Algorithms for Fast Aggregated Convergecast in Sensor Networks

Fast and periodic collection of aggregated data is of considerable interest for mission-critical and continuous monitoring applications in sensor networks. In the many-to-one communication paradigm, referred to as convergecast, we focus on applications wherein data packets are aggregated at each hop en-route to the sink along a tree-based routing topology, and address the problem of minimizing the convergecast schedule length by utilizing multiple frequency channels. The primary hindrance in minimizing the schedule length is the presence of interfering links. We prove that it is NP-complete to determine whether all the interfering links in an arbitrary network can be removed using at most a constant number of frequencies. We give a sufficient condition on the number of frequencies for which all the interfering links can be removed, and propose a polynomial time algorithm that minimizes the schedule length in this case. We also prove that minimizing the schedule length for a given number of frequencies on an arbitrary network is NP-complete, and describe a greedy scheme that gives a constant factor approximation on unit disk graphs. When the routing tree is not given as an input to the problem, we prove that a constant factor approximation is still achievable for degree-bounded trees. Finally, we evaluate our algorithms through simulations and compare their performance under different network parameters

Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypothesis, our two main results are: - For any r larger than some constant, any r-approximation algorithm for the maximum independent set problem must run in at least 2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of 2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time min (k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph pricing problem, where n is the number of vertices in an input graph. This almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and Blum, 2007 and an algorithm in this paper). We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time.Comment: The full version of FOCS 201

Testing bounded arboricity

In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows access to degree queries and neighbor queries, and the distance is defined with respect to the actual number of edges. More specifically, our algorithm distinguishes between graphs that are $\epsilon$-close to having arboricity $\alpha$ and graphs that $c \cdot \epsilon$-far from having arboricity $3\alpha$, where $c$ is an absolute small constant. The query complexity and running time of the algorithm are $\tilde{O}\left(\frac{n}{\sqrt{m}}\cdot \frac{\log(1/\epsilon)}{\epsilon} + \frac{n\cdot \alpha}{m} \cdot \left(\frac{1}{\epsilon}\right)^{O(\log(1/\epsilon))}\right)$ where $n$ denotes the number of vertices and $m$ denotes the number of edges. In terms of the dependence on $n$ and $m$ this bound is optimal up to poly-logarithmic factors since $\Omega(n/\sqrt{m})$ queries are necessary (and $\alpha = O(\sqrt{m}))$. We leave it as an open question whether the dependence on $1/\epsilon$ can be improved from quasi-polynomial to polynomial. Our techniques include an efficient local simulation for approximating the outcome of a global (almost) forest-decomposition algorithm as well as a tailored procedure of edge sampling

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials $\mathcal C$ and show that if $p\in \cal C$ and there is a disk $D$ centered at zero in the complex plane such that $p(G)$ does not vanish on $D$ for all bounded degree graphs $G$, then for each $z$ in the interior of $D$ there exists a deterministic polynomial-time approximation algorithm for evaluating $p(G)$ at $z$. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by. Barvinok, which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In particular a tiny error in Proposition 4.4 has been fixed. The introduction and concluding remarks have also been rewritten to incorporate the most recent developments. Accepted for publication in SIAM Journal on Computatio