# Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs

Save this PDF as:

Size: px
Start display at page:

Download "Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs"

## Transcription

1 Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute & Department of Computer Engineering and Informatics University of Patras, Rio, Greece Abstract. We present new combinatorial approximation algorithms for k-set cover. Previous approaches are based on extending the greedy algorithm by efficiently handling small sets. The new algorithms further extend them by utilizing the natural idea of computing large packings of elements into sets of large size. Our results improve the previously best approximation bounds for the k-set cover problem for all values of k 6. The analysis technique could be of independent interest; the upper bound on the approximation factor is obtained by bounding the objective value of a factor-revealing linear program. Introduction Set cover is a fundamental combinatorial optimization problem with many applications. Instances of the problem consist of a set of elements V and a collection S of subsets of V and the objective is to select a subset of S of minimum cardinality so that every element is covered, i.e., it is contained in at least one of the selected sets. The natural greedy algorithm starts with an empty solution and augments it until all elements are covered by selecting a set that contains the maximum number of elements that are not contained in any of the previously selected sets. Denoting by n the number of elements, it is known by the seminal papers of Johnson [0], Lovasz [3], and Chvátal [] that the greedy algorithm has approximation ratio H n. The tighter analysis of Slavík [4] improves the upper bound on the approximation ratio to ln n ln ln n + O(. Asymptotically, these bounds are tight due to a famous inapproximability result of Feige [3] which states that there is no ( ɛ ln n-approximation algorithm for set cover unless all problems in NP have deterministic algorithms running in subexponential time O ( n polylog(n. An interesting variant is k-set cover where every set of S has size at most k. Without loss of generality, we may assume that S is closed under subsets. In this case, the greedy algorithm can be equivalently expressed as follows: This work was partially supported by the European Union under IST FET Integrated Project AEOLUS.

2 Greedy phases: For i = k down to do: Choose a maximal collection of disjoint i-sets. An i-set is a set that contains exactly i previously uncovered elements and a collection T of disjoint i-sets is called maximal if any other i-set intersects some of the sets in T. A tight bound of H k on the approximation ratio of the greedy algorithm is well known in this case. Since the problem has many applications for particular values of k and due to its interest from a complexity-theoretic viewpoint, designing algorithms with improved second order terms in their approximation ratio has received much attention. Currently, there are algorithms with approximation ratio H k c where c is a constant. Goldschmidt et al. [4] were the first to present a modified greedy algorithm with c = /6. This value was improved to /3 by Halldórsson [6] and to /2 by Duh and Fürer [2]. Recently, Levin [2] further improved the constant to 98/ for k 4. On the negative side, Trevisan [5] has shown that, unless subexponential-time deterministic algorithms for NP exist, no polynomial-time algorithm has an approximation ratio of ln k Ω(ln ln k. The main idea that has been used in order to improve the performance of the greedy algorithm is to efficiently handle small sets. The algorithm of Goldschmidt et al. [4] uses a matching computation to accommodate as many elements as possible in sets of size 2 when no set of size at least 3 contains new elements. The algorithms of Halldórsson [5, 6] and Duh and Fürer [2] handle efficiently sets of size 3. The algorithm of [2] is based on a semi-local optimization technique. Levin s improvement [2] extends the algorithm of [2] by efficiently handling of sets of size 4. A natural but completely different idea is to replace the phases of the greedy algorithm associated with large sets with set-packing phases that also aim to maximize the number of new elements covered by large maximal sets. The approximation factor is not getting worse (due to the maximality condition while it has been left as an open problem in [2] whether it leads to any improvement in the approximation bound. This is the aim of the current paper: we show that by substituting the greedy phases in the algorithms of Duh and Fürer with packing phases, we obtain improved approximation bounds for every k 6 which approaches H k for large values of k. In particular, we will use algorithms for the k-set packing problem which is defined as follows. An instance of k-set packing consists of a set of elements V and a collection S of subsets of V each containing exactly k elements. The objective is to select as many as possible disjoint sets from S. When k = 2, the problem is equivalent to maximum matching in graphs and, hence, it is solvable in polynomial time. For k 3, the problem is APX-hard []; the best known inapproximability bound for large k is asymptotically O ( log k k [7]. Note that any maximal collection of disjoint subsets yields a /k-approximate solution. The best known algorithms have approximation ratio 2 ɛ k for any ɛ > 0 [8] and are based on local search; these are the algorithms used by the packing phases of our algorithms.

3 Our analysis is based on the concept of factor-revealing LPs which has been introduced in a different context in [9] for the analysis of approximation algorithms for facility location. We show that the approximation factor is upperbounded by the maximum objective value of a factor-revealing linear program. Hence, no explicit reasoning about the structure of the solution computed by the algorithms is required. Instead, the performance guarantees of the several phases are used as black boxes in the definition of the constraints of the LP. The rest of the paper is structured as follows. We present the algorithms of Duh and Fürer [2] as well as our modifications in Section 2. The analysis technique is discussed in Section 3 where we present the factor-revealing LP lemmas. Then, in Section 4, we present a part of the proofs of our main results; proofs that have been omitted from this extended abstract will appear in the final version of the paper. We conclude in Section 5. 2 Algorithm description In this section we present the algorithms considered in this paper. We start by giving an overview of the related results in [2]; then, we present our algorithms and main statements. Besides the greedy algorithm, local search algorithms have been used for the k-cover problem, in particular for small values of k. Pure local search starts with any cover and works in steps. In each step, the current solution is improved by replacing a constant number of sets with a (hopefully smaller number of other sets in order to obtain a new cover. Duh and Fürer introduced the technique of semi-local optimization which extends pure local search. In terms of 3-set cover, the main idea behind semi-local optimization is that once the sets of size 3 have been selected, computing the minimum number of sets of size 2 and in order to complete the covering can be done in polynomial time by a matching computation. Hence, a semi-local (s, t-improvement step for 3-set cover consists of the deletion of up to t 3-sets from the current cover and the insertion of up to s 3-sets and the minimum necessary 2-sets and -sets that complete the cover. The quality of an improvement is defined by the total number of sets in the cover while in case of two covers of the same size, the one with the smallest number of -sets is preferable. Semi-local optimization for k 4 is much similar; local improvements are now defined on sets of size at least 3 while 2-sets and -sets are globally changed. The analysis of [2] shows that the best choice of the parameters (s, t is (2,. Theorem (Duh and Fürer [2]. Consider an instance of k-set cover whose optimal solution has a i i-sets. Then, the semi-local (2, -optimization algorithm has cost at most a + a 2 + k i+ 3 a i. Proof (outline. The proof of [2] proceeds as follows. Let b i be the number of i- sets in the solution. First observe that k i= ib i = k i= ia i. Then, the following two properties of semi-local (2, -optimization are proved: b a and b +b 2 k i= a i. The theorem follows by summing the three inequalities.

4 This algorithm has been used as a basis for the following algorithms that approximate k-set cover. We will call them GSLI k,l and GRSLI k,l, respectively. Algorithm GSLI k,l. Greedy phases: For i = k down to l + do: Choose a maximal collection of i-sets. Semi-local optimization phase: Run the semi-local (2, -optimization algorithm on the remaining instance. Theorem 2 (Duh and Fürer [2]. Algorithm GSLI k,4 has approximation ratio H k 5/2. Algorithm GRSLI k,l Greedy phases: For i = k down to l + do: Choose a maximal collection of i-sets. Restricted phases: For i = l down to 4 do: Choose a maximal collection of disjoint i-sets so that the choice of these i-sets does not increase the number of -sets in the final solution. Semi-local optimization phase: Run the semi-local optimization algorithm on the remaining instance. Theorem 3 (Duh and Fürer [2]. Algorithm GRSLI k,5 ratio H k /2. has approximation We will modify the algorithms above by replacing each greedy phase with a packing phase for handling sets of not very small size. We use the local search algorithms of Hurkens and Schrijver [8] in each packing phase. The modified algorithms are called PSLI k,l and PRSLI k,l, respectively. A local search algorithm for set packing uses a constant parameter p (informally, this is an upper bound on the number of local improvements performed at each step and, starting with an empty packing Π, repeatedly updates Π by replacing any set of s < p sets of Π with s + sets so that feasibility is maintained and until no replacement is possible. Clearly, the algorithm runs in polynomial time. It has been analyzed in [8] (see also [5] for related investigations. Theorem 4 (Hurkens and Schrijver [8]. The local search t-set packing algorithm that performs at most p local improvements at each step has approximation ratio ρ t 2(t r t t(t r t, if p = 2r and ρ t 2(t r 2 t(t r 2, if p = 2r. As a corollary, for any constant ɛ > 0, we obtain a 2 ɛ t -approximation algorithm for t-set packing by using p = O(log t /ɛ local improvements. Our algorithms PSLI k,l and PRSLI k,l simply replace each of the greedy phases of the algorithms GSLI k,l and GRSLI k,l, respectively, with the following packing phase: Packing phases: For i = k down to l + do: Select a maximal collection of disjoint i-sets using a 2 ɛ i -approximation local search i-set packing algorithm.

5 Our first main result (Theorem 5 is a statement on the performance of algorithm PSLI k,l (for l = 4 which is the best choice. Theorem 5. For any constant ɛ > 0, algorithm PSLI k,4 has approximation ratio at most H k/ ɛ for even k 6, at most 2H k H k ɛ for k {5, 7, 9,, 3}, and at most H k k k + ɛ for odd k 5. Algorithm PSLI k,4 outperforms algorithm GSLI k,4 for k 5, algorithm GRSLI k,5 for k 9, as well as the improvement of Levin [2] for k 2. For large values of k, the approximation bound approaches H k c with c = ln 2 / Algorithm PSLI k,l has been mainly included here in order to introduce the analysis technique. As we will see in the next section, the factor-revealing LP is simpler in this case. Algorithm PRSLI k,l is even better; its performance (for l = 5 is stated in the following. Theorem 6. For any constant ɛ > 0, algorithm PRSLI k,5 has approximation ratio at most 2H k H k ɛ for odd k 7, at most ɛ for k = 6, and at most 2H k H k/ k k + ɛ for even k 8. Algorithm PRSLI k,5 achieves better approximation ratio than the algorithm of Levin [2] for every k 6. For example, the approximation ratio of for 6-set cover improves the previous bound of H For large values of k, the approximation bound approaches H k c with c = 77/60 ln See Table for a comparison between the algorithms discussed in this section. Table. Comparison of the approximation ratio of the algorithms GSLI k,4, PSLI k,4, GRSLI k,5, the algorithm in [2] and algorithm PRSLI k,5 for several values of k. k GSLI k,4 [2] PSLI k,4 GRSLI k,5 [2] [2] PRSLI k, large k H k H k H k 0.5 H k H k

6 3 Analysis through factor-revealing LPs Our proofs on the approximation guarantee of our algorithms essentially follow by computing upper bounds on the objective value of factor-revealing linear programs whose constraints capture simple invariants maintained in the phases of the algorithms. Consider an instance (V, S of k-set cover. For any phase of the algorithms associated with i (i = l,..., k for algorithm PSLI k,l and i = 3,..., k for algorithm PRSLI k,l, consider the instance (V i, S i where V i contains the elements in V that have not been covered in previous phases and S i contains the sets of S which contain only elements in V i. Denote by OPT i an optimal solution of instance (V i, S i ; we also denote the optimal solution OPT k of (V k, S k = (V, S by OPT. Since S is closed under subsets, without loss of generality, we may assume that OPT i contains disjoint sets. Furthermore, it is clear that OPT i OPT i for i k, i.e., OPT i OPT. For a phase of algorithm PSLI k,l or PRSLI k,l associated with i, denote by a i,j the ratio of the number of j-sets in OPT i over OPT. Since OPT i OPT, we obtain that a i,j. ( The i-set packing algorithm executed on packing phase associated with i includes in i-sets the elements in V i \V i. Since V i V i, their number is i V i \V i = V i V i = ja i,j OPT. (2 ja i,j Denote by ρ i the approximation ratio of the i-set packing algorithm executed on the phase associated with i. Since at the beginning of the packing phase associated with i, there exist at least a i,i OPT i-sets, the i-set packing algorithm computes at least ρ i a i,i OPT i-sets, i.e., covering at least iρ i a i,i OPT elements from sets in OPT i. Hence, V i \V i iρ i a i,i OPT, and (2 yields i i ja i,j ja i,j i( ρ i a i,i 0. (3 So far, we have defined all constraints for the factor-revealing LP of algorithm PSLI k,l. Next, we bound from above the number of sets computed by algorithm PSLI k,l as follows. Let t i be the number of i-sets computed by the i-set packing algorithm executed at the packing phase associated with i l +. Clearly, t i = i V i\v i = ja i,j i ja i,j OPT. (4 i i

7 By Theorem, we have that t l a l, + a l,2 + l j + 3 a l,j OPT. (5 j=3 Hence, by (4 and (5, it follows that the approximation guarantee of algorithm PSLI k,l is k i=4 t i OPT = k i=l+ i i ja i,j ja k,j + + l l + a l,2 + k i=l+ i(i + l ( j + 3 j=3 ja i,j + a l, + a l,2 + ja i,j + l l + a l, j l + l j=3 j + 3 a l,j a l,j (6 Hence, an upper bound on the approximation ratio of algorithm PSLI k,l follows by maximizing the right part of (6 subject to the constraints ( for i = l,..., k and (3 for i = l +,..., k with variables a i,j 0 for i = l,..., k and j =,..., i. Formally, we have proved the following statement. Lemma. The approximation ratio of algorithm PSLI k,l when a ρ i -approximation i-set packing algorithm is used at phase i for i = l +,..., k is upper-bounded by the maximum objective value of the following linear program: maximize k subject to ja k,j + + l l + a l,2 + k i=l+ i(i + l ( j + 3 j=3 a i,j, i = l,..., k ja i,j + l l + a l, j l + a l,j i i ja i,j ja i,j i( ρ i a i,i 0, i = l +,..., k a i,j 0, i = l,..., k, j =,..., i Each packing or restricted phase of algorithm PRSLI k,l satisfies (3; a restricted phase associated with i = 4,..., l computes a maximal i-set packing and, hence, ρ i = /i in this case. In addition, the restricted phases impose extra constraints. Denote by b, b 2, and b 3 the ratio of the number of -sets, 2-sets, and 3-sets computed by the

8 semi-local optimization phase over OPT, respectively. The restricted phases guarantee that the number of the -sets in the final solution does not increase, and, hence, b a i,, for i = 3,..., l. (7 Following the proof of Theorem, we obtain b + b 2 a 3, + a 3,2 + a 3,3 while it is clear that b + 2b 2 + 3b 3 = a 3, + 2a 3,2 + 3a 3,3. We obtain that the number t 3 of sets computed during the semi-local optimization phase of algorithm PRSLI k,l is t 3 = (b + b 2 + b 3 OPT ( b 3 + b + b 2 + b + 2b 2 + 3b ( 3 b a 3, + a 3, a 3,3 OPT OPT. (8 Reasoning as before, we obtain that (4 gives the number t i of i-sets computed during the packing or restricted phase associated with i = 4,..., k. By (4 and (8, we obtain that the performance guarantee of algorithm PRSLI k,l is k t i OPT = k i=4 i i ja i,j ja i,j k ja k,j + i(i + i= a 3, + a 3, a 3,3 + 3 b ja i,j a 3, + 2 a 3, a 3,3 + 3 b (9 Hence, an upper bound on the approximation ratio of algorithm PRSLI k,l follows by maximizing the right part of (9 subject to the constraints ( for i = 3,..., k, (3 for i = 4,..., k, and (7, with variables a i,j 0 for i = 3,..., k and j =,..., i, and b 0. Formally, we have proved the following statement. Lemma 2. The approximation ratio of algorithm PRSLI k,l when a ρ i -approximation i-set packing algorithm is used at phase i for i = l +,..., k is upper-bounded by the maximum objective value of the following linear program: maximize k subject to k ja k,j + i(i + i=4 a i,j, i = 3,..., k ja i,j a 3, + 2 a 3, a 3,3 + 3 b i i ja i,j ja i,j i( ρ i a i,i 0, i = l +,..., k

9 i i ja i,j ja i,j (i a i,i 0, i = 4,..., l b a i, 0, i = 3,..., l a i,j 0, i = 3,..., k, j =,..., i b 0 4 Proofs of main theorems We are now ready to prove our main results. We can show that the maximum objective values of the factor-revealing LPs for algorithms PSLI k,4 and PRSLI k,5 are upper-bounded by the values stated in Theorems 5 and 6. In order to prove this, it suffices to find feasible solutions to the dual LPs that have these values as objective values. In the following, we prove Theorem 5 by considering the case when k is even. The proof for odd k as well as the proof of Theorem 6 (which is slightly more complicated will appear in the final version of the paper. Proof of Theorem 5. The dual of the factor-revealing LP of algorithm PSLI k,4 is: minimize i=4 β i subject to β 4 + γ β 4 + 2γ β 4 + 3γ 5 5 β 4 + 4γ β i + jγ i+ jγ i j, i = 5,..., k, j =,..., i i(i + β i + iγ i+ (i 2 + ɛγ i, i = 5,..., k i + β k jγ k j, j =,..., k k β k (k 2 + ɛγ k β i 0, i = 4,..., k γ i 0, i = 5,..., k We consider only the case of even k. We set γ k = k(k and γ k = 0. If k 8, we set γ i = γ i i(i+(i+2 for i = 5,..., k 2. We also set β k = + (k 2 + ɛγ k, β 4 = 3 5 4γ 5 and β i = i + iγ i+ + (i 2 + ɛγ i

10 for i = 5,..., k. We will show that all the constraints of the dual LP are satisfied. Clearly, γ i 0 for i = 5,..., k. Observe that γ k +γ k = k(k. If k 8, by the definition of γ i for i = 5,..., k 2, we have that and, hence, γ i + γ i+ i(i + = γ 2 i+ + γ i+2 + i(i + (i + 2 i(i + = γ i+ + γ i+2 (i + (i + 2 γ i + γ i+ i(i + = γ k + γ k k(k = 0, i.e., γ i + γ i+ = i(i+ for i = 5,..., k. Now, the definition of β i s yields β i = i + iγ i+ + (i 2 + ɛγ i = i i(i + (i γ i+ + (i γ i + ɛγ i (0 j i(i + jγ i+ + jγ i 0 for i = 5,..., k and j =,..., i. Hence, all the constraints on β i i = 5,..., k are satisfied. The constraints on β k are also maintainted. Since γ k = k(k k for we have that β k = + (k 2 + ɛγ k k k + (k γ k + ɛγ k j k + jγ k 0 for j =,..., k. It remains to show that the constraints on β 4 are also satisfied. It suffices to show that γ 5 /45. This is clear when k = 6. If k 8, consider the equalities γ i + γ i+ = i(i+ for odd i = 5,..., k 3 and γ i γ i+ = i(i+ for even i = 6,..., k 2. Summing them, and since γ k = 0, we obtain that γ 5 = = = k/2 k/2 k/2 ( = k/ i(2i 2i(2i + ( 2i 2i 2i + 2i + ( 2i + k/2 2i + i 2 2i + k H k/

11 = 5 + 2H k 2 H k/ k H k/ = 2H k 2 2H k/2 + k ln /45 The first inequality follows since 2H k 2 2H k/2 + k is increasing on k (this can be easily seen by examining the original definition of γ 5 and since lim t H t / ln t =. We have shown that all the constraints of the dual LP are satisfied, i.e., the solution is feasible. In order to compute the objective value, we use the definition of β i s and equality (0. We obtain i=4 k/2 β i = β 4 + β 5 + (β 2i + β 2i+ + β k = 3 5 4γ γ 5 4γ 6 + k/2 ( 2i + 2iγ 2i+ + (2i 2γ 2i 2i + (2i + (2i + 2 2iγ 2i+2 + 2iγ 2i+ + + (k 2γ k + ɛ k/2 k/2 = 2 + i + 4γ 6 + ((2i 2γ 2i 2iγ 2i+2 + (k 2γ k +ɛ i=5 γ i H k/2 + /6 + ɛ where the last inequality follows since k i=5 γ i k i=5 i(i = k i=5 /4 /k. By duality, k the factor-revealing LP. The theorem follows by Lemma. ( i=5 γ i i i i=4 β i is an upper bound on the maximum objective value of = 5 Extensions We have experimentally verified using Matlab that our upper bounds are tight in the sense that they are the maximum objective values of the factor-revealing LPs (ignoring the ɛ term in the approximation bound. Our analysis technique can also be used to provide simpler proofs of the results in [2] (i.e., Theorems 2 and 3; this is left as an exercise to the reader. The several cases that are considered in the proofs of [2] are actually included as constraints in the factor-revealing

12 LPs which are much simpler than the ones for algorithms PSLI k,l and PRSLI k,l. Furthermore, note that we have not combined our techniques with the recent algorithm of Levin [2] that handles sets of size 4 using a restricted local search phase. It is tempting to conjecture that further improvements are possible. References. V. Chvátal. A greedy hueristic for the set-covering problem. Mathematics of Operations Research, 4, pp , R. Duh and M. Fürer. Approximation of k-set cover by semi local optimization. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC 97, pp , U. Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45(4, pp , O. Goldschmidt, D. Hochbaum, and G. Yu. A modified greedy heuristic for the set covering problem with improved worst case bound. Information Processing Letters, 48, pp , M. M. Halldórsson. Approximating discrete collections via local improvements. In Proceedings of the 6th Annual ACM/SIAM Symposium on Discrete Algorithms (SODA 95, pp , M. M. Halldórsson. Approximating k-set cover and complementary graph coloring. In Proceedings of the 5th Conference on Integer Programming and Combinatorial Optimization (IPCO 96, LNCS 084, Springer, pp. 8-3, E. Hazan, S. Safra, and O. Schwartz. On the complexity of approximating k-set packing. Computational Complexity, 5(, pp , C. A. J. Hurkens and A. Schrijver. On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM Journal on Discrete Mathematics, 2(, pp , K. Jain, M. Mahdian, E. Markakis, A. Saberi, and V. V. Vazirani. Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. Journal of the ACM, 50(6, pp , D. S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9, pp , V. Kann. Maximum bounded 3-dimensional matching is MAX SNP-complete. Information Processing Letters, 37, pp , A. Levin. Approximating the unweighted k-set cover problem: greedy meets local search. In Proceedings of the 4th International Workshop on Approximation and Online Algorithms (WAOA 06, LNCS 4368, Springer, pp , L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 3, pp , P. Slavík. A tight analysis of the greedy algorithm for set cover. Journal of Algorithms, 25, pp , L. Trevisan. Non-approximability results for optimization problems on bounded degree instances. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC 0, pp , 200.

### Energy-efficient communication in multi-interface wireless networks

Energy-efficient communication in multi-interface wireless networks Stavros Athanassopoulos, Ioannis Caragiannis, Christos Kaklamanis, and Evi Papaioannou Research Academic Computer Technology Institute

### Tight Bounds for Selfish and Greedy Load Balancing

Tight Bounds for Selfish and Greedy Load Balancing Ioannis Caragiannis 1, Michele Flammini, Christos Kaklamanis 1, Panagiotis Kanellopoulos 1, and Luca Moscardelli 1 Research Academic Computer Technology

### Competitive Analysis of On line Randomized Call Control in Cellular Networks

Competitive Analysis of On line Randomized Call Control in Cellular Networks Ioannis Caragiannis Christos Kaklamanis Evi Papaioannou Abstract In this paper we address an important communication issue arising

### Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

### Class constrained bin covering

Class constrained bin covering Leah Epstein Csanád Imreh Asaf Levin Abstract We study the following variant of the bin covering problem. We are given a set of unit sized items, where each item has a color

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### The Online Set Cover Problem

The Online Set Cover Problem Noga Alon Baruch Awerbuch Yossi Azar Niv Buchbinder Joseph Seffi Naor ABSTRACT Let X = {, 2,..., n} be a ground set of n elements, and let S be a family of subsets of X, S

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### The Generalized Assignment Problem with Minimum Quantities

The Generalized Assignment Problem with Minimum Quantities Sven O. Krumke a, Clemens Thielen a, a University of Kaiserslautern, Department of Mathematics Paul-Ehrlich-Str. 14, D-67663 Kaiserslautern, Germany

### Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA

A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce

### Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06

CS880: Approximations Algorithms Scribe: Matt Elder Lecturer: Shuchi Chawla Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06 3.1 Set Cover The Set Cover problem is: Given a set of

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### The Conference Call Search Problem in Wireless Networks

The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,

### Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs

Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Leah Epstein Magnús M. Halldórsson Asaf Levin Hadas Shachnai Abstract Motivated by applications in batch scheduling of jobs in manufacturing

### Scheduling to Minimize Power Consumption using Submodular Functions

Scheduling to Minimize Power Consumption using Submodular Functions Erik D. Demaine MIT edemaine@mit.edu Morteza Zadimoghaddam MIT morteza@mit.edu ABSTRACT We develop logarithmic approximation algorithms

### A2 1 10-Approximation Algorithm for a Generalization of the Weighted Edge-Dominating Set Problem

Journal of Combinatorial Optimization, 5, 317 326, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. A2 1 -Approximation Algorithm for a Generalization of the Weighted Edge-Dominating

### Local search for the minimum label spanning tree problem with bounded color classes

Available online at www.sciencedirect.com Operations Research Letters 31 (003) 195 01 Operations Research Letters www.elsevier.com/locate/dsw Local search for the minimum label spanning tree problem with

### The Relative Worst Order Ratio for On-Line Algorithms

The Relative Worst Order Ratio for On-Line Algorithms Joan Boyar 1 and Lene M. Favrholdt 2 1 Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark, joan@imada.sdu.dk

### Short Cycles make W-hard problems hard: FPT algorithms for W-hard Problems in Graphs with no short Cycles

Short Cycles make W-hard problems hard: FPT algorithms for W-hard Problems in Graphs with no short Cycles Venkatesh Raman and Saket Saurabh The Institute of Mathematical Sciences, Chennai 600 113. {vraman

### Online Scheduling with Bounded Migration

Online Scheduling with Bounded Migration Peter Sanders, Naveen Sivadasan, and Martin Skutella Max-Planck-Institut für Informatik, Saarbrücken, Germany, {sanders,ns,skutella}@mpi-sb.mpg.de Abstract. Consider

### Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

### A Note on Maximum Independent Sets in Rectangle Intersection Graphs

A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,

### 9th Max-Planck Advanced Course on the Foundations of Computer Science (ADFOCS) Primal-Dual Algorithms for Online Optimization: Lecture 1

9th Max-Planck Advanced Course on the Foundations of Computer Science (ADFOCS) Primal-Dual Algorithms for Online Optimization: Lecture 1 Seffi Naor Computer Science Dept. Technion Haifa, Israel Introduction

### The multi-integer set cover and the facility terminal cover problem

The multi-integer set cover and the facility teral cover problem Dorit S. Hochbaum Asaf Levin December 5, 2007 Abstract The facility teral cover problem is a generalization of the vertex cover problem.

### An Approximation Algorithm for Bounded Degree Deletion

An Approximation Algorithm for Bounded Degree Deletion Tomáš Ebenlendr Petr Kolman Jiří Sgall Abstract Bounded Degree Deletion is the following generalization of Vertex Cover. Given an undirected graph

### ARTICLE IN PRESS. European Journal of Operational Research xxx (2004) xxx xxx. Discrete Optimization. Nan Kong, Andrew J.

A factor 1 European Journal of Operational Research xxx (00) xxx xxx Discrete Optimization approximation algorithm for two-stage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of

### Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs

Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Leah Epstein Magnús M. Halldórsson Asaf Levin Hadas Shachnai Abstract Motivated by applications in batch scheduling of jobs in manufacturing

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

Load balancing of temporary tasks in the l p norm Yossi Azar a,1, Amir Epstein a,2, Leah Epstein b,3 a School of Computer Science, Tel Aviv University, Tel Aviv, Israel. b School of Computer Science, The

### Ecient approximation algorithm for minimizing makespan. on uniformly related machines. Chandra Chekuri. November 25, 1997.

Ecient approximation algorithm for minimizing makespan on uniformly related machines Chandra Chekuri November 25, 1997 Abstract We obtain a new ecient approximation algorithm for scheduling precedence

Online Adwords Allocation Shoshana Neuburger May 6, 2009 1 Overview Many search engines auction the advertising space alongside search results. When Google interviewed Amin Saberi in 2004, their advertisement

### Improved Algorithms for Data Migration

Improved Algorithms for Data Migration Samir Khuller 1, Yoo-Ah Kim, and Azarakhsh Malekian 1 Department of Computer Science, University of Maryland, College Park, MD 20742. Research supported by NSF Award

### THE SCHEDULING OF MAINTENANCE SERVICE

THE SCHEDULING OF MAINTENANCE SERVICE Shoshana Anily Celia A. Glass Refael Hassin Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single

### Resource Allocation with Time Intervals

Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes

### CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

### Minimum Makespan Scheduling

Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,

### Finding and counting given length cycles

Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected

### Private Approximation of Clustering and Vertex Cover

Private Approximation of Clustering and Vertex Cover Amos Beimel, Renen Hallak, and Kobbi Nissim Department of Computer Science, Ben-Gurion University of the Negev Abstract. Private approximation of search

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

### On Competitiveness in Uniform Utility Allocation Markets

On Competitiveness in Uniform Utility Allocation Markets Deeparnab Chakrabarty Department of Combinatorics and Optimization University of Waterloo Nikhil Devanur Microsoft Research, Redmond, Washington,

### GENERATING LOW-DEGREE 2-SPANNERS

SIAM J. COMPUT. c 1998 Society for Industrial and Applied Mathematics Vol. 27, No. 5, pp. 1438 1456, October 1998 013 GENERATING LOW-DEGREE 2-SPANNERS GUY KORTSARZ AND DAVID PELEG Abstract. A k-spanner

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### A constant-factor approximation algorithm for the k-median problem

A constant-factor approximation algorithm for the k-median problem Moses Charikar Sudipto Guha Éva Tardos David B. Shmoys July 23, 2002 Abstract We present the first constant-factor approximation algorithm

### Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

### Completion Time Scheduling and the WSRPT Algorithm

Completion Time Scheduling and the WSRPT Algorithm Bo Xiong, Christine Chung Department of Computer Science, Connecticut College, New London, CT {bxiong,cchung}@conncoll.edu Abstract. We consider the online

### Online Degree-Bounded Steiner Network Design

Online Degree-Bounded Steiner Network Design The problem of satisfying connectivity demands on a graph while respecting given constraints has been a pillar of the area of network design since the early

### Cycle transversals in bounded degree graphs

Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.

### Cross-Layer Survivability in WDM-Based Networks

Cross-Layer Survivability in WDM-Based Networks Kayi Lee, Member, IEEE, Eytan Modiano, Senior Member, IEEE, Hyang-Won Lee, Member, IEEE, Abstract In layered networks, a single failure at a lower layer

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### A Constant Factor Approximation Algorithm for the Storage Allocation Problem

A Constant Factor Approximation Algorithm for the Storage Allocation Problem Reuven Bar-Yehuda reuven@cs.technion.ac.il Dror Rawitz rawitz@eng.tau.ac.il Michael Beder bederml@cs.technion.ac.il Abstract

### Online Optimization with Uncertain Information

Online Optimization with Uncertain Information Mohammad Mahdian Yahoo! Research mahdian@yahoo-inc.com Hamid Nazerzadeh Stanford University hamidnz@stanford.edu Amin Saberi Stanford University saberi@stanford.edu

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### A Near-linear Time Constant Factor Algorithm for Unsplittable Flow Problem on Line with Bag Constraints

A Near-linear Time Constant Factor Algorithm for Unsplittable Flow Problem on Line with Bag Constraints Venkatesan T. Chakaravarthy, Anamitra R. Choudhury, and Yogish Sabharwal IBM Research - India, New

### JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

Advertisement Allocation for Generalized Second Pricing Schemes Ashish Goel Mohammad Mahdian Hamid Nazerzadeh Amin Saberi Abstract Recently, there has been a surge of interest in algorithms that allocate

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### The Approximability of the Binary Paintshop Problem

The Approximability of the Binary Paintshop Problem Anupam Gupta 1, Satyen Kale 2, Viswanath Nagarajan 2, Rishi Saket 2, and Baruch Schieber 2 1 Dept. of Computer Science, Carnegie Mellon University, Pittsburgh

### A simpler and better derandomization of an approximation algorithm for Single Source Rent-or-Buy

A simpler and better derandomization of an approximation algorithm for Single Source Rent-or-Buy David P. Williamson Anke van Zuylen School of Operations Research and Industrial Engineering, Cornell University,

### Offline sorting buffers on Line

Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

### Algorithm Design and Analysis

Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

### Energy Efficient Monitoring in Sensor Networks

Energy Efficient Monitoring in Sensor Networks Amol Deshpande, Samir Khuller, Azarakhsh Malekian, Mohammed Toossi Computer Science Department, University of Maryland, A.V. Williams Building, College Park,

### Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

### Scheduling Parallel Jobs with Linear Speedup

Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev,m.uetz}@ke.unimaas.nl

### Classification - Examples

Lecture 2 Scheduling 1 Classification - Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking

### Approximating the Online Set Multicover Problems Via Randomized Winnowing

Approximating the Online Set Multicover Problems Via Randomized Winnowing Piotr Berman 1 and Bhaskar DasGupta 2 1 Department of Computer Science and Engineering, Pennsylvania State University, University

### Survey of Terrain Guarding and Art Gallery Problems

Survey of Terrain Guarding and Art Gallery Problems Erik Krohn Abstract The terrain guarding problem and art gallery problem are two areas in computational geometry. Different versions of terrain guarding

### LOCAL SEARCH HEURISTICS FOR k-median AND FACILITY LOCATION PROBLEMS

SIAM J. COMPUT. Vol. 33, No. 3, pp. 544 562 c 2004 Society for Industrial and Applied Mathematics LOCAL SEARCH HEURISTICS FOR k-median AND FACILITY LOCATION PROBLEMS VIJAY ARYA, NAVEEN GARG, ROHIT KHANDEKAR,

### Factoring & Primality

Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount

### A Combinatorial Allocation Mechanism With Penalties For Banner Advertising

A Combinatorial Allocation Mechanism With Penalties For Banner Advertising Uriel Feige Weizmann Institute Rehovot 76100, Israel uriel.feige@weizmann.ac.il Nicole Immorlica Centrum voor Wisunde en Informatica

### Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue

Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue Niv Buchbinder 1, Kamal Jain 2, and Joseph (Seffi) Naor 1 1 Computer Science Department, Technion, Haifa, Israel. 2 Microsoft Research,

### Structural properties of oracle classes

Structural properties of oracle classes Stanislav Živný Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK Abstract Denote by C the class of oracles relative

### CSC2420 Spring 2015: Lecture 3

CSC2420 Spring 2015: Lecture 3 Allan Borodin January 22, 2015 1 / 1 Announcements and todays agenda Assignment 1 due next Thursday. I may add one or two additional questions today or tomorrow. Todays agenda

20 Selfish Load Balancing Berthold Vöcking Abstract Suppose that a set of weighted tasks shall be assigned to a set of machines with possibly different speeds such that the load is distributed evenly among

### Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] 1.1-1.7 Exercises: Do before next class; not to hand in [J] pp. 12-14:

### Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal

Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal ABSTACT Mohit Singh Tepper School of Business Carnegie Mellon University Pittsburgh, PA USA mohits@andrew.cmu.edu In the MINIMUM

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### Scheduling Parallel Jobs with Monotone Speedup 1

Scheduling Parallel Jobs with Monotone Speedup 1 Alexander Grigoriev, Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands, {a.grigoriev@ke.unimaas.nl,

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### A General Purpose Local Search Algorithm for Binary Optimization

A General Purpose Local Search Algorithm for Binary Optimization Dimitris Bertsimas, Dan Iancu, Dmitriy Katz Operations Research Center, Massachusetts Institute of Technology, E40-147, Cambridge, MA 02139,

### CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma

CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma Please Note: The references at the end are given for extra reading if you are interested in exploring these ideas further. You are

### Frequency Capping in Online Advertising

Frequency Capping in Online Advertising (Extended Abstract) Niv Buchbinder 1, Moran Feldman 2, Arpita Ghosh 3, and Joseph (Seffi) Naor 2 1 Open University, Israel niv.buchbinder@gmail.com 2 Computer Science

### Minimize subject to. x S R

Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

### Online Bipartite Perfect Matching With Augmentations

Online Bipartite Perfect Matching With Augmentations Kamalika Chaudhuri, Constantinos Daskalakis, Robert D. Kleinberg, and Henry Lin Information Theory and Applications Center, U.C. San Diego Email: kamalika@soe.ucsd.edu

### ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015

ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Online Scheduling for Cloud Computing and Different Service Levels

2012 IEEE 201226th IEEE International 26th International Parallel Parallel and Distributed and Distributed Processing Processing Symposium Symposium Workshops Workshops & PhD Forum Online Scheduling for

### Minimal Cost Reconfiguration of Data Placement in a Storage Area Network

Minimal Cost Reconfiguration of Data Placement in a Storage Area Network Hadas Shachnai Gal Tamir Tami Tamir Abstract Video-on-Demand (VoD) services require frequent updates in file configuration on the

### Introduction to Algorithms. Part 3: P, NP Hard Problems

Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NP-Completeness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter

### What s next? Reductions other than kernelization

What s next? Reductions other than kernelization Dániel Marx Humboldt-Universität zu Berlin (with help from Fedor Fomin, Daniel Lokshtanov and Saket Saurabh) WorKer 2010: Workshop on Kernelization Nov

### Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

### International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,