197 research outputs found
Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree
Given a simple, undirected graph G=(V,E) and a weight function w:E→ℤ+, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1,k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1+1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2−1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1,k} within a ratio of 3/2 for k=2 (note that this matches the inapproximability bound above), and (2−2/(k+1)) for any k≥3, respectively, in polynomial time
Inferring AS Relationships: Dead End or Lively Beginning?
Recent techniques for inferring business relationships between ASs have
yielded maps that have extremely few invalid BGP paths in the terminology of
Gao. However, some relationships inferred by these newer algorithms are
incorrect, leading to the deduction of unrealistic AS hierarchies. We
investigate this problem and discover what causes it. Having obtained such
insight, we generalize the problem of AS relationship inference as a
multiobjective optimization problem with node-degree-based corrections to the
original objective function of minimizing the number of invalid paths. We solve
the generalized version of the problem using the semidefinite programming
relaxation of the MAX2SAT problem. Keeping the number of invalid paths small,
we obtain a more veracious solution than that yielded by recent heuristics
Graph orientation with splits
The Minimum Maximum Outdegree Problem (MMO) is to assign a direction to every edge in an input undirected, edge-weighted graph so that the maximum weighted outdegree taken over all vertices becomes as small as possible. In this paper, we introduce a new variant of MMO called the p-Split Minimum Maximum Outdegree Problem (p-Split-MMO) in which one is allowed to perform a sequence of p split operations on the vertices before orienting the edges, for some specified non-negative integer p, and study its computational complexity
Size bounds and parallel algorithms for networks
SIGLEAvailable from British Library Document Supply Centre- DSC:D34009/81 / BLDSC - British Library Document Supply CentreGBUnited Kingdo
Parameterized Orientable Deletion
A graph is d-orientable if its edges can be oriented so that the maximum in-degree of the resulting digraph is at most d. d-orientability is a well-studied concept with close connections to fundamental graph-theoretic notions and applications as a load balancing problem. In this paper we consider the d-Orientable Deletion problem: given a graph G=(V,E), delete the minimum number of vertices to make G d-orientable. We contribute a number of results that improve the state of the art on this problem. Specifically:
- We show that the problem is W[2]-hard and log n-inapproximable with respect to k, the number of deleted vertices. This closes the gap in the problem\u27s approximability.
- We completely characterize the parameterized complexity of the problem on chordal graphs: it is FPT parameterized by d+k, but W-hard for each of the parameters d,k separately.
- We show that, under the SETH, for all d,epsilon, the problem does not admit a (d+2-epsilon)^{tw}, algorithm where tw is the graph\u27s treewidth, resolving as a special case an open problem on the complexity of PseudoForest Deletion.
- We show that the problem is W-hard parameterized by the input graph\u27s clique-width. Complementing this, we provide an algorithm running in time d^{O(d * cw)}, showing that the problem is FPT by d+cw, and improving the previously best know algorithm for this case
Parameterized Approximation Schemes using Graph Widths
Combining the techniques of approximation algorithms and parameterized
complexity has long been considered a promising research area, but relatively
few results are currently known. In this paper we study the parameterized
approximability of a number of problems which are known to be hard to solve
exactly when parameterized by treewidth or clique-width. Our main contribution
is to present a natural randomized rounding technique that extends well-known
ideas and can be used for both of these widths. Applying this very generic
technique we obtain approximation schemes for a number of problems, evading
both polynomial-time inapproximability and parameterized intractability bounds
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