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Minimal Feedback Vertex/Arc Set
Aggregating Inconsistency and democracy?
Algorithmic Perspective
2007/3/28 2
Minimum Feedback Vertex Set
▪ NP-‐complete ▪ Compliment: Maximum Acyclic Sub-‐graph
2007/3/28 3
Minimum Feedback Vertex Set
▪ NP-‐complete ▪ Compliment: Maximum Acyclic Sub-‐graph
2007/3/28 3
Minimum Feedback Arc Set
▪ NP-‐complete
2007/3/28 4
Minimum Feedback Arc Set
▪ NP-‐complete
2007/3/28 4
MFVS/MFAS Applications
▪ Deadlock recovery ▪ VLSI design ▪ Loop Cut-‐set of Bayesian networks
2007/3/28 5
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
2007/3/28 6
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
2007/3/28 7
Weighted In-‐degrees Algorithm
2007/3/28 8
1.6
1.8
1.4
1.2.5
.8
.6
.9
.5
.7
Weighted In-‐degrees Algorithm
▪ Sort by in-‐degrees ▫ How many weighted edges go into a vertex?
▪ 5-‐approximation to vertex numbers
2007/3/28 8
1.6
1.8
1.4
1.2.5
.8
.6
.9
.5
.7
Weighted In-‐degrees Algorithm
▪ Sort by in-‐degrees ▫ How many weighted edges go into a vertex?
▪ 5-‐approximation to vertex numbers
2007/3/28 8
1.6
1.8
1.4
1.2.5
.8
.6
.9
.5
.7
Weighted In-‐degrees Algorithm
▪ Sort by in-‐degrees ▫ How many weighted edges go into a vertex?
▪ 5-‐approximation to vertex numbers
2007/3/28 8
1.6
1.4
1.2
.8
.5
.7
Pivoting Algorithm
• Steps:– Find a pivot• Randomly• Deterministically
– Branch losers to right, winners to left
– Recursion
• Similar to:
2007/3/28 9
.5
.8
.6
.9
.5
.7
Pivoting Algorithm
• Steps:– Find a pivot• Randomly• Deterministically
– Branch losers to right, winners to left
– Recursion
• Similar to:
2007/3/28 9
.5
.6
.9
.5
Pivoting Algorithm
• Steps:– Find a pivot• Randomly• Deterministically
– Branch losers to right, winners to left
– Recursion
• Similar to:
2007/3/28 9
.5
.6.5
Pivoting Algorithm
• Steps:– Find a pivot• Randomly• Deterministically
– Branch losers to right, winners to left
– Recursion
• Similar to:– Quick Sort
2007/3/28 9
.5
.6.5
Pivoting Algorithm
• Steps:– Find a pivot• Randomly• Deterministically
– Branch losers to right, winners to left
– Recursion
• Similar to:– Quick Sort– Minimum Spanning Tree
2007/3/28 9
.5
.6.5
On Bipartite Tournaments
▪ No cycles in length of 3 but 4
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.5
.9
.5
.7
On Bipartite Tournaments
▪ No cycles in length of 3 but 4
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.5
.9
.5
.7
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
2007/3/28 11
Syllable-‐boundary Decision
2007/3/28 12
zhi1-‐shi4
tou2-‐shi4
dao4-‐shi4
shi4-‐wei4
shi4-‐li4
shi4-‐shang4
General Modeling Approach
2007/3/28 13
Inconsistency
▪ Different Rankings on a List ▫ Personal preferences ▫ Search engine orders ▫ Ambiguous natural language usages ▫ Protein sequence alignments
2007/3/28 14
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
2007/3/28 15
2007/3/28 16
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]
2007/3/28 16
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]
• Spearman footrule distance – | 1 – 2 | + | 2 – 1 | + | 3 – 3 | = 2
2007/3/28 16
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]
• 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
2007/3/28 16
Distance Measures
Kemeny Optimal Aggregation
▪ Minimize disagreements, i.e. Kendall tau ▫ NP-‐hard when voters = 4 and candidates = 3
2007/3/28 17
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
2007/3/28 18
Aggregating Inconsistency:Ranking
▪ FAS-‐Tournament ▫ Apply pivoting algorithm
▪ Rank-‐Aggregation ▫ Convert to weighted FAS-‐Tournament ▫ Convert to un-‐weighted FAS-‐Tournament ◾Majority tournament
2007/3/28 19
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.
2007/3/28 20
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
2007/3/28 21
Political Economy View
2007/3/28 22
Democracy: Rule by People
▪ Voting System: Competitive▪ Consensus Decision Making: Co-‐operative
2007/3/28 23
Democracy: Rule by People
▪ Voting System: Competitive▪ Consensus Decision Making: Co-‐operative▪ Off-‐topic for the time being
2007/3/28 23
Condorcet and Borda
▪ Condorcet Winner/Loser in tournaments ▪ Extended Condorcet Criterion▫ Associative Law
2007/3/28 24
Condorcet and Borda
▪ Condorcet Winner/Loser in tournaments ▪ Extended Condorcet Criterion▫ Associative Law◾Not always true
2007/3/28 24
Condorcet and Borda
▪ Condorcet Winner/Loser in tournaments ▪ Extended Condorcet Criterion▫ Associative Law◾Not always true
▫ “Local Kemenization”
2007/3/28 24
Condorcet and Borda
▪ Condorcet Winner/Loser in tournaments ▪ Extended Condorcet Criterion▫ Associative Law◾Not always true
▫ “Local Kemenization”
▪ Borda Count
2007/3/28 24
Condorcet and Borda
▪ Condorcet Winner/Loser in tournaments ▪ Extended Condorcet Criterion▫ Associative Law◾Not always true
▫ “Local Kemenization”
▪ Borda Count▫ Descending ranking
2007/3/28 24
Condorcet and Borda
▪ Condorcet Winner/Loser in tournaments ▪ Extended Condorcet Criterion▫ Associative Law◾Not always true
▫ “Local Kemenization”
▪ Borda Count▫ Descending ranking▫ “Weighted In-‐degrees Algorithm”
2007/3/28 24
Voting System
▪ Aggregating Different Rankings ▫ Preliminary: Rational Choices ▫ Property: Arrow’s Impossibility Theorem
2007/3/28 25
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.
2007/3/28 26
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
2007/3/28 27
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?
2007/3/28 28
The EndThank you.