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Register a new userOptimal Average Satisfaction and Extended Justified Representation in Polynomial Time
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Added by
Martin Lackner
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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In this short note, we describe an approval-based committee selection rule that admits a polynomial-time algorithm and satisfies the Extended Justified Representation (EJR) axiom. This rule is based on approximately maximizing the PAV score, by means of local search. Our proof strategy is to show that this rule provides almost optimal average satisfaction to all cohesive groups of voters, and that high average satisfaction for cohesive groups implies extended justified representation.
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The goal of multi-winner elections is to choose a fixed-size committee based on voters preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. Recently, Aziz et al. (2015a;2017) proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). In this paper, we extend the work of Aziz et al. in several directions. First, we answer an open question of Aziz et al., by showing that Reweighted Approval Voting satisfies JR for $k=3, 4, 5$, but fails it for $kge 6$. Second, we observe that EJR is incompatible with the Perfect Representation criterion, which is important for many applications of multi-winner voting, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect representation, and a committee that provides PJR can be computed in polynomial time if the committee size divides the number of voters. Moreover, just like EJR, PJR can be used to characterize the classic PAV rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.
We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant--optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm. The problem of determining Dodgson winners is known to be complete for parallel access to NP [HHR97]. Homan and Hemaspaandra [HH06] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is a ``frequently self-knowingly correct algorithm. We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenscheins junta distributions [PR07].
In the late 19th century, Swedish mathematician Lars Edvard Phragm{e}n proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants -- one minimizing the maximal load and one minimizing the variance of loads -- and a sequential variant. We study Phragm{e}ns methods from an axiomatic point of view, focusing on properties capturing proportional representation. We show that the sequential variant satisfies proportional justified representation, which is a rare property for committee monotonic methods. Moreover, we show that the optimization variants satisfy perfect representation. We also analyze the computational complexity of Phragm{e}ns methods and provide mixed-integer programming based algorithms for computing them.
An average-time game is played on the infinite graph of configurations of a finite timed automaton. The two players, Min and Max, construct an infinite run of the automaton by taking turns to perform a timed transition. Player Min wants to minimise the average time per transition and player Max wants to maximise it. A solution of average-time games is presented using a reduction to average-price game on a finite graph. A direct consequence is an elementary proof of determinacy for average-time games. This complements our results for reachability-time games and partially solves a problem posed by Bouyer et al., to design an algorithm for solving average-price games on priced timed automata. The paper also establishes the exact computational complexity of solving average-time games: the problem is EXPTIME-complete for timed automata with at least two clocks.
We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $mge 2$ and $fin mathbb Z[x]$ any squarefree polynomial of degree $dge 3$, along with a positive integer $N$. It can compute $#X(mathbb F_p)$ for all $ple N$ not dividing $mmathrm{lc}(f)mathrm{disc}(f)$ in time $O(md^3 Nlog^3 Nloglog N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$.
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