102,777 research outputs found
A Systematic Approach to Constructing Incremental Topology Control Algorithms Using Graph Transformation
Communication networks form the backbone of our society. Topology control
algorithms optimize the topology of such communication networks. Due to the
importance of communication networks, a topology control algorithm should
guarantee certain required consistency properties (e.g., connectivity of the
topology), while achieving desired optimization properties (e.g., a bounded
number of neighbors). Real-world topologies are dynamic (e.g., because nodes
join, leave, or move within the network), which requires topology control
algorithms to operate in an incremental way, i.e., based on the recently
introduced modifications of a topology. Visual programming and specification
languages are a proven means for specifying the structure as well as
consistency and optimization properties of topologies. In this paper, we
present a novel methodology, based on a visual graph transformation and graph
constraint language, for developing incremental topology control algorithms
that are guaranteed to fulfill a set of specified consistency and optimization
constraints. More specifically, we model the possible modifications of a
topology control algorithm and the environment using graph transformation
rules, and we describe consistency and optimization properties using graph
constraints. On this basis, we apply and extend a well-known constructive
approach to derive refined graph transformation rules that preserve these graph
constraints. We apply our methodology to re-engineer an established topology
control algorithm, kTC, and evaluate it in a network simulation study to show
the practical applicability of our approachComment: This document corresponds to the accepted manuscript of the
referenced journal articl
Analysis of the Min-Sum Algorithm for Packing and Covering Problems via Linear Programming
Message-passing algorithms based on belief-propagation (BP) are successfully
used in many applications including decoding error correcting codes and solving
constraint satisfaction and inference problems. BP-based algorithms operate
over graph representations, called factor graphs, that are used to model the
input. Although in many cases BP-based algorithms exhibit impressive empirical
results, not much has been proved when the factor graphs have cycles.
This work deals with packing and covering integer programs in which the
constraint matrix is zero-one, the constraint vector is integral, and the
variables are subject to box constraints. We study the performance of the
min-sum algorithm when applied to the corresponding factor graph models of
packing and covering LPs.
We compare the solutions computed by the min-sum algorithm for packing and
covering problems to the optimal solutions of the corresponding linear
programming (LP) relaxations. In particular, we prove that if the LP has an
optimal fractional solution, then for each fractional component, the min-sum
algorithm either computes multiple solutions or the solution oscillates below
and above the fraction. This implies that the min-sum algorithm computes the
optimal integral solution only if the LP has a unique optimal solution that is
integral.
The converse is not true in general. For a special case of packing and
covering problems, we prove that if the LP has a unique optimal solution that
is integral and on the boundary of the box constraints, then the min-sum
algorithm computes the optimal solution in pseudo-polynomial time.
Our results unify and extend recent results for the maximum weight matching
problem by [Sanghavi et al.,'2011] and [Bayati et al., 2011] and for the
maximum weight independent set problem [Sanghavi et al.'2009]
Hardness of Graph Pricing through Generalized Max-Dicut
The Graph Pricing problem is among the fundamental problems whose
approximability is not well-understood. While there is a simple combinatorial
1/4-approximation algorithm, the best hardness result remains at 1/2 assuming
the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate
within a factor better than 1/4 under the UGC, so that the simple combinatorial
algorithm might be the best possible. We also prove that for any , there exists such that the integrality gap of
-rounds of the Sherali-Adams hierarchy of linear programming for
Graph Pricing is at most 1/2 + .
This work is based on the effort to view the Graph Pricing problem as a
Constraint Satisfaction Problem (CSP) simpler than the standard and complicated
formulation. We propose the problem called Generalized Max-Dicut(), which
has a domain size for every . Generalized Max-Dicut(1) is
well-known Max-Dicut. There is an approximation-preserving reduction from
Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and
both our results are achieved through this reduction. Besides its connection to
Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own
right since in most arity two CSPs studied in the literature, SDP-based
algorithms perform better than LP-based or combinatorial algorithms --- for
this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page
Approximating Unique Games Using Low Diameter Graph Decomposition
We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints.
Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games
Evolving Graphs by Graph Programming
Graphs are a ubiquitous data structure in computer science and can be used to represent solutions to difficult problems in many distinct domains. This motivates the use of Evolutionary Algorithms to search over graphs and efficiently find approximate solutions. However, existing techniques often represent and manipulate graphs in an ad-hoc manner. In contrast, rule-based graph programming offers a formal mechanism for describing relations over graphs.
This thesis proposes the use of rule-based graph programming for representing and implementing genetic operators over graphs. We present the Evolutionary Algorithm Evolving Graphs by Graph Programming and a number of its extensions which are capable of learning stateful and stateless digital circuits, symbolic expressions and Artificial Neural Networks. We demonstrate that rule-based graph programming may be used to implement new and effective constraint-respecting mutation operators and show that these operators may strictly generalise others found in the literature. Through our proposal of Semantic Neutral Drift, we accelerate the search process by building plateaus into the fitness landscape using domain knowledge of equivalence. We also present Horizontal Gene Transfer, a mechanism whereby graphs may be passively recombined without disrupting their fitness.
Through rigorous evaluation and analysis of over 20,000 independent executions of Evolutionary Algorithms, we establish numerous benefits of our approach. We find that on many problems, Evolving Graphs by Graph Programming and its variants may significantly outperform other approaches from the literature. Additionally, our empirical results provide further evidence that neutral drift aids the efficiency of evolutionary search
Justifications in Constraint Handling Rules for Logical Retraction in Dynamic Algorithms
We present a straightforward source-to-source transformation that introduces
justifications for user-defined constraints into the CHR programming language.
Then a scheme of two rules suffices to allow for logical retraction (deletion,
removal) of constraints during computation. Without the need to recompute from
scratch, these rules remove not only the constraint but also undo all
consequences of the rule applications that involved the constraint. We prove a
confluence result concerning the rule scheme and show its correctness. When
algorithms are written in CHR, constraints represent both data and operations.
CHR is already incremental by nature, i.e. constraints can be added at runtime.
Logical retraction adds decrementality. Hence any algorithm written in CHR with
justifications will become fully dynamic. Operations can be undone and data can
be removed at any point in the computation without compromising the correctness
of the result. We present two classical examples of dynamic algorithms, written
in our prototype implementation of CHR with justifications that is available
online: maintaining the minimum of a changing set of numbers and shortest paths
in a graph whose edges change.Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
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