370 research outputs found
The Complexity of Manipulating -Approval Elections
An important problem in computational social choice theory is the complexity
of undesirable behavior among agents, such as control, manipulation, and
bribery in election systems. These kinds of voting strategies are often
tempting at the individual level but disastrous for the agents as a whole.
Creating election systems where the determination of such strategies is
difficult is thus an important goal.
An interesting set of elections is that of scoring protocols. Previous work
in this area has demonstrated the complexity of misuse in cases involving a
fixed number of candidates, and of specific election systems on unbounded
number of candidates such as Borda. In contrast, we take the first step in
generalizing the results of computational complexity of election misuse to
cases of infinitely many scoring protocols on an unbounded number of
candidates. Interesting families of systems include -approval and -veto
elections, in which voters distinguish candidates from the candidate set.
Our main result is to partition the problems of these families based on their
complexity. We do so by showing they are polynomial-time computable, NP-hard,
or polynomial-time equivalent to another problem of interest. We also
demonstrate a surprising connection between manipulation in election systems
and some graph theory problems
Computational Complexity of Covering Multigraphs with Semi-Edges: Small Cases
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for graphs with semi-edges. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello et al. asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding k-tuples of pairwise disjoint perfect matchings in regular graphs, or finding equitable partitions of regular bipartite graphs)
Cluster formation in mesoscopic systems
Graph-theoretical approach is used to study cluster formation in mesocsopic
systems. Appearance of these clusters are due to discrete resonances which are
presented in the form of a multigraph with labeled edges. This presentation
allows to construct all non-isomorphic clusters in a finite spectral domain and
generate corresponding dynamical systems automatically. Results of MATHEMATICA
implementation are given and two possible mechanisms of cluster destroying are
discussed
List covering of regular multigraphs
A graph covering projection, also known as a locally bijective homomorphism,
is a mapping between vertices and edges of two graphs which preserves
incidencies and is a local bijection. This notion stems from topological graph
theory, but has also found applications in combinatorics and theoretical
computer science.
It has been known that for every fixed simple regular graph of valency
greater than 2, deciding if an input graph covers is NP-complete. In recent
years, topological graph theory has developed into heavily relying on multiple
edges, loops, and semi-edges, but only partial results on the complexity of
covering multigraphs with semi-edges are known so far. In this paper we
consider the list version of the problem, called \textsc{List--Cover}, where
the vertices and edges of the input graph come with lists of admissible
targets. Our main result reads that the \textsc{List--Cover} problem is
NP-complete for every regular multigraph of valency greater than 2 which
contains at least one semi-simple vertex (i.e., a vertex which is incident with
no loops, with no multiple edges and with at most one semi-edge). Using this
result we almost show the NP-co/polytime dichotomy for the computational
complexity of \textsc{ List--Cover} of cubic multigraphs, leaving just five
open cases.Comment: Accepted to IWOCA 202
Strong cliques and equistability of EPT graphs
In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-intersection graphs of paths in a tree. This result generalizes a previously known characterization of equistable line graphs. Our approach is based on the combinatorial features of triangle graphs and general partition graphs. We also show that, in EPT graphs, testing whether a given clique is strong is co-NP-complete. We obtain this hardness result by first showing hardness of the problem of determining whether a given graph has a maximal matching disjoint from a given edge cut. As a positive result, we prove that the problem of testing whether a given clique is strong is polynomial in the class of local EPT graphs, which are defined as the edge intersection graphs of paths in a star and are known to coincide with the line graphs of multigraphs.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones CientĂficas y TĂ©cnica
Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat).
The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem
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