[Paper Review] Approximation and Parameterized Complexity of Minimax Approval Voting

Marek Cygan, Łukasz Kowalik+2 more|arXiv (Cornell University)|Jul 26, 2016

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TL;DR

This paper establishes tight parameterized complexity bounds for Minimax Approval Voting, proving that the best-known algorithm is essentially optimal under the Exponential Time Hypothesis. It introduces a parameterized approximation scheme running in time $\mathcal{O}^*( (3/\epsilon)^{2d} )$, and presents a new randomized polynomial-time approximation scheme with near-optimal running time, nearly matching the fastest known PTAS for Closest String.

ABSTRACT

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 $\mathcal{O}^\star(\left({3}/ε ight)^{2d})$, which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for Minimax Approval Voting, which runs in time $n^{\mathcal{O}(1/ε^2 \cdot \log(1/ε))} \cdot \mathrm{poly}(m)$, almost matching the running time of the fastest known PTAS for Closest String due to Ma and Sun [SIAM J. Comp. 2009].

Motivation & Objective

• To understand the parameterized complexity of Minimax Approval Voting when parameterized by Hamming distance $d$ from the solution to the votes.
• To close the gap between known upper and lower bounds for parameterized algorithms in this setting.
• To develop a fast, randomized polynomial-time approximation scheme (PTAS) for Minimax Approval Voting with near-optimal running time.
• To explore the possibility of derandomizing the PTAS and improving the running time of existing PTASs for Minimax Approval Voting and related problems like Closest String.

Proposed method

• Proves that no algorithm can solve Minimax Approval Voting in time $\mathcal{O}^*(2^{o(d\log d)})$ unless the Exponential Time Hypothesis (ETH) fails, establishing tight lower bounds.
• Designs a parameterized approximation scheme (PAS) running in time $\mathcal{O}^*( (3/\epsilon)^{2d} )$, which is essentially optimal under ETH.
• Introduces a randomized PTAS that runs in time $n^{\mathcal{O}(1/\epsilon^2 \cdot \log(1/\epsilon))} \cdot \text{poly}(m)$, nearly matching the fastest known PTAS for Closest String.
• Employs random rounding of a linear programming solution and uses concentration inequalities (e.g., Chernoff bounds) to bound the probability of deviation from expected values.
• Combines a deterministic algorithm for small $d$ with a randomized PTAS to achieve a high-probability $ (1+\epsilon) $-approximation.
• Uses probabilistic analysis to show that with high probability, the Hamming distance to all votes is within $ (1+\epsilon) \cdot \text{OPT} $, and the number of ones in the solution is close to $k$.

Experimental results

Research questions

• RQ1 Can the $\mathcal{O}^*(d^{2d})$ algorithm for Minimax Approval Voting be improved, and what are the limits of such improvements?
• RQ2 Is there a parameterized approximation scheme for Minimax Approval Voting that runs in time $\mathcal{O}^*( (3/\epsilon)^{2d} )$, and is this bound tight?
• RQ3 Can a randomized PTAS for Minimax Approval Voting be constructed with running time nearly matching the fastest known PTAS for Closest String?
• RQ4 Is it possible to derandomize the proposed randomized PTAS while maintaining similar running time?
• RQ5 What are the tightest possible running time bounds for PTAS in Minimax Approval Voting and Closest String under ETH?

Key findings

• Minimax Approval Voting admits no algorithm running in time $\mathcal{O}^*(2^{o(d\log d)})$ unless the Exponential Time Hypothesis (ETH) fails, proving that the $\mathcal{O}^*(d^{2d})$ algorithm by Misra et al. is essentially optimal.
• A parameterized approximation scheme (PAS) is presented with running time $\mathcal{O}^*( (3/\epsilon)^{2d} )$, which is essentially tight under ETH.
• A new randomized polynomial-time approximation scheme (PTAS) is developed with running time $n^{\mathcal{O}(1/\epsilon^2 \cdot \log(1/\epsilon))} \cdot \text{poly}(m)$, nearly matching the fastest known PTAS for Closest String.
• The PTAS achieves a $ (1+\epsilon) $-approximation with probability at least $ 1-r $, and the total running time is bounded by $ n^{\mathcal{O}(\log(1/\epsilon)/\epsilon^2)} \cdot \text{poly}(m) $.
• Under ETH, no PTAS for Minimax Approval Voting can run in time $ f(\epsilon) \cdot n^{o(1/\epsilon)} $, implying a fundamental lower bound on the running time of any PTAS.
• The analysis shows that with probability at least $ 1/2 $, the Hamming distance to all votes is within $ (1+\epsilon) \cdot \text{OPT} $, and the number of ones in the solution is within $ \frac{1}{2}\epsilon \cdot \text{OPT} $ of $ k $, using concentration bounds on $ E_i $ and $ F_i $.