On Coloring Resilient Graphs

We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we study $r$-resiliently $k$-colorable graphs, which are those $k$-colorable graphs that remain $k$-colorable even after the addition of any $r$ new edges. We prove lower bounds on the NP-hardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs. We also analyze the corresponding notion of resilience for $k$-SAT. This notion of resilience suggests an array of open questions for graph coloring and other combinatorial problems.Comment: Appearing in MFCS 201

Algorithmic Aspects of Energy-Delay Tradeoff in Multihop Cooperative Wireless Networks

We consider the problem of energy-efficient transmission in delay constrained cooperative multihop wireless networks. The combinatorial nature of cooperative multihop schemes makes it difficult to design efficient polynomial-time algorithms for deciding which nodes should take part in cooperation, and when and with what power they should transmit. In this work, we tackle this problem in memoryless networks with or without delay constraints, i.e., quality of service guarantee. We analyze a wide class of setups, including unicast, multicast, and broadcast, and two main cooperative approaches, namely: energy accumulation (EA) and mutual information accumulation (MIA). We provide a generalized algorithmic formulation of the problem that encompasses all those cases. We investigate the similarities and differences of EA and MIA in our generalized formulation. We prove that the broadcast and multicast problems are, in general, not only NP hard but also o(log(n)) inapproximable. We break these problems into three parts: ordering, scheduling and power control, and propose a novel algorithm that, given an ordering, can optimally solve the joint power allocation and scheduling problems simultaneously in polynomial time. We further show empirically that this algorithm used in conjunction with an ordering derived heuristically using the Dijkstra's shortest path algorithm yields near-optimal performance in typical settings. For the unicast case, we prove that although the problem remains NP hard with MIA, it can be solved optimally and in polynomial time when EA is used. We further use our algorithm to study numerically the trade-off between delay and power-efficiency in cooperative broadcast and compare the performance of EA vs MIA as well as the performance of our cooperative algorithm with a smart noncooperative algorithm in a broadcast setting.Comment: 12 pages, 9 figure

Baby-Step Giant-Step Algorithms for the Symmetric Group

We study discrete logarithms in the setting of group actions. Suppose that $G$ is a group that acts on a set $S$. When $r,s \in S$, a solution $g \in G$ to $r^g = s$ can be thought of as a kind of logarithm. In this paper, we study the case where $G = S_n$, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets $A, B \subseteq S_n$ such that every permutation of $S_n$ can be written as a product $ab$ of elements $a \in A$ and $b \in B$. Our deterministic procedure is optimal up to constant factors, in the sense that $A$ and $B$ can be computed in optimal asymptotic complexity, and $|A|$ and $|B|$ are a small constant from $\sqrt{n!}$ in size. We also analyze randomized "collision" algorithms for the same problem

Multiflow Transmission in Delay Constrained Cooperative Wireless Networks

This paper considers the problem of energy-efficient transmission in multi-flow multihop cooperative wireless networks. Although the performance gains of cooperative approaches are well known, the combinatorial nature of these schemes makes it difficult to design efficient polynomial-time algorithms for joint routing, scheduling and power control. This becomes more so when there is more than one flow in the network. It has been conjectured by many authors, in the literature, that the multiflow problem in cooperative networks is an NP-hard problem. In this paper, we formulate the problem, as a combinatorial optimization problem, for a general setting of $k$-flows, and formally prove that the problem is not only NP-hard but it is $o(n^{1/7-\epsilon})$ inapproxmiable. To our knowledge*, these results provide the first such inapproxmiablity proof in the context of multiflow cooperative wireless networks. We further prove that for a special case of k = 1 the solution is a simple path, and devise a polynomial time algorithm for jointly optimizing routing, scheduling and power control. We then use this algorithm to establish analytical upper and lower bounds for the optimal performance for the general case of $k$ flows. Furthermore, we propose a polynomial time heuristic for calculating the solution for the general case and evaluate the performance of this heuristic under different channel conditions and against the analytical upper and lower bounds.Comment: 9 pages, 5 figure