13925350
Embedding k-outerplanar graphs into ℓ1
We show that the shortest-path metric of any k-outerplanar graph, for any fixed k, can be approximated by a probability distribution over tree metrics with constant distortion, and hence also embedded into l1 with constant distortion. These graphs play a central role in polynomial time approximation schemes for many NP-hard optimization problems on general planar graphs, and include the family of weighted k × n planar grids.This result implies a constant upper bound on the ratio between the sparsest cut and the maximum concurrent flow in multicommodity networks for k-outerplanar graphs, thus extending a classical theorem of Okamura and Seymour [26] for outerplanar graphs, and of Gupta et al. [17] for treewidth-2 graphs. In addition, we obtain improved approximation ratios for k-outerplanar graphs on various problems for which approximation algorithms are based on probabilistic tree embeddings. We also conjecture that our random tree embeddings for k-outerplanar graphs can serve as a building block for more powerful l1 embeddings in future.
100323
Tree based MPLS routing
MPLS (MultiProtocol Label Switching) is a new technology proposed by the IETF [4,10] for network routing, and is being increasingly deployed by the largest Internet service providers. The MPLS technology differs from conventional network protocols in a crucial way: instead of reading the entire packet header at all switching points, the analysis of the packet header is done just once, when the packet header is assigned a stack of labels, and thenceforth, each switching point or router just gets to look at the label at the top of the stack (and the ingress edge), and based only on this information, it has to make a decision about the next-hop node [17,16]. In another departure from conventional routing and in particular from IP source routing, where the entire packet route is explicitly put in the header and popped off along the route, the router can not only pop the top label, it can push other labels on top of the stack.The two parameters of interest in designing MPLS routing protocols are the number of labels used, and the depth of the stack used for routing. Clearly, both cannot be simultaneously minimized, and there is often an interesting trade-off between label size and stack depth: it is obvious that if k labels are used, one must have a stack depth of logk n.In fact, it was not known whether this bound could be achieved even for trees; the best stack depth previously achieved with a constant number of labels was O(log2 n). In this paper, we show that one can indeed get asymptotically optimal upper bounds and route on trees using k labels and a maximum stack depth of O(logk n), and that this trade-off can be achieved using a simpler and more intuitive protocol than the one given in [9]. These tree-routing ideas are then shown to give better routing protocols for planar graphs as well. In particular, we show how to route along near-shortest paths (of length at most (1 + ε) times the shortest-path) with O(log2 n/ε) labels and logarithmic stack depth. We also apply them to graphs with smaller separators, including most large ISP networks.
47205636
Approximation Algorithms for the Unsplittable Flow Problem
We present approximation algorithms for the unsplittable flow problem (UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the non-uniform capacity case in which the edge capacities can vary arbitrarily over the graph. Our results are: - For undirected graphs we obtain a O(??-1 log2 n) approximation ratio, where n is the number of vertices, ? the maximum degree, and ? the expansion of the graph. Our ratio is capacity independent and improves upon the earlier O(??-1(cmax/cmin) log n) bound [15] for large values of cmax/cmin. Furthermore, if we specialize to the case where all edges have the same capacity, our algorithm gives an O(??-1 log n) approximation, which matches the performance of the best-known algorithm [15] for this special case. - For certain strong constant-degree expanders considered by Frieze [10] we obtain an O(? log n) approximation for the uniform capacity case, improving upon the current O(log n) approximation. - For UFP on the line and the ring, we give the first constant-factor approximation algorithms. Previous results addressed only the uniform capacity case. - All of the above results improve if the maximum demand is bounded away from the minimum capacity.Our results are based on randomized rounding followed by greedy alteration and are inspired by the use of this idea in recent work [21, 9].
37922436
Improved Bandwidth Approximation for Trees and Chordal Graphs
A linear arrangement of an n-vertex graph G=(V,E) is a one?one mapping f of the vertex set V onto the set n]={0,1,?,n?1}. The bandwidth of this linear arrangement is the maximum difference between the images of the endpoints of any edge in E(G). When the input graph G is a tree, the best known approximation algorithms for the minimum bandwidth linear arrangement (which are based on the principle of volume respecting embeddings) output a linear arrangement which has bandwidth within O(log3n) of the optimal bandwidth. In this paper, we present a simple randomized O(log2.5n)-approximation algorithm for bandwidth minimization on trees. We also show that a close variant of this algorithm gives a similar performance guarantee for chordal graphs.
411060
Approximating unique games
The UNIQUE GAMES problem is the following: we are given a graph <i>G</i> = (<i>V, E</i>), with each edge <i>e</i> = (<i>u, v</i>) having a weight <i>w</i><inf><i>e</i></inf> and a permutation π<inf><i>uv</i></inf> on [<i>k</i>]. The objective is to find a labeling of each vertex <i>u</i> with a label <i>f</i><inf><i>u</i></inf> ∈ [<i>k</i>] to minimize the weight of unsatisfied edges---where an edge (<i>u, v</i>) is <i>satisfied</i> if <i>f</i><inf><i>v</i></inf> = π<inf><i>uv</i></inf>(<i>f</i><inf><i>u</i></inf>).The <i>Unique Games Conjecture</i> of Khot [8] essentially says that for each ε > 0, there is a <i>k</i> such that it is NP-hard to distinguish instances of Unique games with (1-ε) satisfiable edges from those with only ε satisfiable edges. Several hardness results have recently been proved based on this assumption, including optimal ones for Max-Cut, Vertex-Cover and other problems, making it an important challenge to prove or refute the conjecture.In this paper, we give an <i>O</i>(log <i>n</i>)-approximation algorithm for the problem of minimizing the number of unsatisfied edges in any Unique game. Previous results of Khot [8] and Trevisan [12] imply that if the optimal solution has OPT = ε<i>m</i> unsatisfied edges, semidefinite relaxations of the problem could give labelings with min {<i>k</i><sup>2</sup>ε<sup>1/5</sup>, (ε log <i>n</i>)<sup>1/2</sup>}<i>m</i> unsatisfied edges. In this paper we show how to round a LP relaxation to get an <i>O</i>(log <i>n</i>)-approximation to the problem; i.e., to find a labeling with only <i>O</i>(ε<i>m</i> log <i>n</i>) = <i>O</i>(OPT log <i>n</i>) unsatisfied edges.
11033759
Oblivious network design
Consider the following network design problem: given a network <i>G = (V, E)</i>, source-sink pairs {<i>s</i><inf><i>i</i></inf>, <i>t</i><inf><i>i</i></inf>} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge <i>e</i> carries a total of <i>f</i><inf><i>e</i></inf> flow (from all the terminal pairs), the cost is given by Σ <inf><i>e</i></inf><i>l</i>(<i>f</i><inf><i>e</i></inf>), where <i>l</i> is some concave cost function; the goal is to minimize the total cost incurred. However, we want the routing to be <i>oblivious</i>: when terminal pair {<i>s</i><inf><i>i</i></inf>, <i>t</i><inf><i>i</i></inf>} makes its routing decisions, it does not know the current flow on the edges of the network, nor the identity of the other pairs in the system. Moreover, it does not even know the identity of the function <i>l</i>, merely knowing that <i>l</i> is a concave function of the total flow on the edge. How should it (obliviously) route its one unit of flow? Can we get competitive algorithms for this problem?In this paper, we develop a framework to model <i>oblivious network design</i> problems (of which the above problem is a special case), and give algorithms with poly-logarithmic competitive ratio for problems in this framework (and hence for this problem). Abstractly, given a problem like the one above, the solution is a multicommodity flow producing a "load" on each edge of <i>L</i><inf><i>e</i></inf> = <i>l</i>(<i>f</i><inf>1</inf>(<i>e</i>),<i>f</i><inf>2</inf>(<i>e</i>), ..., <i>f</i><inf><i>k</i></inf>(<i>e</i>)), and the total cost is given by an "aggregation function" agg (<i>L</i><inf><i>e</i>1</inf>,...,<i>L</i><inf><i>em</i></inf>) of the loads of all edges. Our goal is to develop oblivious algorithms that approximately minimize the total cost of the routing, knowing the aggregation function agg, but merely knowing that <i>l</i> lies in some class C, and having no other information about the current state of the network. Hence we want algorithms that are simultaneously "function-oblivious" as well as "traffic-oblivious".The aggregation functions we consider are the max and σ objective functions, which correspond to the well-known measures of <i>congestion</i> and <i>total cost</i> of a network; in this paper, we prove the following:• If the aggregation function is Σ, we give an oblivious algorithm with <i>O</i>(log<sup>2</sup> <i>n</i>) competitive ratio whenever the load function <i>l</i> is in the class of <i>monotone sub-additive functions</i>. (Recall that our algorithm is also "function-oblivious"; it works whenever each edge has a load function <i>l</i> in the class.)• For the case when the aggregation function is max, we give an oblivious algorithm with <i>O</i>(log<sup>2</sup> <i>n</i> log log <i>n</i>) competitive ratio, when the load function <i>l</i> is a <i>norm;</i> we also show that such a competitive ratio is not possible for general sub-additive functions.These are the first such general results about oblivious algorithms for network design problems, and we hope the ideas and techniques will lead to more and improved results in this area.
20946285
Auction Based Resource Allocation in Grids
Auctions have been used as the market mechanism for allocating resources to users in a grid. However, most of the existing work either consider only homogeneous resources, or pick a resource randomly for bidding to increase the number of jobs finishing within their deadlines. Random selection does not consider resource capabilities and hence, do not optimize other important metrics like the average turnaround time and the average budget spent per job etc. In this paper, we consider the resource allocation problem in a grid with heterogeneous resources. We present allocation policies using sealed-bid auction that reduce the average turnaround time and the average budget spent per job, while still maintaining a high number of jobs finishing within their deadlines. Simulation results are presented to evaluate the performance of the policies.
2981830
Characterizing the Complexity of Subgraph Isomorphism for Graphs of Bounded Path-Width
We show that the complexity of the subgraph isomorphism problem on graphs of bounded path-width is inherently dependent on the connectivity of the source and target graphs. In particular, for the problem of determining whether a source graph G of path-width k is a subgraph of a target graph H of path-width k, we present an O(n3) algorithm for G and H both k-connected, for n the sum of the sizes of the graphs, and NP-completeness results for connectivity less than k. In previous polynomial-time algorithms, the degree of the polynomial in the running time was a function of k. In contrast, we show that when neither G nor H is k-connected, the problem becomes NP-complete. The same result also holds if one of the graphs has at least k vertices of unbounded degree. Since bounded path-width graphs are also bounded tree-width graphs, our hardness results immediately extend to this larger class. A further NP-completeness result applies to the situation in which both graphs have tree-width k, but only the target graph is k-connected. This provides a complete characterization of the subgraph isomorphism problem on bounded tree-width graphs, thus answering an open question of Matousek and Thomas.
209398811
Life Science Research and Data Management.
A distance measuring arrangement for use, for example, in a camera and the like equipped with an automatic focus control system which includes an illuminance intensity detecting portion for detecting the illuminance intensity of a target object and a distance signal correcting portion for correcting the distance signal from a distance measuring portion according to the output of the illuminance intensity detecting portion so as to obtain the distance signal not affected by the illuminance intensity of the target object.
38885821
Embeddings of k-Connected Graphs of Pathwidth k
We present O(n3) embedding algorithms (generalizing subgraph isomorphism) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the first polynomialtime algorithm for minor containment and the first O(nc) algorithm (c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more efficient solutions.
715438
Non-linear Dimensionality Reduction by Locally Linear Isomaps
Algorithms for nonlinear dimensionality reduction (NLDR) find meaningful hidden low-dimensional structures in a high-dimensional space. Current algorithms for NLDR are Isomaps, Local Linear Embedding and Laplacian Eigenmaps. Isomaps are able to reliably recover low-dimensional nonlinear structures in high-dimensional data sets, but suffer from the problem of short-circuiting, which occurs when the neighborhood distance is larger than the distance between the folds in the manifolds. We propose a new variant of Isomap algorithm based on local linear properties of manifolds to increase its robustness to short-circuiting. We demonstrate that the proposed algorithm works better than Isomap algorithm for normal, noisy and sparse data sets.