236 research outputs found
Sequential coloring versus Welsh-Powell bound
AbstractWe comment in this note on the relations between sequential coloring and the Welsh-Powell upper bound for the chromatic number of a graph
Breaking Instance-Independent Symmetries In Exact Graph Coloring
Code optimization and high level synthesis can be posed as constraint
satisfaction and optimization problems, such as graph coloring used in register
allocation. Graph coloring is also used to model more traditional CSPs relevant
to AI, such as planning, time-tabling and scheduling. Provably optimal
solutions may be desirable for commercial and defense applications.
Additionally, for applications such as register allocation and code
optimization, naturally-occurring instances of graph coloring are often small
and can be solved optimally. A recent wave of improvements in algorithms for
Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests
generic problem-reduction methods, rather than problem-specific heuristics,
because (1) heuristics may be upset by new constraints, (2) heuristics tend to
ignore structure, and (3) many relevant problems are provably inapproximable.
Problem reductions often lead to highly symmetric SAT instances, and
symmetries are known to slow down SAT solvers. In this work, we compare several
avenues for symmetry breaking, in particular when certain kinds of symmetry are
present in all generated instances. Our focus on reducing CSPs to SAT allows us
to leverage recent dramatic improvement in SAT solvers and automatically
benefit from future progress. We can use a variety of black-box SAT solvers
without modifying their source code because our symmetry-breaking techniques
are static, i.e., we detect symmetries and add symmetry breaking predicates
(SBPs) during pre-processing.
An important result of our work is that among the types of
instance-independent SBPs we studied and their combinations, the simplest and
least complete constructions are the most effective. Our experiments also
clearly indicate that instance-independent symmetries should mostly be
processed together with instance-specific symmetries rather than at the
specification level, contrary to what has been suggested in the literature
Heuristic Coloring Algorithm for the Composite Graph Coloring Problem
A composite graph is a finite undirected graph in which a positive integer known as a chromaticity is associated with each vertex of the graph. The composite graph coloring problem (CGCP) is the problem of finding the chromatic number of a composite graph, i.e., the minimum number of colors (positive integers) required to assign a sequence of consecutive colors to each vertex of the graph in a manner such that adjacent vertices are not assigned sequences with colors in common and the sequence assigned to a vertex has the number of colors indicated by the chromaticity of the vertex. The CGCP problem is an NP-complete problem that has applications to scheduling and resource allocation problems in which the tasks to be scheduled are of unequal durations.
The pigeonhole principle gives rise to a problem reduction technique for the CGCP and a vertex ordering used in the vertex-sequentia1-with-interchange (VSI) algorithm. LFPHI. An upper bound on the chromatic number of a composite graph is obtained from the definition of a color-sequential coloring algorithm for the CGCP.
The performances of twelve heuristic coloring algorithms are compared on a variety of random composite graphs. Three VSI algorithms (LF1I, LFPHI, and LFCDI) performed superior to the other algorithms on graphs having the lower numbers of vertices and low edge densities while two color-sequential algorithms (RLF1 and RLFD1) were superior on graphs having the higher numbers of vertices and high edge densities
Exact Algorithms for Maximum Clique: a computational study
We investigate a number of recently reported exact algorithms for the maximum
clique problem (MCQ, MCR, MCS, BBMC). The program code used is presented and
critiqued showing how small changes in implementation can have a drastic effect
on performance. The computational study demonstrates how problem features and
hardware platforms influence algorithm behaviour. The minimum width order
(smallest-last) is investigated, and MCS is broken into its consituent parts
and we discover that one of these parts degrades performance. It is shown that
the standard procedure used for rescaling published results is unsafe.Comment: 40 pages, 14 figures, 10 tables, 12 short java program listings, code
afailable to download at
http://www.dcs.gla.ac.uk/~pat/maxClique/distribution
Chromatic scheduling of dynamic data-graph computations
Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 67-73).Data-graph computations are a parallel-programming model popularized by programming systems such as Pregel, GraphLab, PowerGraph, and GraphChi. A fundamental issue in parallelizing data-graph computations is the avoidance of races between computation occurring on overlapping regions of the graph. Common solutions such as locking protocols and bulk-synchronous execution often sacrifice performance, update atomicity, or determinism. A known alternative is chromatic scheduling which uses a vertex coloring of the conflict graph to divide data-graph updates into sets which may be parallelized without races. To date, however, only static data-graph computations, which do not schedule updates at runtime, have employed chromatic scheduling. I introduce PRISM, a work-efficient scheduling algorithm for dynamic data-graph computations that uses chromatic scheduling. For a collection of four application benchmarks on a modern multicore machine, chromatic scheduling approximately doubles the performance of the lock-based GraphLab implementation, and triples the performance of GraphChi's update execution phase when enforcing determinism. Chromatic scheduling motivates the development of efficient deterministic parallel coloring algorithms. New analysis of the Jones-Plassmann message-passing algorithm shows that only O([Delta] + In A in V/ In ln V) rounds are needed to color a graph G = (V, E) with max vertex degree [Delta], generalizing previous results for bounded degree graphs. A new log-degree ordering heuristic is described which can reduce the number of colors used in practice, while only increasing the number of rounds by a logrithmic factor. An efficient implementation for the shared-memory setting is described and analyzed using the CRQW contention model, showing that this algorithm performs [Theta](V + E) work and has expected span O([Delta] In [Delta]A + 1n 2[Delta] In V/In In V). Benchmarks on a set of real world graphs show that, in practice, these parallel algorithms achieve modest speedup over optimized serial code (around 4x on a 12-core machine).by Tim Kaler.M. Eng
Parallelization techniques for quantum simulation of fermionic systems
Mapping fermionic operators to qubit operators is an essential step for
simulating fermionic systems on a quantum computer. We investigate how the
choice of such a mapping interacts with the underlying qubit connectivity of
the quantum processor to enable (or impede) parallelization of the resulting
Hamiltonian-simulation algorithm. It is shown that this problem can be mapped
to a path coloring problem on a graph constructed from the particular choice of
encoding fermions onto qubits and the fermionic interactions onto paths. The
basic version of this problem is called the weak coloring problem. Taking into
account the fine-grained details of the mapping yields what is called the
strong coloring problem, which leads to improved parallelization performance. A
variety of illustrative analytical and numerical examples are presented to
demonstrate the amount of improvement for both weak and strong coloring-based
parallelizations. Our results are particularly important for implementation on
near-term quantum processors where minimizing circuit depth is necessary for
algorithmic feasibility.Comment: 27 pages, 12 figures; (v2) corrected a misplaced figure; (v3) updated
for publication with minor change
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