from minimal feedback vertex set to democracy

Minimal Feedback Vertex/Arc Set

Aggregating Inconsistency and democracy?

Algorithmic Perspective

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Minimum Feedback Vertex Set

▪ NP-‐complete ▪ Compliment: Maximum Acyclic Sub-‐graph

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Minimum Feedback Arc Set

▪ NP-‐complete

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MFVS/MFAS Applications

▪ Deadlock recovery ▪ VLSI design ▪ Loop Cut-‐set of Bayesian networks

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On Tournaments

▪ NP-‐hard ▫ Tournament = Complete graph

▪ Preliminaries for approximation ▫ Break ties arbitrarily: right hand side always wins. ▫ Probability constraint: mostly required ▫ Triangle inequality: optional

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Tournament Applications

▪ Tennis games ▪ Aggregating ranking functions ▫ SQL statement GROUP BY, ORDER BY, etc ◾SELECT boy FROM human ORDER BY height DESC, weight, income DESC

▫ Meta-‐search ◾Merge and re-‐rank Google, Yahoo, and Baidu results on the same keyword.

▪ Election

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Weighted In-‐degrees Algorithm

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1.6

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1.4

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▪ Sort by in-‐degrees ▫ How many weighted edges go into a vertex?

▪ 5-‐approximation to vertex numbers

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1.6

1.8

1.4

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▪ Sort by in-‐degrees ▫ How many weighted edges go into a vertex?

▪ 5-‐approximation to vertex numbers

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1.6

1.4

1.2

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Pivoting Algorithm

• Steps:– Find a pivot• Randomly• Deterministically

– Branch losers to right, winners to left

– Recursion

• Similar to:

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Pivoting Algorithm



– Recursion

• Similar to:

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Pivoting Algorithm



– Recursion

• Similar to:

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Pivoting Algorithm



– Recursion

• Similar to:– Quick Sort

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Pivoting Algorithm



– Recursion

• Similar to:– Quick Sort– Minimum Spanning Tree

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On Bipartite Tournaments

▪ No cycles in length of 3 but 4

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Bipartite Tournament Applications

▪ Stable marriage problem ▫ Hospitals/residents problem with couples

▪ Automatic syllable-‐to-‐word conversion ▫ Boundary-‐decision around /shi4/ ◾A low-‐rank player may defeat a high-‐rank one

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Syllable-‐boundary Decision

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zhi1-‐shi4

tou2-‐shi4

dao4-‐shi4

shi4-‐wei4

shi4-‐li4

shi4-‐shang4

General Modeling Approach

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Inconsistency

▪ Different Rankings on a List ▫ Personal preferences ▫ Search engine orders ▫ Ambiguous natural language usages ▫ Protein sequence alignments

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Aggregation

▪ Denote voters as V, candidates as C ▫ Large V, small C ◾Elections

▫ Small V, modest C ◾Games

▫ Small V, large C ◾Meta-‐search ◾Travel plan

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Distance Measures

• Vectors: Bus := 1, Van := 2, Train := 3 – A prefers Bus > Van > Train: [1, 2, 3] – B prefers Van > Bus > Train: [2, 1, 3]

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Distance Measures


• Spearman footrule distance – | 1 – 2 | + | 2 – 1 | + | 3 – 3 | = 2

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Distance Measures


• Spearman footrule distance – | 1 – 2 | + | 2 – 1 | + | 3 – 3 | = 2

• Kendall tau distance: disagreement – I(1, 2) + I(1, 3) + I(2, 3) = 1 + 0 + 0 = 1 • Normalize: 1 / (3 * (3 – 1) / 2) = 1/3 • Bubble sort

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Distance Measures

Kemeny Optimal Aggregation

▪ Minimize disagreements, i.e. Kendall tau ▫ NP-‐hard when voters = 4 and candidates = 3

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Heuristics

▪ Footrule optimization ▫ K(σ,τ) ≤ F(σ,τ) ≤ 2 K(σ,τ)

▪ Local Kemenization ▫ An aggregated partial list

▪ Markov chain ▫ State := candidates ▫ Transition probability := rankings ▫ Result := stationary distribution

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Aggregating Inconsistency:Ranking

▪ FAS-‐Tournament ▫ Apply pivoting algorithm

▪ Rank-‐Aggregation ▫ Convert to weighted FAS-‐Tournament ▫ Convert to un-‐weighted FAS-‐Tournament ◾Majority tournament

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Aggregating Inconsistency: Clustering

▪ Correlation-‐Clustering ▫ Minimize disagreement pairs on edges with +/-‐. ◾Disagreement: ‘+’ in different clusters or ‘-‐’ in the same one.

▪ Consensus-‐Clustering ▫ Minimize distances between each given clusters and the merged one. ◾Distance: unordered pairs are clustered together by one and separated by the other.

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Related Works

▪ Approximating Minimum Feedback sets and Multicuts in Directed Graphs ▫ The first approximation on ratio O(lognloglogn)

▪ Deterministic pivoting algorithms for constrained ranking and clustering problems ▫ Hierarchical clustering

▪ Fixed-‐Parameter Tractability Results for Feedback Set Problems in Tournaments ▫ Iterative compression

▪ Approximation Algorithms for the Feedback Vertex Set Problem with Applications to Constraint Satisfaction and Bayesian Inference

▪ Stable Marriage Problem and College Admission ▫ By Feedback Vertex Set

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Political Economy View

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Democracy: Rule by People

▪ Voting System: Competitive▪ Consensus Decision Making: Co-‐operative

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Democracy: Rule by People

▪ Voting System: Competitive▪ Consensus Decision Making: Co-‐operative▪ Off-‐topic for the time being

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Condorcet and Borda

▪ Condorcet Winner/Loser in tournaments ▪ Extended Condorcet Criterion▫ Associative Law

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Condorcet and Borda

▪ Condorcet Winner/Loser in tournaments ▪ Extended Condorcet Criterion▫ Associative Law◾Not always true

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Condorcet and Borda


▫ “Local Kemenization”

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Condorcet and Borda



▪ Borda Count

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Condorcet and Borda



▪ Borda Count▫ Descending ranking

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Condorcet and Borda



▪ Borda Count▫ Descending ranking▫ “Weighted In-‐degrees Algorithm”

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Voting System

▪ Aggregating Different Rankings ▫ Preliminary: Rational Choices ▫ Property: Arrow’s Impossibility Theorem

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Rational Choice

▪ Definition: Transitive Law ▫ Acyclic: “Minimal Feedback Arc Set”

▪ Example: At the moment of voting, my preference for beverage is coffee > juice > tea.

At the same time, if a waitress ask me: “coffee or tea, sir?”

A rational choice cannot be tea.

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Arrow’s Paradox

Statements Non-‐dictatorship Universality

Unrestricted domain, deterministic Independence of irrelevant alternatives (IIA)

Acyclic Monotonicity

positive association of social and individual values Non-‐imposition

Onto No fair system if candidates ≥ 3

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Inspiration

▪ How hard to conduct a perfect voting system? ▪ How good we can reach on voting problem? ▪ How do computer scientists and economists help each other?

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The EndThank you.

from minimal feedback vertex set to democracy

Technology

pivotingalgorithm steps

largec metasearch travelplan

quicksort minimumspanningtree

modestc games smallv

smallc elections smallv

candidatesasc largev

aggregation denotevotersasv

general modeling approach