Greedy Set Cover Estimations

1 Greedy Set Cover E stimations H. Aslanyan Yerevan State University 1 A.Manoukyan, 375049, Yerevan, Armenia Abstract . More pr ecise estimation of the greedy algori thm comple xity for a special case of the set cover problem is given in this pap er. Introduction The greedy heuristic is the most u sed for o ptimization problems. The general approach is as f ollows: repeatedly, a procedure th at minim izes (m aximizes) the loc al i ncrease of the objective function is applied. In some cases the greedy strategy guarantees the optimal solution (minimal spanning trees, the shortest path, etc.), in so me o thers - it provides a cceptable approximations (e.g. disjunctive fo rms an d tests). Typically, the greedy a lgorithms use simple structures that require minimal computat ional resource – time and memory. Consider the set cover problem. Given a finite set } , , { 1 n a a A L = an d a fa mily of subsets of A , } , , { 1 m A A F L = , such that every element of A belongs to at least one subset from F , - F covers A . The problem is in finding a collection F C ⊆ of minimal size, that covers A . The set cover p roblem is on e of the most typical NP -complete problems. It has proven that there is n o constant factor approximation to this problem (unless P=NP) [2]. The p roblem can be represented in terms o f (0,1)-matrices. The elements of A correspo nd to t he columns, and e ach subset from F correspo nds to a row of the matrix. The problem is in findin g t he minimal number of rows that co ver at least one “1” in each column. There are kno wn reasonable approximation greedy algorithms for this problem. Consider the following set cover greedy algo rithm: – at the first st ep the algorithm selects the row that contains the maximal number of “1”’ s (covers maximal number o f elements o f A ). At the current step the row, which co vers the greatest number of uncovered yet elements, has been selected. It is clea r that continuing this process (at most till the last row selection) all elements will be covered. Probably it may occur b efore. We consider the scheme [1] where a part of elements of A ha s been covered by the greedy steps, and then, ea ch uncovered element is being cove red by taking some new row having “1” in that column. The problem is in estimating t he number of all se lected rows (the size of cover). We consider the estimation given for a special case in [1], and give a more precise formula for this case. Formula Improvem ent Given a (0,1)-matrix of size n m × ( m is t he n umber of rows). Each co lumn contains a t least m γ “1”’s (special case). The problem is to find minimal number of rows that con tain at least one “1” on each column. Then the number of “1”’ s of the whole matrix is not less th an mn γ , and there is a row with a t least n γ “1”s. A t th e first step the greedy a lgorithm selects the row with maximal numbe r of “1”’s, therefore the se lected row will co ver at le ast n γ elements. Let we have done k similar steps, and let the number of uncovered yet ele ments does n ot exceed n k δ . The estimate ) 1 ( 1 γ − δ ≤ δ + k k n n is the main related re sult, ob tained in [1], Part 3, Chapter 3.5, p. 136-137.The table b elow ou tlines th e pa rt of rows, selected du ring th e first k greedy step s. The shaded columns do not contain “1”’s and h ence the corresponding elements are not covered yet. 2 1 a 2 a … … ) 1 ( k n a δ − 1 ) 1 ( + δ − k n a … … … … n a 1 A 2 A … k A 1 + k A m A The summary number of “1”’s in rows, from 1 + k A to m A and pa rt of uncovered elements (columns) is at least n m k δ γ . The fo rmula improvement takes in to account, that this “1”s ca n’t b e situ ated in the shaded part of the table. Hence, among the subsets 1 + k A to m A there exists one with at least k m n m k − δ γ “1”s. Therefore, . 1 ) 1 ( 1 ) 1 ( ) 1 )( ( ) 1 ( ) 1 ( 1 m k m k n k m k m m n k m m n k m mn n n k k k k k k − γ − − γ − δ = γ − − − γ − γ − δ = − γ − δ = − δ γ − δ ≤ δ + Using recurrently the above relation we get: ∏ ∏ = = +       −         γ − − γ − ≤ δ k i k i k k m i m i 1 1 1 1 ) 1 ( 1 ) 1 ( , which is the proposed improvement of the set cover complexity estimate in the above given special case of the problem. The additional coefficient is easy to maintain using the following formulae: ∏ − = − −         − ≤         − ≤         − 1 1 1 2 1 2 1 1 1 x i x x y x y i y x for y x ≤ positive integers ( γ is supposed justified to obey this). As a result it follows that in asymptotes, when ∞ → n and ∞ → m , the additional term is close to 1 wh en 0 → γ , but it is 1 << otherwise, providing the sensitive improvement of the set cover estimate. Bibliog raphy: 1. Äèñêðåòíàÿ ìàòåìàòèêà è ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè, ò.1, Ì., Íàóêà, 1974. 2. Garey M., Johnson D., Computers and intrac tabilit y, Fre eman, 1979.