184 research outputs found
The Complexity of Kings
A king in a directed graph is a node from which each node in the graph can be
reached via 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 [Lan53]. Recently,
kings have proven useful in theoretical computer science, in particular in the
study of the complexity of the semifeasible sets [HNP98,HT05] and in the study
of the complexity of reachability problems [Tan01,NT02].
In this paper, we study the complexity of recognizing kings. For each
succinctly specified family of tournaments, the king problem is known to belong
to [HOZZ]. We prove that this bound is optimal: We construct a
succinctly specified tournament family whose king problem is
-complete. It follows easily from our proof approach that the problem
of testing kingship in succinctly specified graphs (which need not be
tournaments) is -complete. We also obtain -completeness
results for k-kings in succinctly specified j-partite tournaments, , and we generalize our main construction to show that -completeness
holds for testing k-kingship in succinctly specified families of tournaments
for all
Reachability analysis of first-order definable pushdown systems
We study pushdown systems where control states, stack alphabet, and
transition relation, instead of being finite, are first-order definable in a
fixed countably-infinite structure. We show that the reachability analysis can
be addressed with the well-known saturation technique for the wide class of
oligomorphic structures. Moreover, for the more restrictive homogeneous
structures, we are able to give concrete complexity upper bounds. We show ample
applicability of our technique by presenting several concrete examples of
homogeneous structures, subsuming, with optimal complexity, known results from
the literature. We show that infinitely many such examples of homogeneous
structures can be obtained with the classical wreath product construction.Comment: to appear in CSL'1
Enumerating Cyclic Orientations of a Graph
Acyclic and cyclic orientations of an undirected graph have been widely
studied for their importance: an orientation is acyclic if it assigns a
direction to each edge so as to obtain a directed acyclic graph (DAG) with the
same vertex set; it is cyclic otherwise. As far as we know, only the
enumeration of acyclic orientations has been addressed in the literature. In
this paper, we pose the problem of efficiently enumerating all the
\emph{cyclic} orientations of an undirected connected graph with vertices
and edges, observing that it cannot be solved using algorithmic techniques
previously employed for enumerating acyclic orientations.We show that the
problem is of independent interest from both combinatorial and algorithmic
points of view, and that each cyclic orientation can be listed with
delay time. Space usage is with an additional setup cost
of time before the enumeration begins, or with a setup cost of
time
P-Selectivity, Immunity, and the Power of One Bit
We prove that P-sel, the class of all P-selective sets, is EXP-immune, but is
not EXP/1-immune. That is, we prove that some infinite P-selective set has no
infinite EXP-time subset, but we also prove that every infinite P-selective set
has some infinite subset in EXP/1. Informally put, the immunity of P-sel is so
fragile that it is pierced by a single bit of information.
The above claims follow from broader results that we obtain about the
immunity of the P-selective sets. In particular, we prove that for every
recursive function f, P-sel is DTIME(f)-immune. Yet we also prove that P-sel is
not \Pi_2^p/1-immune
On reachability in graphs with bounded independence number
Abstract. We study the reachability problem for finite directed graphs whose independence number is bounded by some constant k. This problem is a generalisation of the reachability problem for tournaments. We show that the problem is first-order definable for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable. Also in contrast, first-order definability does not carry over to the infinite version of the problem. We prove that the number of strongly connected components in a graph with bounded independence number can be computed using TC 0 -circuits, but cannot be computed using AC 0 -circuits. We also study the succinct version of the problem and show that it is Î P 2 -complete for all k
Computing Shortest Paths in Series-Parallel Graphs in Logarithmic Space
Series-parallel graphs, which are built by repeatedly applying
series or parallel composition operations to paths, play an
important role in computer science as they model the flow of
information in many types of programs. For directed series-parallel
graphs, we study the problem of finding a shortest path between two
given vertices. Our main result is that we can find such a path in
logarithmic space, which shows that the distance problem for
series-parallel graphs is L-complete. Previously, it was known
that one can compute some path in logarithmic space; but for
other graph types, like undirected graphs or tournament graphs,
constructing some path between given vertices is possible in
logarithmic space while constructing a shortest path is
NL-complete
Flow Games
In the traditional maximal-flow problem, the goal is to transfer maximum flow in a network by directing, in each vertex in the network, incoming flow into outgoing edges. While the problem has been extensively used in order to optimize the performance of networks in numerous application areas, it corresponds to a setting in which the authority has control on all vertices of the network.
Today\u27s computing environment involves parties that should be considered adversarial.
We introduce and study {em flow games}, which capture settings in which the authority can control only part of the vertices. In these games, the vertices are partitioned between two players: the authority and the environment. While the authority aims at maximizing the flow, the environment need not cooperate. We argue that flow games capture many modern settings, such as partially-controlled pipe or road systems or hybrid software-defined communication networks.
We show that the problem of finding the maximal flow as well as an optimal strategy for the authority in an acyclic flow game is -complete, and is already -hard to approximate. We study variants of the game: a restriction to strategies that ensure no loss of flow, an extension to strategies that allow non-integral flows, which we prove to be stronger, and a dynamic setting in which a strategy for a vertex is chosen only once flow reaches the vertex.
We discuss additional variants and their applications, and point to several interesting open problems
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