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 the Average Case Communication Complexity for Detection in Sensor Networks

The problem of sensor-network-based distributed intrusion detection in the presence of clutter is considered. It is argued that sensing is best regarded as a local phenomenon in that only sensors in the immediate vicinity of an intruder are triggered. In such a setting, lack of knowledge of intruder location gives rise to correlated sensor readings. A signal-space viewpoint is introduced in which the noise-free sensor readings associated to intruder and clutter appear as surfaces $\mathcal{S_I}$ and $\mathcal{S_C}$ and the problem reduces to one of determining in distributed fashion, whether the current noisy sensor reading is best classified as intruder or clutter. Two approaches to distributed detection are pursued. In the first, a decision surface separating $\mathcal{S_I}$ and $\mathcal{S_C}$ is identified using Neyman-Pearson criteria. Thereafter, the individual sensor nodes interactively exchange bits to determine whether the sensor readings are on one side or the other of the decision surface. Bounds on the number of bits needed to be exchanged are derived, based on communication complexity (CC) theory. A lower bound derived for the two-party average case CC of general functions is compared against the performance of a greedy algorithm. The average case CC of the relevant greater-than (GT) function is characterized within two bits. In the second approach, each sensor node broadcasts a single bit arising from appropriate two-level quantization of its own sensor reading, keeping in mind the fusion rule to be subsequently applied at a local fusion center. The optimality of a threshold test as a quantization rule is proved under simplifying assumptions. Finally, results from a QualNet simulation of the algorithms are presented that include intruder tracking using a naive polynomial-regression algorithm.

Sequence design for symbol-asynchronous CDMA with power or rate constraints

Sequence design and resource allocation for a symbol-asynchronous chip-synchronous code division multiple access (CDMA) system is considered in this paper. A simple lower bound on the minimum sum-power required for a non-oversized system, based on the best achievable for a non-spread system, and an analogous upper bound on the sum rate are first summarised. Subsequently, an algorithm of Sundaresan and Padakandla is shown to achieve the lower bound on minimum sum power (upper bound on sum rate, respectively). Analogous to the synchronous case, by splitting oversized users in a system with processing gain N, a system with no oversized users is easily obtained, and the lower bound on sum power (upper bound on sum rate, respectively) is shown to be achieved by using N orthogonal sequences. The total number of splits is at most N - 1.

Limit analysis of rectangular R.C. slabs on ground

A method is presented for obtaining lower bound on the carrying capacity of reinforced concrete foundation slab-structures subject to non-uniform contact pressure distributions. Functional approach suggested by Vallance for simply supported square slabs subject to uniform pressure distribution has been extended to simply supported rectangular slabs subject to symmetrical non-uniform pressure distributions. Radial solutions, ideally suited for rotationally symmetric problems, are shown to be adoptable for regular polygonal slabs subject to contact pressure paraboloids with constant edge pressures. The functional approach has been shown to be well suited even when the pressure is varying along the edges.

A non-orthogonal distributed space-time coded protocol - Part 1:Signal model and design criteria

In this two-part series of papers, a generalized non-orthogonal amplify and forward (GNAF) protocol which generalizes several known cooperative diversity protocols is proposed. Transmission in the GNAF protocol comprises of two phases - the broadcast phase and the cooperation phase. In the broadcast phase, the source broadcasts its information to the relays as well as the destination. In the cooperation phase, the source and the relays together transmit a space-time code in a distributed fashion. The GNAF protocol relaxes the constraints imposed by the protocol of Jing and Hassibi on the code structure. In Part-I of this paper, a code design criteria is obtained and it is shown that the GNAF protocol is delay efficient and coding gain efficient as well. Moreover GNAF protocol enables the use of sphere decoders at the destination with a non-exponential Maximum likelihood (ML) decoding complexity. In Part-II, several low decoding complexity code constructions are studied and a lower bound on the Diversity-Multiplexing Gain tradeoff of the GNAF protocol is obtained.

On the maximal rate of (n+1) x n and (n+2) x n complex orthogonal designs

For p x n complex orthogonal designs in k variables, where p is the number of channels uses and n is the number of transmit antennas, the maximal rate L of the design is asymptotically half as n increases. But, for such maximal rate codes, the decoding delay p increases exponentially. To control the delay, if we put the restriction that p = n, i.e., consider only the square designs, then, the rate decreases exponentially as n increases. This necessitates the study of the maximal rate of the designs with restrictions of the form p = n+1, p = n+2, p = n+3 etc. In this paper, we study the maximal rate of complex orthogonal designs with the restrictions p = n+1 and p = n+2. We derive upper and lower bounds for the maximal rate for p = n+1 and p = n+2. Also for the case of p = n+1, we show that if the orthogonal design admit only the variables, their negatives and multiples of these by root-1 and zeros as the entries of the matrix (other complex linear combinations are not allowed), then the maximal rate always equals the lower bound.

Limit Analysis of Slabs with Free Edge

The general method earlier developed by the writers for obtaining valid lower bound solutions to slabs under uniformly distributed load and supported along all edges is extended to the slabs with a free edge. Lower bound solutions with normal moment criterion are presented for six cases of orthotropically reinforced slabs, with one of the short edges being free and the other three edges being any combination of fixed and simply supported conditions. The expressions for moment field and collapse load are given for each slab. The lower bounds have been compared with the corresponding upper bound values obtained from the yield line theory with simple straight yield line modes of failure. They are also compared with Nielsen’s solutions available for two cases with isotropic reinforcement.

Triangular Ising antiferromagnet in a staggered field

We study the equilibrium properties of the nearest-neighbor Ising antiferromagnet on a triangular lattice in the presence of a staggered field conjugate to one of the degenerate ground states. Using a mapping of the ground states of the model without the staggered field to dimer coverings on the dual lattice, we classify the ground states into sectors specified by the number of "strings." We show that the effect of the staggered field is to generate long-range interactions between strings. In the limiting case of the antiferromagnetic coupling constant J becoming infinitely large, we prove the existence of a phase transition in this system and obtain a finite lower bound for the transition temperature. For finite J, we study the equilibrium properties of the system using Monte Carlo simulations with three different dynamics. We find that in all the three cases, equilibration times for low-field values increase rapidly with system size at low temperatures. Due to this difficulty in equilibrating sufficiently large systems at low temperatures, our finite-size scaling analysis of the numerical results does not permit a definite conclusion about the existence of st phase transition for finite values of J. A surprising feature in the system is the fact that unlike usual glassy systems; a zero-temperature quench almost always leads to the ground state, while a slow cooling does not.

Max-Coloring Paths: Tight Bounds and Extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G = (V, E) with a non-negative weight function w on V such that Sigma(k)(i=1) max(v epsilon Ci) w(v(i)) is minimized, where C-1, ... , C-k are the various color classes. Max-coloring general graphs is as hard as the classical vertex coloring problem, a special case where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring abroad class of trees and show it can be solved in time O(vertical bar V vertical bar+time for sorting the vertex weights). When vertex weights belong to R, we show a matching lower bound of Omega(vertical bar V vertical bar log vertical bar V vertical bar) in the algebraic computation tree model.

Task assignment in a multiprocessor system

One of the key problems in the design of any incompletely connected multiprocessor system is to appropriately assign the set of tasks in a program to the Processing Elements (PEs) in the system. The task assignment problem has proven difficult both in theory and in practice. This paper presents a simple and efficient heuristic algorithm for assigning program tasks with precedence and communication constraints to the PEs in a Message-based Multiple-bus Multiprocessor System, M3, so that the total execution time for the program is minimized. The algorithm uses a cost function: “Minimum Distance and Parallel Transfer” to minimize the completion time. The effectiveness of the algorithm has been demonstrated by comparing the results with (i) the lower bound on the execution time of a program (task) graph and (ii) a random assignment.

An optimal local approximation algorithm for max-min linear programs

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0 for nonnegative matrices A and C. We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The approximation ratio of our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.

Random-Restart Reactive Tabu Search Algorithm for Detection in Large-MIMO Systems

We present a low-complexity algorithm based on reactive tabu search (RTS) for near maximum likelihood (ML) detection in large-MIMO systems. The conventional RTS algorithm achieves near-ML performance for 4-QAM in large-MIMO systems. But its performance for higher-order QAM is far from ML performance. Here, we propose a random-restart RTS (R3TS) algorithm which achieves significantly better bit error rate (BER) performance compared to that of the conventional RTS algorithm in higher-order QAM. The key idea is to run multiple tabu searches, each search starting with a random initial vector and choosing the best among the resulting solution vectors. A criterion to limit the number of searches is also proposed. Computer simulations show that the R3TS algorithm achieves almost the ML performance in 16 x 16 V-BLAST MIMO system with 16-QAM and 64-QAM at significantly less complexities than the sphere decoder. Also, in a 32 x 32 V-BLAST MIMO system, the R3TS performs close to ML lower bound within 1.6 dB for 16-QAM (128 bps/Hz), and within 2.4 dB for 64-QAM (192 bps/Hz) at 10(-3) BER.

A non-cooperative game-theoretic approach to channel assignment in multi-channel multi-radio wireless networks

Channel assignment in multi-channel multi-radio wireless networks poses a significant challenge due to scarcity of number of channels available in the wireless spectrum. Further, additional care has to be taken to consider the interference characteristics of the nodes in the network especially when nodes are in different collision domains. This work views the problem of channel assignment in multi-channel multi-radio networks with multiple collision domains as a non-cooperative game where the objective of the players is to maximize their individual utility by minimizing its interference. Necessary and sufficient conditions are derived for the channel assignment to be a Nash Equilibrium (NE) and efficiency of the NE is analyzed by deriving the lower bound of the price of anarchy of this game. A new fairness measure in multiple collision domain context is proposed and necessary and sufficient conditions for NE outcomes to be fair are derived. The equilibrium conditions are then applied to solve the channel assignment problem by proposing three algorithms, based on perfect/imperfect information, which rely on explicit communication between the players for arriving at an NE. A no-regret learning algorithm known as Freund and Schapire Informed algorithm, which has an additional advantage of low overhead in terms of information exchange, is proposed and its convergence to the stabilizing outcomes is studied. New performance metrics are proposed and extensive simulations are done using Matlab to obtain a thorough understanding of the performance of these algorithms on various topologies with respect to these metrics. It was observed that the algorithms proposed were able to achieve good convergence to NE resulting in efficient channel assignment strategies.

A Flexible Class of Regenerating Codes for Distributed Storage

In the distributed storage setting introduced by Dimakis et al., B units of data are stored across n nodes in the network in such a way that the data can be recovered by connecting to any k nodes. Additionally one can repair a failed node by connecting to any d nodes while downloading at most beta units of data from each node. In this paper, we introduce a flexible framework in which the data can be recovered by connecting to any number of nodes as long as the total amount of data downloaded is at least B. Similarly, regeneration of a failed node is possible if the new node connects to the network using links whose individual capacity is bounded above by beta(max) and whose sum capacity equals or exceeds a predetermined parameter gamma. In this flexible setting, we obtain the cut-set lower bound on the repair bandwidth along with a constructive proof for the existence of codes meeting this bound for all values of the parameters. An explicit code construction is provided which is optimal in certain parameter regimes.

On Network-Error Correcting Convolutional Codes under the BSC Edge Error Model

Convolutional network-error correcting codes (CNECCs) are known to provide error correcting capability in acyclic instantaneous networks within the network coding paradigm under small field size conditions. In this work, we investigate the performance of CNECCs under the error model of the network where the edges are assumed to be statistically independent binary symmetric channels, each with the same probability of error pe(0 <= p(e) < 0.5). We obtain bounds on the performance of such CNECCs based on a modified generating function (the transfer function) of the CNECCs. For a given network, we derive a mathematical condition on how small p(e) should be so that only single edge network-errors need to be accounted for, thus reducing the complexity of evaluating the probability of error of any CNECC. Simulations indicate that convolutional codes are required to possess different properties to achieve good performance in low p(e) and high p(e) regimes. For the low p(e) regime, convolutional codes with good distance properties show good performance. For the high p(e) regime, convolutional codes that have a good slope ( the minimum normalized cycle weight) are seen to be good. We derive a lower bound on the slope of any rate b/c convolutional code with a certain degree.