground set X, and k is odd, the kmajority tournament generated by Π has vertex set X and has an edge from u to v if and only if a majority of the orders in Π rank u before v. Let fk(n) be the minimum, over all kmajority tournaments with n vertices, of the maximum order of an induced transitive subtournament. We prove that f3(n) ≥ √ n always and that f3(n) ≤ 2 √ n − 1 when n is a perfect square. We also prove that f5(n) ≥ n1/4. For general k, we prove that nck ≤ fk(n) ≤ ndk, where ck = 3−(k−1)/2 and dk = (ln ln k)+2 ln k (ln k)−1 (1 + n).
On the learnability of majority rule
, 2007
"... We establish how large a sample of past decisions is required to predict future decisions of a committee with few members. The committee uses majority rule to choose between pairs of alternatives. Each member’s vote is derived from a linear ordering over all the alternatives. We prove that there are ..."
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We establish how large a sample of past decisions is required to predict future decisions of a committee with few members. The committee uses majority rule to choose between pairs of alternatives. Each member’s vote is derived from a linear ordering over all the alternatives. We prove that there are cases in which an observer cannot predict precisely any decision of a committee based on its past decisions. Nonetheless, approximate prediction is possible after observing relatively few random past decisions.
Largest digraphs contained in all ntournaments, Combinatorica 3
, 1983
"... Let f(n) (resp. g(n)) be the largest m such that there is a digraph (resp. a spanning weakly connected igraph) on nvertices and m edges which is a subgraph of every tournament on nvertices. We prove that n log2 ncxn>=f(n) ~_g(n) ~ n log ~ nc..n loglog n. A directed graph G is an unavoidabl ..."
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Let f(n) (resp. g(n)) be the largest m such that there is a digraph (resp. a spanning weakly connected igraph) on nvertices and m edges which is a subgraph of every tournament on nvertices. We prove that n log2 ncxn>=f(n) ~_g(n) ~ n log ~ nc..n loglog n. A directed graph G is an unavoidable subgraph of all ntournaments or, simply nunavoidable, if every tournament on n vertices contain san isomorphic copy of G, i.e., for each ntournament T there exists an edge preserving injection of the vertices of G into the vertices of T. The problem of showing certain types of graphs to be nunavoidable has been the subject of several papers, for example, it is known that every ntournament contains a Hamiltonian path ([7]), an antidirected Hamiltonian path ([4]) and a transitive subtournament on [log2 n] vertices ([6]). Results of this type are also found in [1], [3], and [8]. In this paper we answer the following question: what is the maximum number of edges that an nunavoidable subgraph can have? Our graph theoretic terminology is standard. For a vertex v of a digraph G=(V, E) we let G+(v)={w[(v, w)EE}. All logarithms are base 2. Let f (n) (resp. g(n)) be the largest m such that there exists a digraph resp. spanmng, weakly connected iagraph) with m edges that is nunavoidable subgraph. Trivially f (n)~g(n). Our main resullt is Theorem. There exists positive constants c ~ and c2 such that for all positi~e integers n,
More measures of vulnerability: Splinter sets and directed toughness
"... We introduce the splinter set of a graph which records all triples (s, m, k) such that there exists a set S of s vertices whose removal leaves k components with maximum component order m. From this set several parameters can be calculated such as connectivity, toughness and integrity. For a parame ..."
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We introduce the splinter set of a graph which records all triples (s, m, k) such that there exists a set S of s vertices whose removal leaves k components with maximum component order m. From this set several parameters can be calculated such as connectivity, toughness and integrity. For a parameter ψ that can be calculated from the splinter set, we define ψ for directed graphs by replacing component by strong component in the definition, and define the directed value ¯ ψ for an undirected graph as the maximum value of ψ over all orientations. As an example we explore the properties of directed toughness.