7,861 research outputs found
Pliable Index Coding via Conflict-Free Colorings of Hypergraphs
In the pliable index coding (PICOD) problem, a server is to serve multiple
clients, each of which possesses a unique subset of the complete message set as
side information and requests a new message which it does not have. The goal of
the server is to do this using as few transmissions as possible. This work
presents a hypergraph coloring approach to the PICOD problem. A
\textit{conflict-free coloring} of a hypergraph is known from literature as an
assignment of colors to its vertices so that each edge of the graph contains
one uniquely colored vertex. For a given PICOD problem represented by a
hypergraph consisting of messages as vertices and request-sets as edges, we
present achievable PICOD schemes using conflict-free colorings of the PICOD
hypergraph. Various graph theoretic parameters arising out of such colorings
(and some new variants) then give a number of upper bounds on the optimal PICOD
length, which we study in this work. Our achievable schemes based on hypergraph
coloring include scalar as well as vector linear PICOD schemes. For the scalar
case, using the correspondence with conflict-free coloring, we show the
existence of an achievable scheme which has length where
refers to a parameter of the hypergraph that captures the maximum
`incidence' number of other edges on any edge. This result improves upon known
achievability results in PICOD literature, in some parameter regimes.Comment: 21 page
An Exact Algorithm for the Generalized List -Coloring Problem
The generalized list -coloring is a common generalization of many graph
coloring models, including classical coloring, -labeling, channel
assignment and -coloring. Every vertex from the input graph has a list of
permitted labels. Moreover, every edge has a set of forbidden differences. We
ask for such a labeling of vertices of the input graph with natural numbers, in
which every vertex gets a label from its list of permitted labels and the
difference of labels of the endpoints of each edge does not belong to the set
of forbidden differences of this edge. In this paper we present an exact
algorithm solving this problem, running in time ,
where is the maximum forbidden difference over all edges of the input
graph and is the number of its vertices. Moreover, we show how to improve
this bound if the input graph has some special structure, e.g. a bounded
maximum degree, no big induced stars or a perfect matching
Acyclic orientations with path constraints
Many well-known combinatorial optimization problems can be stated over the
set of acyclic orientations of an undirected graph. For example, acyclic
orientations with certain diameter constraints are closely related to the
optimal solutions of the vertex coloring and frequency assignment problems. In
this paper we introduce a linear programming formulation of acyclic
orientations with path constraints, and discuss its use in the solution of the
vertex coloring problem and some versions of the frequency assignment problem.
A study of the polytope associated with the formulation is presented, including
proofs of which constraints of the formulation are facet-defining and the
introduction of new classes of valid inequalities
Approximate Graph Coloring by Semidefinite Programming
We consider the problem of coloring k-colorable graphs with the fewest
possible colors. We present a randomized polynomial time algorithm that colors
a 3-colorable graph on vertices with min O(Delta^{1/3} log^{1/2} Delta log
n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any
vertex. Besides giving the best known approximation ratio in terms of n, this
marks the first non-trivial approximation result as a function of the maximum
degree Delta. This result can be generalized to k-colorable graphs to obtain a
coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)}
log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and
Williamson who used an algorithm for semidefinite optimization problems, which
generalize linear programs, to obtain improved approximations for the MAX CUT
and MAX 2-SAT problems. An intriguing outcome of our work is a duality
relationship established between the value of the optimum solution to our
semidefinite program and the Lovasz theta-function. We show lower bounds on the
gap between the optimum solution of our semidefinite program and the actual
chromatic number; by duality this also demonstrates interesting new facts about
the theta-function
Rainbow Coloring Hardness via Low Sensitivity Polymorphisms
A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs.
Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover
Learning-Based Constraint Satisfaction With Sensing Restrictions
In this paper we consider graph-coloring problems, an important subset of
general constraint satisfaction problems that arise in wireless resource
allocation. We constructively establish the existence of fully decentralized
learning-based algorithms that are able to find a proper coloring even in the
presence of strong sensing restrictions, in particular sensing asymmetry of the
type encountered when hidden terminals are present. Our main analytic
contribution is to establish sufficient conditions on the sensing behaviour to
ensure that the solvers find satisfying assignments with probability one. These
conditions take the form of connectivity requirements on the induced sensing
graph. These requirements are mild, and we demonstrate that they are commonly
satisfied in wireless allocation tasks. We argue that our results are of
considerable practical importance in view of the prevalence of both
communication and sensing restrictions in wireless resource allocation
problems. The class of algorithms analysed here requires no message-passing
whatsoever between wireless devices, and we show that they continue to perform
well even when devices are only able to carry out constrained sensing of the
surrounding radio environment
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