Greed is go o d Approximating indep endent sets in sparse and y b oundeddegree graphs z x Magns M Halldrsson Jaikumar Radhakrishnan Abstract The minimumdegree greedy algorithm or Greedy for short is a simple and well studied metho d for nding indep endent sets in graphs We show that it achieves a p erformance ratio of ( + 2)3 for approximating indep endent sets in graphs with degree b ounded by The analysis yields a precise characterization of the size of the indep endent sets found by the algorithm as a function of the indep endence numb er as well as a generalization of Turns b ound We also analyze the algorithm when run in combination with a known prepro cessing d + 3)5 p erformance ratio on graphs with average technique and obtain an improved (2 degree d improving on the previous b est (d + 1)2 of Ho chbaum Finally we present an ecient parallel and distributed algorithm attaining the p erformance guarantees of Greedy Intro duction An independent set in a graph is a collection of vertices that are mutually nonadjacent The problem of nding an indep endent set of maximum cardinality is one of the fundamental com binatorial problems It is known to b e NP complete even for b oundeddegree graphs and therefore no ecient algorithms are in sight Given the hardness of exact computation we are interested in approximation algorithms for the indep endent set problem in b oundeddegree graphs In particular we seek an algorithm with a go o d performance ratio which is a b ound on the maximum ratio b etween the optimal solution size ie the indep endence numb er and the size of the solution found by the heuristic One of the most ubiquitous heuristic metho ds for this problem is the greedy algorithm which selects a vertex of minimum degree deletes that vertex and all of its neighb ors from the graph and rep eats this pro cess until the graph b ecomes empty As a delightfully simple and ecient algorithm the Greedy metho d deserves a particularly detailed analysis It is already known to p ossess several imp ortant qualities attaining the Turn b ound and its generalization in terms of degree sequences almost always obtaining a solution at least half the size of an optimal solution in a random graph yielding a nontrivial graph coloring approximation and a light coloring with a small chromatic sum when applied iteratively as a coloring metho d and nding optimal indep endent sets in trees seriesparallel cographs and graphs of degree at most While the p erformance ratio of Greedy has b een analyzed b efore to some extent the true extent of its p erformance has apparently not b een determined b efore The b est ratio previously Gordon Gekko y A preliminary version of this pap er app eared in the th ACM Symp osium on Theory of Computing z Science Institute University of Iceland IS Reykjavik Iceland email mmhrhihiis Work done at Japan Advanced Institute of Science and Technology Hokuriku x Theoretical Computer Science Group Tata Institute of Fundamental Research Bombay India email jaikumartcstifrresin Work done while visiting JAIST claimed for Greedy was on graphs with maximum degree and d on graphs of average degree d Our main result is that Greedy is much b etter than previously claimed We obtain a tight p erformance ratio of in terms of maximum degree and an asymptotically optimal d in terms of average degree In the pro cess we give a natural extension b ound of of Turns b ound that incorp orates the actual indep endence numb er of the graph and give a general tight expression of the size of the solution found as a function of the indep endenc e numb er and the numb er of vertices We further analyze Greedy extended with a prepro cessing metho d of Ho chbaum We use d but it to improve the b est p erformance ratio known in terms of average degree to show it to b e of limited use in terms of maximum degree It follows from our analysis that globally minimum degree is not required for Greedy to achieve the p erformance guarantees claimed ab ove in fact it holds for any vertex whose degree is at most the average of the degrees of it and its neighb ors This is a lo cally evaluated prop erty that naturally leads to a parallel and distributed algorithm inheriting the approximative prop erties of Greedy for the rst nontrivial such approximations known to us The remainder of the pap er is organized as follows In Section we present the Greedy algorithm and some of its prop erties including the Turn b ound and review other results on approximating this problem We analyze Greedy in detail in Section starting with a general ization of the Turn b ound in Section followed by tight p erformance ratios in Section and limitations on its p erformance in Section In Section we consider improvements ob tained by additionally applying a prepro cessing technique and describ e in Section a parallel algorithm attaining the b ounds proved for Greedy Notation We use fairly standard graph notation and terminology For the graph in question usually denoted G V E n denotes the numb er of vertices the maximum degree d the average degree the indep endence numb er the size of the largest indep endent set and the inde p endence fraction that is n For a vertex v dv denotes the degree of v and N v the set of neighb ors of v For an indep endent set algorithm A AG is the size of the solution obtained by A on graph G V E The p erformance ratio of A is dened by A G n max A A GjGj=n AG We fo cus on Greedy denoted by Gr The Greedy Algorithm and Related Results The Greedy Algorithm The minimumdegree greedy algorithm or Greedy for short incrementally constructs an inde p endent set by selecting some vertex of minimum degree removing it and its neighb ors from the graph and iterating on the remaining graph until empty GreedyG I while G do Cho ose v such that dv min dw w V (G) I I fv g G G fv g N v o d Output I end The algorithm can b e implemented in time linear in the numb er of edges and vertices indep endent of the degree We call a no de critical if its degree is at most the average of the degrees of it and its neighb ors That is v is critical if it satises X 2 d v dw w N (v ) A vertex of minimum degree is critical hence such a no de always exists Although we state Greedy with this minimumdegree pivoting rule the only prop erty that we shall use is that the selected vertex is critical Consider an execution of the algorithm to b e a sequence of reductions each corresp onding to an iteration In a reduction a vertex is selected added to the solution and then removed along with its neighb orho o d from the graph Let t denote the numb er of reductions and d the i degree in the remaining graph of the ith vertex selected i t The numb er of vertices removed in the ith reduction is thus d i The main prop erty of the algorithm that we shall b e using in our analysis is that the sum of the degrees of the d vertices removed in the ith reduction must b e at least d d i i i This allows us to b ound from b elow the numb er of edges removed in each step Previous results on Greedy A classical theorem in graph theory due to Turn states that for any graph G G n d and that the inequality is tight only for the graph consisting of G disjoint cliques of size as equal as p ossible Wei see proved an extension X G dv v V which he showed was attained by Greedy This result actually follows from an earlier theorem of Erds see which states that if a graph contains no indep endent set of size k then there is graph consisting of k disjoint cliques whose degree sequence dominates that of the original graph The pro of of his theorem can also b e seen to indirectly refer to Greedy We include a pro of here of the fact that Greedy attains the Turn b ound since the pro ofs of our central results build directly up on it n Theorem Gr d Proof The pro of is a variation of the one given by Ho chbaum The Greedy algorithm p erforms a sequence of t reductions each time picking a vertex and deleting it and its neighb ors from the graph We count the numb er of vertices and edges deleted in each reduction The removal of vertices in each reduction partitions the vertex set thus t X d n i i=1 Since Greedy always selects a critical vertex the sum of the degrees of the vertices deleted in step i is at least d d and thus the numb er of edges deleted is at least half that amount i i Summing over all the reductions t X d d i n jE j i=1 We now add and twice and obtain t X 2 d n d i i=1 Using the CauchySchwarz inequality and we get 2 d n n t Rearranging the inequality we obtain the desired b ound on t which is precisely the numb er of vertices found by Greedy Another simple algorithm op erates by a rule that is the inverse of Greedy it deletes a vertex of maximum degree until no edge remains Surprisingly this algorithm also attains the Turn b ound as proved indep endently by Griggs and Chvtal and McDiarmid Its approximative prop erties are however weaker On the graph with s vertices that is complete bipartite less a single p erfect matching the algorithm may nd only a two vertex indep endent set for an approximation ratio of s As a result we do not consider this metho d further An upp er b ound of of the p erformance ratio of Greedy with the minimumdegree rule can b e obtained by rudimentary arguments Observe that in a graph with minimum degree the indep endence numb er is b ounded from ab ove by n This is b ecause at least jI j edges must exit an indep endent set I while at most n jI