Experimental Evaluation of the Greedy and Random Algorithms for Finding Independent Sets in Random Graphs

Abstract

Abstract. This work is motivated by the long-standing open problem of designing a polynomial-time algorithm that with high probability constructs an asymptotically maximum independent set in a random graph. We present the results of an experimental investigation of the comparative performance of several efficient heuristics for constructing maximal independent sets. Among the algorithms that we evaluate are the well known randomized heuristic, the greedy heuristic, and a modification of the latter which breaks ties in a novel way. All algorithms deliver online upper bounds on the size of the maximum independent set for the specific input-graph. In our experiments, we consider random graphs parameterized by the number of vertices n and the average vertex degree d. Our results provide strong experimental evidence in support of the following conjectures:1. for d = c · n (c is a constant), the greedy and random algorithms are asymptotically equivalent; 2. for fixed d, the greedy algorithms are asymptotically superior to the random algorithm; 3. for graphs with d ≤ 3, the approximation ratio of the modified greedy algorithm is asymptotically < 1.005. We also consider random 3-regular graphs, for which non-trivial lower and upper bounds on the size of a maximum independent set are known. Our experiments suggest that the lower bound is asymptotically tight.

Towards maximum independent sets on massive graphs

“…Note that we now cannot obtain the exact independent number if NP ̸ = P. Therefore we develop an algorithm to calculate its theoretical upper bound. The detail of our algorithm is shown in the Appendix (see Algorithm 5), which is modified for the semi-external model from the approach in [13]. In our implementation, for each β value, we generate ten random graphs and compute their optimal bounds with Algorithm 5.…”

“…Then the algorithm proceeds in three phases: pre-swap (Line 7-14), swap (Line 15-19) and post-swap (Line 20-28). (1) In the pre-swap phase, there are three cases for vertex u: (i) u has conflicted with other swap candidates (Line 9-10); or (ii) there is a new 1-2 swap skeleton for u (Line 11-12); or (iii) there will be a 0-1 swap for u (Line [13][14]. (2) In the swap phase, more vertices are added to the independent set.…”

“…Its significance is not just limited to graph theory but also in numerous real-world applications, such as indexing techniques for shortest path and distance queries [11,17], automated labeling of maps [22], information coding [9], signal transmission analysis [24], social network analysis [13], and computer vision [27]. In particular, the stateof-the-art techniques [11,17] for shortest path and distance queries require building some graph indexing structures, the construction of which requires repeatedly invoking a sub-routine for solving the MIS problem.…”

On the Analysis of Efficient Hybrid MAC Protocol for Wireless Sensor Networks

“…The MNLS graphic model G (n, m, p) can be transformed into the MIS graphic model G(m, p) with m vertices, and each pair of nodes generate an edge with the probability p. For G(m, p), it is known in [13] that, for fixed p, the standard randomized algorithm 3 outputs an independent set of size log 1/(1−p) m, with the probability near 1.…”