A constant factor approximation algorithm for boxicity of circular arc graphs

Boxicity of a graph G(V, E) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional axis parallel boxes in Rk. Equivalently, it is the minimum number of interval graphs on the vertex set V such that the intersection of their edge sets is E. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below O(n0.5-ε)-factor, for any ε > 0 in polynomial time unless NP = ZPP. Till date, there is no well known graph class of unbounded boxicity for which even an nε-factor approximation algorithm for computing boxicity is known, for any ε < 1. In this paper, we study the boxicity problem on Circular Arc graphs - intersection graphs of arcs of a circle. We give a (2+ 1/k)-factor polynomial time approximation algorithm for computing the boxicity of any circular arc graph along with a corresponding box representation, where k ≥ 1 is its boxicity. For Normal Circular Arc(NCA) graphs, with an NCA model given, this can be improved to an additive 2-factor approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity is O(mn+n2) in both these cases and in O(mn+kn2) which is at most O(n3) time we also get their corresponding box representations, where n is the number of vertices of the graph and m is its number of edges. The additive 2-factor algorithm directly works for any Proper Circular Arc graph, since computing an NCA model for it can be done in polynomial time.

A shortest path tree based algorithm for relay placement in a wireless sensor network and its performance analysis

In this paper, we study a problem of designing a multi-hop wireless network for interconnecting sensors (hereafter called source nodes) to a Base Station (BS), by deploying a minimum number of relay nodes at a subset of given potential locations, while meeting a quality of service (QoS) objective specified as a hop count bound for paths from the sources to the BS. The hop count bound suffices to ensure a certain probability of the data being delivered to the BS within a given maximum delay under a light traffic model. We observe that the problem is NP-Hard. For this problem, we propose a polynomial time approximation algorithm based on iteratively constructing shortest path trees and heuristically pruning away the relay nodes used until the hop count bound is violated. Results show that the algorithm performs efficiently in various randomly generated network scenarios; in over 90% of the tested scenarios, it gave solutions that were either optimal or were worse than optimal by just one relay. We then use random graph techniques to obtain, under a certain stochastic setting, an upper bound on the average case approximation ratio of a class of algorithms (including the proposed algorithm) for this problem as a function of the number of source nodes, and the hop count bound. To the best of our knowledge, the average case analysis is the first of its kind in the relay placement literature. Since the design is based on a light traffic model, we also provide simulation results (using models for the IEEE 802.15.4 physical layer and medium access control) to assess the traffic levels up to which the QoS objectives continue to be met. (C) 2014 Elsevier B.V. All rights reserved.

A constant factor approximation algorithm for boxicity of circular arc graphs

The boxicity (resp. cubicity) of a graph G(V, E) is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes (resp. cubes) in R-k. Equivalently, it is the minimum number of interval graphs (resp. unit interval graphs) on the vertex set V, such that the intersection of their edge sets is E. The problem of computing boxicity (resp. cubicity) is known to be inapproximable, even for restricted graph classes like bipartite, co-bipartite and split graphs, within an O(n(1-epsilon))-factor for any epsilon > 0 in polynomial time, unless NP = ZPP. For any well known graph class of unbounded boxicity, there is no known approximation algorithm that gives n(1-epsilon)-factor approximation algorithm for computing boxicity in polynomial time, for any epsilon > 0. In this paper, we consider the problem of approximating the boxicity (cubicity) of circular arc graphs intersection graphs of arcs of a circle. Circular arc graphs are known to have unbounded boxicity, which could be as large as Omega(n). We give a (2 + 1/k) -factor (resp. (2 + log n]/k)-factor) polynomial time approximation algorithm for computing the boxicity (resp. cubicity) of any circular arc graph, where k >= 1 is the value of the optimum solution. For normal circular arc (NCA) graphs, with an NCA model given, this can be improved to an additive two approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity (resp. cubicity) is O(mn + n(2)) in both these cases, and in O(mn + kn(2)) = O(n(3)) time we also get their corresponding box (resp. cube) representations, where n is the number of vertices of the graph and m is its number of edges. Our additive two approximation algorithm directly works for any proper circular arc graph, since their NCA models can be computed in polynomial time. (C) 2014 Elsevier B.V. All rights reserved.

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R-1 x R-2 x...x R-k where R-i (for 1 <= i <= k) is a closed interval of the form [a(i), b(i)] on the real line. For a graph G, its boxicity box(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a left perpendicular1 + 1/c log n right perpendicular(d-1) approximation ratio for any constant c >= 1 when d >= 2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard. We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in left perpendicular(Delta + 2) ln nright perpendicular dimensions, where Delta is the maximum degree of G. This algorithm implies that box(G) <= left perpendicular(Delta + 2) ln nright perpendicular for any graph G. Our bound is tight up to a factor of ln n. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree Delta, we show that for almost all graphs on n vertices, their boxicity is O(d(av) ln n) where d(av) is the average degree.

On the Cubicity of AT-Free Graphs and Circular-Arc Graphs

A unit cube in k dimensions (k-cube) is defined as the Cartesian product R-1 x R-2 x ... x R-k where R-i (for 1 <= i <= k) is a closed interval of the form [a(i), a(i) + 1] on the real line. A graph G on n nodes is said to be representable as the intersection of k-cubes (cube representation in k dimensions) if each vertex of C can be mapped to a k-cube such that two vertices are adjacent in G if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G denoted as cub(G) is the minimum k for which G can be represented as the intersection of k-cubes. An interesting aspect about cubicity is that many problems known to be NP-complete for general graphs have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step. We give an O(bw . n) algorithm to compute the cube representation of a general graph G in bw + 1 dimensions given a bandwidth ordering of the vertices of G, where bw is the bandwidth of G. As a consequence, we get O(Delta) upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and cocomparability graphs which have O(Delta) bandwidth. Thus we have: 1. cub(G) <= 3 Delta - 1, if G is an AT-free graph. 2. cub(G) <= 2 Delta + 1, if G is a circular-arc graph. 3. cub(G) <= 2 Delta, if G is a cocomparability graph. Also for these graph classes, there axe constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with O(Delta) width. We can thus generate the cube representation of such graphs in O(Delta) dimensions in polynomial time.

Hardness of approximation results for the problem of finding the stopping distance in Tanner graphs

Tanner Graph representation of linear block codes is widely used by iterative decoding algorithms for recovering data transmitted across a noisy communication channel from errors and erasures introduced by the channel. The stopping distance of a Tanner graph T for a binary linear block code C determines the number of erasures correctable using iterative decoding on the Tanner graph T when data is transmitted across a binary erasure channel using the code C. We show that the problem of finding the stopping distance of a Tanner graph is hard to approximate within any positive constant approximation ratio in polynomial time unless P = NP. It is also shown as a consequence that there can be no approximation algorithm for the problem achieving an approximation ratio of 2(log n)(1-epsilon) for any epsilon > 0 unless NP subset of DTIME(n(poly(log n))).

Algorithms for Colouring Random k-colourable Graphs

The k-colouring problem is to colour a given k-colourable graph with k colours. This problem is known to be NP-hard even for fixed k greater than or equal to 3. The best known polynomial time approximation algorithms require n(delta) (for a positive constant delta depending on k) colours to colour an arbitrary k-colourable n-vertex graph. The situation is entirely different if we look at the average performance of an algorithm rather than its worst-case performance. It is well known that a k-colourable graph drawn from certain classes of distributions can be ii-coloured almost surely in polynomial time. In this paper, we present further results in this direction. We consider k-colourable graphs drawn from the random model in which each allowed edge is chosen independently with probability p(n) after initially partitioning the vertex set into ii colour classes. We present polynomial time algorithms of two different types. The first type of algorithm always runs in polynomial time and succeeds almost surely. Algorithms of this type have been proposed before, but our algorithms have provably exponentially small failure probabilities. The second type of algorithm always succeeds and has polynomial running time on average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work as long as p(n) greater than or equal to n(-1+is an element of) where is an element of is a constant greater than 1/4.

Efficient algorithms for domination and Hamilton circuit problems on permutation graphs

The domination and Hamilton circuit problems are of interest both in algorithm design and complexity theory. The domination problem has applications in facility location and the Hamilton circuit problem has applications in routing problems in communications and operations research.The problem of deciding if G has a dominating set of cardinality at most k, and the problem of determining if G has a Hamilton circuit are NP-Complete. Polynomial time algorithms are, however, available for a large number of restricted classes. A motivation for the study of these algorithms is that they not only give insight into the characterization of these classes but also require a variety of algorithmic techniques and data structures. So the search for efficient algorithms, for these problems in many classes still continues.A class of perfect graphs which is practically important and mathematically interesting is the class of permutation graphs. The domination problem is polynomial time solvable on permutation graphs. Algorithms that are already available are of time complexity O(n2) or more, and space complexity O(n2) on these graphs. The Hamilton circuit problem is open for this class.We present a simple O(n) time and O(n) space algorithm for the domination problem on permutation graphs. Unlike the existing algorithms, we use the concept of geometric representation of permutation graphs. Further, exploiting this geometric notion, we develop an O(n2) time and O(n) space algorithm for the Hamilton circuit problem.

New Approximation Algorithms for Minimum Cycle Bases of Graphs

We consider the problem of computing an approximate minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over F-2 generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Cycle bases of low weight are useful in a number of contexts, e.g. the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction. Although in most such applications any cycle basis can be used, a low weight cycle basis often translates to better performance and/or numerical stability. Despite the fact that the problem can be solved exactly in polynomial time, we design approximation algorithms since the performance of the exact algorithms may be too expensive for some practical applications. We present two new algorithms to compute an approximate minimum cycle basis. For any integer k >= 1, we give (2k - 1)-approximation algorithms with expected running time O(kmn(1+2/k) + mn((1+1/k)(omega-1))) and deterministic running time O(n(3+2/k) ), respectively. Here omega is the best exponent of matrix multiplication. It is presently known that omega < 2.376. Both algorithms are o(m(omega)) for dense graphs. This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Theta(m(omega) ) bound. We also present a 2-approximation algorithm with expected running time O(M-omega root n log n), a linear time 2-approximation algorithm for planar graphs and an O(n(3)) time 2.42-approximation algorithm for the complete Euclidean graph in the plane.

The existence of homeomorphic subgraphs in chordal graphs

We establish conditions for the existence, in a chordal graph, of subgraphs homeomorphic to K-n (n greater than or equal to 3), K-m,K-n (m,n greater than or equal to 2), and wheels W-r (r greater than or equal to 3). Using these results, we develop a simple linear time algorithm for testing planarity of chordal graphs. We also show how these results lead to simple polynomial time algorithms for the Fixed Subgraph Homeomorphism problem on chordal graphs for some special classes of pattern graphs.

Delay constrained optimal relay placement for planned wireless sensor networks

In this paper, we study the problem of wireless sensor network design by deploying a minimum number of additional relay nodes (to minimize network design cost) at a subset of given potential relay locationsin order to convey the data from already existing sensor nodes (hereafter called source nodes) to a Base Station within a certain specified mean delay bound. We formulate this problem in two different ways, and show that the problem is NP-Hard. For a problem in which the number of existing sensor nodes and potential relay locations is n, we propose an O(n) approximation algorithm of polynomial time complexity. Results show that the algorithm performs efficiently (in over 90% of the tested scenarios, it gave solutions that were either optimal or exceeding optimal just by one relay) in various randomly generated network scenarios.

Geometric representations of graphs in low dimension

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in $1.5 (\Delta + 2) \ln n$ dimensions, where $\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (\Delta + 2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree $\Delta$, we show that for almost all graphs on n vertices, its boxicity is upper bound by $c\cdot(d_{av} + 1) \ln n$ where d_{av} is the average degree and c is a small constant. Also, we show that for any graph G, $\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b \sqrt{\ln n}$ for a constant b.

Cubicity and Bandwidth

A unit cube in (or a k-cube in short) is defined as the Cartesian product R (1) x R (2) x ... x R (k) where R (i) (for 1 a parts per thousand currency sign i a parts per thousand currency sign k) is a closed interval of the form a (i) , a (i) + 1] on the real line. A k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that two vertices in G are adjacent if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G is the minimum k such that G has a k-cube representation. From a geometric embedding point of view, a k-cube representation of G = (V, E) yields an embedding such that for any two vertices u and v, ||f(u) - f(v)||(a) a parts per thousand currency sign 1 if and only if . We first present a randomized algorithm that constructs the cube representation of any graph on n vertices with maximum degree Delta in O(Delta ln n) dimensions. This algorithm is then derandomized to obtain a polynomial time deterministic algorithm that also produces the cube representation of the input graph in the same number of dimensions. The bandwidth ordering of the graph is studied next and it is shown that our algorithm can be improved to produce a cube representation of the input graph G in O(Delta ln b) dimensions, where b is the bandwidth of G, given a bandwidth ordering of G. Note that b a parts per thousand currency sign n and b is much smaller than n for many well-known graph classes. Another upper bound of b + 1 on the cubicity of any graph with bandwidth b is also shown. Together, these results imply that for any graph G with maximum degree Delta and bandwidth b, the cubicity is O(min{b, Delta ln b}). The upper bound of b + 1 is used to derive upper bounds for the cubicity of circular-arc graphs, cocomparability graphs and AT-free graphs in terms of the maximum degree Delta.

Efficient Methods for Robust Classification Under Uncertainty in Kernel Matrices

In this paper we study the problem of designing SVM classifiers when the kernel matrix, K, is affected by uncertainty. Specifically K is modeled as a positive affine combination of given positive semi definite kernels, with the coefficients ranging in a norm-bounded uncertainty set. We treat the problem using the Robust Optimization methodology. This reduces the uncertain SVM problem into a deterministic conic quadratic problem which can be solved in principle by a polynomial time Interior Point (IP) algorithm. However, for large-scale classification problems, IP methods become intractable and one has to resort to first-order gradient type methods. The strategy we use here is to reformulate the robust counterpart of the uncertain SVM problem as a saddle point problem and employ a special gradient scheme which works directly on the convex-concave saddle function. The algorithm is a simplified version of a general scheme due to Juditski and Nemirovski (2011). It achieves an O(1/T-2) reduction of the initial error after T iterations. A comprehensive empirical study on both synthetic data and real-world protein structure data sets show that the proposed formulations achieve the desired robustness, and the saddle point based algorithm outperforms the IP method significantly.

Solving MIN ONES 2-SAT as fast as VERTEX COVER

The problem of finding a satisfying assignment that minimizes the number of variables that are set to 1 is NP-complete even for a satisfiable 2-SAT formula. We call this problem MIN ONES 2-SAT. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k. In this paper, we present a polynomial-time reduction from MIN ONES 2-SAT to VERTEX COVER without increasing the parameter and ensuring that the number of vertices in the reduced instance is equal to the number of variables of the input formula. Consequently, we conclude that this problem also has a simple 2-approximation algorithm and a 2k - c logk-variable kernel subsuming (or, in the case of kernels, improving) the results known earlier. Further, the problem admits algorithms for the parameterized and optimization versions whose runtimes will always match the runtimes of the best-known algorithms for the corresponding versions of vertex cover. Finally we show that the optimum value of the LP relaxation of the MIN ONES 2-SAT and that of the corresponding VERTEX COVER are the same. This implies that the (recent) results of VERTEX COVER version parameterized above the optimum value of the LP relaxation of VERTEX COVER carry over to the MIN ONES 2-SAT version parameterized above the optimum of the LP relaxation of MIN ONES 2-SAT. (C) 2013 Elsevier B.V. All rights reserved.