1,261 research outputs found
A nonmonotone GRASP
A greedy randomized adaptive search procedure (GRASP) is an itera-
tive multistart metaheuristic for difficult combinatorial optimization problems. Each
GRASP iteration consists of two phases: a construction phase, in which a feasible
solution is produced, and a local search phase, in which a local optimum in the
neighborhood of the constructed solution is sought. Repeated applications of the con-
struction procedure yields different starting solutions for the local search and the
best overall solution is kept as the result. The GRASP local search applies iterative
improvement until a locally optimal solution is found. During this phase, starting from
the current solution an improving neighbor solution is accepted and considered as the
new current solution. In this paper, we propose a variant of the GRASP framework that
uses a new “nonmonotone” strategy to explore the neighborhood of the current solu-
tion. We formally state the convergence of the nonmonotone local search to a locally
optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP
on three classical hard combinatorial optimization problems: the maximum cut prob-
lem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and
the quadratic assignment problem (QAP)
A Solution Merging Heuristic for the Steiner Problem in Graphs Using Tree Decompositions
Fixed parameter tractable algorithms for bounded treewidth are known to exist
for a wide class of graph optimization problems. While most research in this
area has been focused on exact algorithms, it is hard to find decompositions of
treewidth sufficiently small to make these al- gorithms fast enough for
practical use. Consequently, tree decomposition based algorithms have limited
applicability to large scale optimization. However, by first reducing the input
graph so that a small width tree decomposition can be found, we can harness the
power of tree decomposi- tion based techniques in a heuristic algorithm, usable
on graphs of much larger treewidth than would be tractable to solve exactly. We
propose a solution merging heuristic to the Steiner Tree Problem that applies
this idea. Standard local search heuristics provide a natural way to generate
subgraphs with lower treewidth than the original instance, and subse- quently
we extract an improved solution by solving the instance induced by this
subgraph. As such the fixed parameter tractable algorithm be- comes an
efficient tool for our solution merging heuristic. For a large class of sparse
benchmark instances the algorithm is able to find small width tree
decompositions on the union of generated solutions. Subsequently it can often
improve on the generated solutions fast
Seeding with Costly Network Information
We study the task of selecting nodes in a social network of size , to
seed a diffusion with maximum expected spread size, under the independent
cascade model with cascade probability . Most of the previous work on this
problem (known as influence maximization) focuses on efficient algorithms to
approximate the optimal seed set with provable guarantees, given the knowledge
of the entire network. However, in practice, obtaining full knowledge of the
network is very costly. To address this gap, we first study the achievable
guarantees using influence samples. We provide an approximation
algorithm with a tight (1-1/e){\mbox{OPT}}-\epsilon n guarantee, using
influence samples and show that this dependence on
is asymptotically optimal. We then propose a probing algorithm that queries
edges from the graph and use them to find a seed set with the
same almost tight approximation guarantee. We also provide a matching (up to
logarithmic factors) lower-bound on the required number of edges. To address
the dependence of our probing algorithm on the independent cascade probability
, we show that it is impossible to maintain the same approximation
guarantees by controlling the discrepancy between the probing and seeding
cascade probabilities. Instead, we propose to down-sample the probed edges to
match the seeding cascade probability, provided that it does not exceed that of
probing. Finally, we test our algorithms on real world data to quantify the
trade-off between the cost of obtaining more refined network information and
the benefit of the added information for guiding improved seeding strategies
Markov-Chain-Based Heuristics for the Minimum Feedback Vertex Set Problem
Let G be a directed graph. A vertex set F is called feedback vertex set (FVS) if its removal from G results in an acyclic graph. Because determining a minimum cardinality FVS is known to be NP hard, [Karp72], one is interested in designing fast approximation algorithms determining near-optimum FVSs. The paper presents deterministic and randomised heuristics based on Markov chains. In this regard, an earlier approximation algorithm developed in [Speckenmeyer89] is revisited and refined. Experimental results demonstrate the overall performance superiority of our algorithms compared to other algorithms known from literature with respect to both criteria, the sizes of solutions determined, as well as the consumed runtimes
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