Greedy approximations of independent sets in low degree graphs

Abstract

Abstract. We investigate the power of a family of greedy algorithms for the independent set problem graphs of maximum degree three. These algorithm iteratively select vertices of minimum degree, but differ in the secondary rule for choosing among many candidates. We present two such algorithms that run in linear time, and show their performance ratios to be 3/2 and 9/7 ~ 1.28, respectively. This also translates to good ratios for other classes of low-degree graphs. We also show certain inherent limitations in the power of this family of algorithm: any algorithm that greedily selects vertices of minimum degree has a performance ratio at least 1.25 on degree-three graphs, even if given an oracle to choose among candidate vertices of minimum degree. IntroductionAn independent set of a graph G is a subset of vertices in which no two are adjacent. The MAX INDEPENDENT SET problem is that of finding an independent set of maximum cardinality. It is one of the core A/P-hard problems [4], and thus, polynomial time exact algorithms are unlikely to exist. It is therefore interesting to explore algorithms that produce solutions that are not always optimal but are close to optimal. The quality of an approximation algorithm is generally measured by the performance ratio, or the maximum ratio of the size of an optimal solution (the size of the maximum independent set) to the size of the solution found by the algorithm. In this paper we focus on a central case of bounded-degree graphs, namely when the maxinmm degree is at most three. Since the independent set problem is polynomial solvable when maximum degree is two, this problem can be thought of as the initial frontier of A/P-hardness of the problem. Also, many of the results for higher degrees use reductions to lower degree cases, in which the degree-three plays the role of the basis case [3,7,5,2], and improvements for that case translate to improvements for all odd degrees.Let us review the known results about approximating independent sets in degree-three graphs. Hochbaum [9] presented an algorithm with a 1.5 ratio, that runs in time proportional to bipartite matching or O(nl5

Models of Greedy Algorithms for Graph Problems

“…Independent Set We consider the independent set problem for graphs of small degree, in the node model. For degree-3 graphs, the standard greedy algorithm achieves an approximation ratio of 5 3 , [15]. This algorithm fits in the ADAP-TIVE priority model.…”

“…The known greedy approximation [16] for the weighted vertex cover problem (WVC) can be classified as an ADAPTIVE priority algorithm. The greedy approximation for the independent set problem [15] also fits our model. As noted in [4], the best known greedy approximation algorithm for the set cover problem also fits the framework of priority algorithms, and similarly the greedy algorithms for the facility location in arbitrary and metric spaces have priority models.…”

“…15 Proof To prove the theorem it suffices to show that any winning strategy for the Solver for the Memoryless game for a fixed ρ, in which accepting and rejecting decisions are not separated into distinct phases, can be converted to a winning strategy for the game with same ρ, in which the Solver never accepts after she has rejected a data item. Intuitively, since is a maximization problem, whose decision options are {accept, reject}, the Solver gains nothing after rejecting a data item (the value of the objective function of her solution depends only on the data items accepted and hence does not increase when a data item is rejected).…”

Ultimate greedy approximation of independent sets in subcubic graphs

“…• We completely resolve the open problem from the paper of Halldórsson and Yoshihara [28] and design a fast, ultimate advice for greedy obtaining a 5/4-approximation, that is, the best possible greedy ratio for MIS on subcubic graphs. A lower bound of 5/4 on the ratio of greedy with any, even exponential time, advice on such graphs was proved in [28], and the best previously known ratio of greedy was 3/2 [28]. Halldórsson and Radhakrishnan [27] also prove a lower bound of 5/3 for any greedy algorithm that does not use any advice for MIS on subcubic graphs.…”

“…Halldórsson and Yoshihara [28] asked in their paper the following fundamental question: what is the power of the greedy algorithm when we augment it with an advice, that is, a fast method that tells the greedy which minimum degree vertex to choose if there are many? They, for instance, proved that no advice can imply a better than 5/4-approximation of greedy for MIS with ∆ = 3.…”