Classical network design models, such as the Survivable Network Design problem (SNDP), are (partly) motivated by robustness to faults under the assumption that any subset of edges upto a specific number can fail. We c...
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ISBN:
(纸本)9783959772785
Classical network design models, such as the Survivable Network Design problem (SNDP), are (partly) motivated by robustness to faults under the assumption that any subset of edges upto a specific number can fail. We consider non-uniform fault models where the subset of edges that fail can be specified in different ways. Our primary interest is in the flexible graph connectivity model [1, 3, 4, 8], in which the edge set is partitioned into safe and unsafe edges. Given parameters p, q ≥ 1, the goal is to find a cheap subgraph that remains p-connected even after the failure of q unsafe edges. We also discuss the bulk-robust model [6, 2] and the relative survivable network design model [19]. While SNDP admits a 2-approximation [32], the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions. Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when p and q are fixed, we consider two special cases. For the s-t case we obtain an approximation ratio that depends only on p, q whenever p + q > pq/2 which includes (p, 2) and (2, q) for all p, q ≥ 1. For the global connectivity case we obtain an O(q) approximation for (2, q), and an O(p) approximation for (p, 2) and (p, 3) for any p ≥ 1, and for (p, 4) when p is even. These are based on an augmentation framework and decomposing the families of cuts that need to be covered into a small number of uncrossable families. Our second result is a poly-logarithmic approximation for a generalization of the bulk-robust model when the "width" of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. We utilize a recent framework due to Chen et al. [17] that was designed for handling group co
approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hami...
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We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {Li}i∈F and a set D of cli...
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ISBN:
(纸本)9783959770132
We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {Li}i∈F and a set D of clients located in a common metric space {c(i, j)}i,j∈F∪D, and bounds k, m. A feasible solution is a pair (S ⊆ F, σ : D → S ∪ {out}), where σ specifies the client assignments, such that |S| ≤ k, |σ-1(i)| ≥ Li for all i ∈ S, and |σ-1(out)| ≤ m. In the lPower-bounded min-sum-of-radii with outliers (LBkSRO) problem, the objective is to minimize Σi∈S maxj∈σ-1(i) c(i, j), and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize maxi∈S maxj∈σ-1(i) c(i, j). We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| ≤ k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplier-preserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.
In clustering problems, one has to partition a given set of objects into pairwise disjoint subsets (clusters) taking into account only similarity of objects. In the graph cluster editing problem similarity relation on...
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Given a simple connected graph, we seek to partition the vertex set V into k non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximu...
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We present polynomial{time approximation algorithms for string folding problems over any finite alphabet. Our idea is the following: describe a class of feasible solutions by means of an ambiguous contextfree grammar ...
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We consider the problem of partitioning the nodes of a com-plete edge weighted graph into k clusters so as to minimize the sum of the diameters of the clusters. Since the problem is NP-complete, our focus is on the de...
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A tree cover is a collection of subtrees of a graph such that each vertex is a part of at least one subtree. The bounded tree cover problem (BTC) requires to find a tree cover with minimum number of subtrees of bounde...
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We introduce the following optimization version of the classical pattern matching problem (referred to as the maximum pattern matching problem). Given a two-dimensional rectangular text and a 2- dimensional rectangula...
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Amajor task of telecommunication network planners is deciding where spare capacity is needed, and howmuch, so that interrupted traffic may be rerouted in the event of a failure. Planning the spare capacity so as to mi...
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