Asymptotic and finite size parameters for phase transitions: Hamiltonian circuit as a case study

Frank, Jeremy; Gent, Ian P.; Walsh, Toby

doi:10.1016/s0020-0190(97)00222-6

“…1.62 m is the published worst-case bound for the recursive algorithm used to compute a manipulation [3]. Similar phase transition studies have been used to identify hard instances of NP-hard problems like propositional satisfiability [12,17,18], constraint satisfaction [19][20][21][22][23], number partitioning [24][25][26], Hamiltonian circuit [27,28], and the traveling salesperson problem [29,30]. Phase transition studies have also been used to study polynomial problems [31,32] as well as higher complexity classes [33,34] and optimization problems [35,36].…”

Section: Resultsmentioning

confidence: 99%

Is computational complexity a barrier to manipulation?

Walsh

¹

2011

Ann Math Artif Intell

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Abstract. When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by misreporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NPhardness only bounds the worst-case complexity. Recent theoretical results suggest that manipulation may often be easy in practice. To address this issue, I suggest studying empirically if computational complexity is in practice a barrier to manipulation. The basic tool used in my investigations is the identification of computational "phase transitions". Such an approach has been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems. I show that phase transition behaviour gives insight into the hardness of manipulating voting rules, increasing concern that computational complexity is indeed any sort of barrier. Finally, I look at the problem of computing manipulation of other, related problems like stable marriage and tournament problems.

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“…Di erent researchers (Cheeseman et al, 1991;Frank & Martel, 1995;Frank, Gent, & Walsh, 1998) have examined phase transitions on random graphs for the Hamiltonian cycle problem. The obvious constraint parameter is the average degree (or average connectivity) of the graph.…”

Section: What Is Hardness?mentioning

confidence: 99%

The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem

Vandegriend¹,

Culberson²

1998

jair

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Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian cycle instances with the G n;m phase transition between Hamiltonicity and non-Hamiltonicity.Instead all tested graphs of 100 to 1500 vertices are easily solved.When we arti cially restrict the degree sequence with a bounded maximum degree, although there is some increase in di culty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O1 to On. We h a ve so far found no class of graphs correlated with the G n;m phase transition which asymptotically produces a high frequency of hard instances.

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“…The number of agents n is fixed and we vary the number of candidates m. The y-axis measures the probability that the manipulator can make a random candidate win. Similar phase transition studies have been used to identify hard instances of NP-hard problems like propositional satisfiability [12,17,18], constraint satisfaction [19][20][21][22][23], number partitioning [24][25][26], Hamiltonian circuit [27,28], and the traveling salesperson problem [29,30]. Phase transition studies have also been used to study polynomial problems [31,32] as well as higher complexity classes [33,34] and optimization problems [35,36].…”

Section: Resultsmentioning

confidence: 99%

Is Computational Complexity a Barrier to Manipulation?

Walsh

¹

2010

Lecture Notes in Computer Science

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0

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Abstract. When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by misreporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NPhardness only bounds the worst-case complexity. Recent theoretical results suggest that manipulation may often be easy in practice. To address this issue, I suggest studying empirically if computational complexity is in practice a barrier to manipulation. The basic tool used in my investigations is the identification of computational "phase transitions". Such an approach has been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems. I show that phase transition behaviour gives insight into the hardness of manipulating voting rules, increasing concern that computational complexity is indeed any sort of barrier. Finally, I look at the problem of computing manipulation of other, related problems like stable marriage and tournament problems.

show abstract

Asymptotic and finite size parameters for phase transitions: Hamiltonian circuit as a case study

Cited by 10 publications

References 10 publications

Is computational complexity a barrier to manipulation?

Is computational complexity a barrier to manipulation?

The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem

Is Computational Complexity a Barrier to Manipulation?

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