PTAS for Minimax Approval Voting

We consider Approval Voting systems where each voter decides on a subset to candidates he/she approves. We focus on the optimization problem of finding the committee of fixed size k minimizing the maximal Hamming distance from a vote. In this paper we give a PTAS for this problem and hence resolve the open question raised by Carragianis et al. [AAAI'10]. The result is obtained by adapting the techniques developed by Li et al. [JACM'02] originally used for the less constrained Closest String problem. The technique relies on extracting information and structural properties of constant size subsets of votes.Comment: 15 pages, 1 figur

Lower bounds for approximation schemes for Closest String

In the Closest String problem one is given a family $\mathcal S$ of equal-length strings over some fixed alphabet, and the task is to find a string $y$ that minimizes the maximum Hamming distance between $y$ and a string from $\mathcal S$. While polynomial-time approximation schemes (PTASes) for this problem are known for a long time [Li et al., J. ACM'02], no efficient polynomial-time approximation scheme (EPTAS) has been proposed so far. In this paper, we prove that the existence of an EPTAS for Closest String is in fact unlikely, as it would imply that $\mathrm{FPT}=\mathrm{W}[1]$, a highly unexpected collapse in the hierarchy of parameterized complexity classes. Our proof also shows that the existence of a PTAS for Closest String with running time $f(\varepsilon)\cdot n^{o(1/\varepsilon)}$, for any computable function $f$, would contradict the Exponential Time Hypothesis

Approximation and Parameterized Complexity of Minimax Approval Voting

We present three results on the complexity of Minimax Approval Voting. First, we study Minimax Approval Voting parameterized by the Hamming distance $d$ from the solution to the votes. We show Minimax Approval Voting admits no algorithm running in time $\mathcal{O}^\star(2^{o(d\log d)})$, unless the Exponential Time Hypothesis (ETH) fails. This means that the $\mathcal{O}^\star(d^{2d})$ algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time $\mathcal{O}^\star(\left({3}/{\epsilon}\right)^{2d})$, which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for Minimax Approval Voting, which runs in time $n^{\mathcal{O}(1/\epsilon^2 \cdot \log(1/\epsilon))} \cdot \mathrm{poly}(m)$, almost matching the running time of the fastest known PTAS for Closest String due to Ma and Sun [SIAM J. Comp. 2009].Comment: 14 pages, 3 figures, 2 pseudocode

Mathematical Programming formulations for the efficient solution of the kk-sum approval voting problem

In this paper we address the problem of electing a committee among a set of $m$ candidates and on the basis of the preferences of a set of $n$ voters. We consider the approval voting method in which each voter can approve as many candidates as she/he likes by expressing a preference profile (boolean $m$-vector). In order to elect a committee, a voting rule must be established to `transform' the $n$ voters' profiles into a winning committee. The problem is widely studied in voting theory; for a variety of voting rules the problem was shown to be computationally difficult and approximation algorithms and heuristic techniques were proposed in the literature. In this paper we follow an Ordered Weighted Averaging approach and study the $k$-sum approval voting (optimization) problem in the general case $1 \leq k <n$. For this problem we provide different mathematical programming formulations that allow us to solve it in an exact solution framework. We provide computational results showing that our approach is efficient for medium-size test problems ($n$ up to 200, $m$ up to 60) since in all tested cases it was able to find the exact optimal solution in very short computational times

Penalty-Based Aggregation of Strings

International Summer School on Aggregation Operators (2019. Olomouc, Czech Republic

Measuring diversity of socio-cognitively inspired ACO search

In our recent research, we implemented an enhancement of Ant Colony Optimization incorporating the socio-cognitive dimension of perspective taking. Our initial results suggested that increasing the diversity of ant population - introducing different pheromones, different species and dedicated inter-species relations - yielded better results. In this paper, we explore the diversity issue by introducing novel diversity measurement strategies for ACO. Based on these strategies we compare both classic ACO and its socio-cognitive variation

Faster Binary Mean Computation Under Dynamic Time Warping

Many consensus string problems are based on Hamming distance. We replace Hamming distance by the more flexible (e.g., easily coping with different input string lengths) dynamic time warping distance, best known from applications in time series mining. Doing so, we study the problem of finding a mean string that minimizes the sum of (squared) dynamic time warping distances to a given set of input strings. While this problem is known to be NP-hard (even for strings over a three-element alphabet), we address the binary alphabet case which is known to be polynomial-time solvable. We significantly improve on a previously known algorithm in terms of worst-case running time. Moreover, we also show the practical usefulness of one of our algorithms in experiments with real-world and synthetic data. Finally, we identify special cases solvable in linear time (e.g., finding a mean of only two binary input strings) and report some empirical findings concerning combinatorial properties of optimal means

On Finding the Jaccard Center

We initiate the study of finding the Jaccard center of a given collection N of sets. For two sets X,Y, the Jaccard index is defined as |Xcap Y|/|Xcup Y| and the corresponding distance is 1-|Xcap Y|/|Xcup Y|. The Jaccard center is a set C minimizing the maximum distance to any set of N. We show that the problem is NP-hard to solve exactly, and that it admits a PTAS while no FPTAS can exist unless P = NP. Furthermore, we show that the problem is fixed parameter tractable in the maximum Hamming norm between Jaccard center and any input set. Our algorithms are based on a compression technique similar in spirit to coresets for the Euclidean 1-center problem. In addition, we also show that, contrary to the previously studied median problem by Chierichetti et al. (SODA 2010), the continuous version of the Jaccard center problem admits a simple polynomial time algorithm

Population Diversity in Ant-inspired Optimization Algorithms

Finding a balance between exploration and exploitation is very important in the case of metaheuristics optimization, especially in the systems leveraging population of individuals expressing (as in Evolutionary Algorithms, etc.) or constructing (as in Ant Colony Optimization) solutions. Premature convergence is a real problem and finding means of its automatic detection and counteracting are of great importance. Measuring diversity in Evolutionary Algorithms working in real-value search space is often computationally complex, but feasible while measuring diversity in combinatorial domain is practically impossible (cf. Closest String Problem). Nevertheless, we propose several practical and feasible diversity measurement techniques dedicated to Ant Colony Optimization algorithms, leveraging the fact that even though analysis of the search space is at least an NP problem, we can focus on the pheromone table, where the direct outcomes of the search are expressed and can be analyzed. Besides proposing the measurement techniques, we apply them to assess the diversity of several variants of ACO, and closely analyze their features for the classic ACO. The discussion of the results is the first step towards applying the proposed measurement techniques in auto-adaptation of the parameters affecting directly the exploitation and exploration features in ACO in the future