Succinct representations of graphs
Information and Control
Kings in k-partite tournaments
Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Word problems requiring exponential time(Preliminary Report)
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
The Complexity of Finding Paths in Graphs with Bounded Independence Number
SIAM Journal on Computing
The Complexity of Finding Top-Toda-Equivalence-Class Members
Theory of Computing Systems
Generalizations of tournaments: A survey
Journal of Graph Theory
The equivalence problem for regular expressions with squaring requires exponential space
SWAT '72 Proceedings of the 13th Annual Symposium on Switching and Automata Theory (swat 1972)
Llull and Copeland voting computationally resist bribery and constructive control
Journal of Artificial Intelligence Research
P-Selectivity, immunity, and the power of one bit
SOFSEM'06 Proceedings of the 32nd conference on Current Trends in Theory and Practice of Computer Science
Hi-index | 5.23 |
A king in a directed graph is a vertex from which each vertex in the graph can be reached through paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of reachability problems and semifeasible sets. In this article, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is already known to belong to @P"2^p. We prove that the complexity of kingship problems is a rich enough vocabulary to pinpoint every nontrivial many-one degree in @P"2^p. That is, we show that every set in @P"2^p other than 0@? and @S^* is equivalent to a king problem under @?"m^p-reductions. Indeed, we show that the equivalence can even be realized by relatively simple padding, and holds even if the notion of kings is redefined to refer to k-kings (for any fixed k=2)-vertices from which all vertices can be reached through paths of length at most k. In contrast, we prove that for each succinctly specified family of tournaments the source problem (the problem of deciding whether a given vertex v has the property that there exists a k such that v is a k-king) also falls within @P"2^p, yet cannot be @P"2^p-complete-or even NP-hard-unless P=NP. Using these and related techniques, we obtain a broad range of additional results about the complexity of king problems, diameter problems, and radius problems. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is @P"2^p-complete. We show that the radius problem for arbitrary succinctly represented graphs is @S"3^p-complete, but that in contrast the diameter problem for arbitrary succinctly represented graphs (or even tournaments) is @P"2^p-complete.