Optimisation Problems in Wireless Sensor Networks : Local Algorithms and Local Graphs

This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs.

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.

Acyclic edge coloring of subcubic graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.

A note on the Hadwiger number of circular arc graphs

The intention of this note is to motivate the researchers to study Hadwiger's conjecture for circular arc graphs. Let η(G) denote the largest clique minor of a graph G, and let χ(G) denote its chromatic number. Hadwiger's conjecture states that η(G)greater-or-equal, slantedχ(G) and is one of the most important and difficult open problems in graph theory. From the point of view of researchers who are sceptical of the validity of the conjecture, it is interesting to study the conjecture for graph classes where η(G) is guaranteed not to grow too fast with respect to χ(G), since such classes of graphs are indeed a reasonable place to look for possible counterexamples. We show that in any circular arc graph G, η(G)less-than-or-equals, slant2χ(G)−1, and there is a family with equality. So, it makes sense to study Hadwiger's conjecture for this family.

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.

d-Regular Graphs of Acyclic Chromatic Index at Least d+2

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Suclakov and Zaks (and earlier by Fiamcik) that a'(G) <= Delta+2, where Delta = Delta(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require Delta+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d-regular graphs with 2n vertices and d>n, requires at least d+2 colors. We also show that a'(K-n,K-n) >= n+2, when n is odd using a more non-trivial argument. (Here K-n,K-n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn,n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d >= 5, n >= 2d+3 and dn even, there exist d-regular graphs which require at least d+2-colors to be acyclically edge colored. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226-230, 2010.

Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number eta(G) of a graph G is the largest integer n for which the complete graph K-n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, eta(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G square H of graphs. As the main result of this paper, we prove that eta(G(1) square G(2)) >= h root 1 (1 - o(1)) for any two graphs G(1) and G(2) with eta(G(1)) = h and eta(G(2)) = l. We show that the above lower bound is asymptotically best possible when h >= l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let G = G(1) square G(2) square ... square G(k) be the ( unique) prime factorization of G. Then G satisfies Hadwiger's conjecture if k >= 2 log log chi(G) + c', where c' is a constant. This improves the 2 log chi(G) + 3 bound in [2] 2. Let G(1) and G(2) be two graphs such that chi(G1) >= chi(G2) >= clog(1.5)(chi(G(1))), where c is a constant. Then G1 square G2 satisfies Hadwiger's conjecture. 3. Hadwiger's conjecture is true for G(d) (Cartesian product of G taken d times) for every graph G and every d >= 2. This settles a question by Chandran and Sivadasan [2]. ( They had shown that the Hadiwger's conjecture is true for G(d) if d >= 3).

On a Special Co-cycle Basis of Graphs

In this paper we consider the problems of computing a minimum co-cycle basis and a minimum weakly fundamental co-cycle basis of a directed graph G. A co-cycle in G corresponds to a vertex partition (S,V ∖ S) and a { − 1,0,1} edge incidence vector is associated with each co-cycle. The vector space over ℚ generated by these vectors is the co-cycle space of G. Alternately, the co-cycle space is the orthogonal complement of the cycle space of G. The minimum co-cycle basis problem asks for a set of co-cycles that span the co-cycle space of G and whose sum of weights is minimum. Weakly fundamental co-cycle bases are a special class of co-cycle bases, these form a natural superclass of strictly fundamental co-cycle bases and it is known that computing a minimum weight strictly fundamental co-cycle basis is NP-hard. We show that the co-cycle basis corresponding to the cuts of a Gomory-Hu tree of the underlying undirected graph of G is a minimum co-cycle basis of G and it is also weakly fundamental.

New approximation algorithms for minimum cycle bases of graphs

We consider the problem of computing an approximate minimum cycle basis of an undirected 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. 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 0(kmn(1+2/k) + mn((1+1/k)(omega-1))) and deterministic running time 0(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 O(m(omega) root n log n) expected running time, 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.

Boxicity and cubicity of asteroidal triple free graphs

The boxicity of a graph G, denoted box(G), is the least integer d such that G is the intersection graph of a family of d-dimensional (axis-parallel) boxes. The cubicity, denoted cub(G), is the least dsuch that G is the intersection graph of a family of d-dimensional unit cubes. An independent set of three vertices is an asteroidal triple if any two are joined by a path avoiding the neighbourhood of the third. A graph is asteroidal triple free (AT-free) if it has no asteroidal triple. The claw number psi(G) is the number of edges in the largest star that is an induced subgraph of G. For an AT-free graph G with chromatic number chi(G) and claw number psi(G), we show that box(G) <= chi(C) and that this bound is sharp. We also show that cub(G) <= box(G)([log(2) psi(G)] + 2) <= chi(G)([log(2) psi(G)] + 2). If G is an AT-free graph having girth at least 5, then box(G) <= 2, and therefore cub(G) <= 2 [log(2) psi(G)] + 4. (c) 2010 Elsevier B.V. All rights reserved.

A new class of separators and planarity of chordal graphs

We introduce a new class of clique separators, called base sets, for chordal graphs. Base sets of a chordal graph closely reflect its structure. We show that the notion of base sets leads to structural characterizations of planar k-trees and planar chordal graphs. Using these characterizations, we develop linear time algorithms for recognizing planar k-trees and planar chordal graphs. These algorithms are extensions of the Lexicographic_Breadth_First_Search algorithm for recognizing chordal graphs and are much simpler than the general planarity checking algorithm. Further, we use the notion of base sets to prove the equivalence of hamiltonian 2-trees and maximal outerplanar graphs.

On the complexity of finding stopping set size in Tanner Graphs

The problem of determining whether a Tanner graph for a linear block code has a stopping set of a given size is shown to be NT-complete.

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))).

Computing complete test graphs for hierarchical systems

Conformance testing focuses on checking whether an implementation. under test (IUT) behaves according to its specification. Typically, testers are interested it? performing targeted tests that exercise certain features of the IUT This intention is formalized as a test purpose. The tester needs a "strategy" to reach the goal specified by the test purpose. Also, for a particular test case, the strategy should tell the tester whether the IUT has passed, failed. or deviated front the test purpose. In [8] Jeron and Morel show how to compute, for a given finite state machine specification and a test purpose automaton, a complete test graph (CTG) which represents all test strategies. In this paper; we consider the case when the specification is a hierarchical state machine and show how to compute a hierarchical CTG which preserves the hierarchical structure of the specification. We also propose an algorithm for an online test oracle which avoids a space overhead associated with the CTG.