I. Dinur and S. Safra. On the importance of being biased. Proceedings of the 34rd ACM Symposium on Theory of Computing, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....k center is 46 n) hard to approximate, unless NP DT IME(n ) We note here that Halperin, Kortsarz, and Krauthgamer [7] have informed us that they have obtained a similar result. Our results build on the sequence of recent results leading to a (k 1 ) hardness for k hypergraph covering [6, 5, 3, 4]. The result of [4] can be viewed as a construction of an instance of Set Cover from an instance of Gap 3SAT(3) with a strong bi criterion gap. If the formula is satis able then a 1= k 1 ) fraction of the sets are sucient to cover all the elements. If the formula is unsatis able then a fraction ....

I. Dinur and S. Safra. The importance of being biased. Proceedings of the 34 ACM Symposium on Theory of Computing, 2002.

....vertex cover is NP hard to approximate within 2 for any 0. This is one of the major open questions in the eld of approximation algorithms. In [11] H astad showed that approximating vertex cover within constant factors less than 6 is NP hard. This was recently improved by Dinur and Safra [5] to 1:36. In a related result, Arora et al. 1] considered algorithms based on linear programming. They showed an integrality gap of 2 for a large family of linear programs for vertex cover. This implies that many linear programming based algorithms cannot obtain an approximation ratio better ....

....a 2 prover 1 round game and the Inner PCP is based on long codes and often, the Fourier analysis of long codes. However, this standard recipe hasn t been very successful in attacking the vertex cover problem. H astad s 6 hardness remained the best known result for a long time. Dinur and Safra [5] were able to break this barrier by relying on techniques from extremal combinatorics. However, their approach still doesn t succeed in getting a hardness factor better than 1:36. Khot [15] observed that the bottleneck in getting hardness results for vertex cover and a number of other problems ....

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I. Dinur and S. Safra. On the importance of being biased. In Proc. 34th ACM Symp. on Theory of Computing (STOC), 2002.

....in combinatorial optimization. A great number of researchers believe that there is no polynomial time approximation algorithm achieving an approximation factor strictly smaller than 2 #, for a positive constant #, unless P = NP . Currently, the best known lower bound for this factor is 1. 36 [DS02] and the best upper bound is 2 which can be obtained easily. The best current fixed parameter tractable algorithm has time O(1.271 k V ) CKJ99] In this section, we present an exponentially faster algorithm for this problem on clique sum graphs. Without loss of generality, we can ....

Irit Dinur and Shmuel Safra. The importance of being biased. In Annual ACM Symposium on Theory of Computing (Montreal, 2002), pages 33--42. 2002.

....can be naturally posed as questions concerning the simultaneous solvability of families of equations over finite groups. This connection has been exploited to achieve a variety of strong inapproximability results for problems such as Max Cut, Max Di Cut, Exact Satisfiability, and Vertex Cover [8,10,11,15 17,20,26]. A chief technical ingredient in these hardness results is a tight lower bound on the approximability of Email addresses: enge kth.se (Lars Engebretsen) joho kth.se (Jonas Holmerin) acr cse.uconn.edu (Alexander Russell) Research partly performed at MIT with support from the Marcus ....

I. Dinur, S. Safra, The importance of being biased, in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, Montral, Quebc, Canada, 2002, pp. 33--42.

....problems. However, for many other problems including all four problems mentioned in the opening paragraph, the PCP based results are fairly weak or nonexistent, and fall well below the integrality gaps of the best relaxations. The best hardness result for Vertex Cover due to Dinur and Safra [6], who improved upon a long line of work only shows that 1.36 approximation is NP hard. The best hardness result for metric TSP only shows that 1.01approximation is NP hard [16] yet decades of work has failed to yield a relaxation with integrality gap better than 1.5 [20] or 4 3, if one ....

....(namely, they apply to LPs that we do not know how to solve in 2 time) that they may be seen as complementary to PCPbased results. Even if it were shown using PCPs that (2 #) approximation to vertex cover is NPhard, the proof would probably involve even more complex reductions than those in [6]. Thus it might reduce 3SAT formulae of size n to vertex cover on graphs of size n c , where c is astronomical. Even if we assume 3SAT has no 2 time algorithms, such a reduction would not rule out integrality gap of 1.1 (say) for the relaxations in Section 2. In other words, even in a world ....

I. Dinur and S. Safra. The importance of being biased. Proc. ACM STOC, 2002.

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I. Dinur and S. Safra, On the importance of being biased, In Proc. 34th ACM Symp. on Theory of Computing, 2002.

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I. Dinur, S. Safra, On the Importance of Being Biased, Proc. of the 34 ACM STOC, ACM Press (2002), 33--42.

....inapproximability results for small k values may be more di#cult. This is the case for low bound values instances of other problems (for example, for the Vertex Cover problem, a long line of complex proofs and ideas let to the current state of inapproximability factor slightly larger then [DS02] versus an approximation ratio of 2) The current gap between known approximation ratios and unapproximability factors for k DM is significant. For k = 4 the current gap is 2 vs. and for k = 3 the known approximation ratio is but no explicit inapproximability factor is known to date . A ....

I. Dinur and S. Safra. The importance of being biased. In Proc. of the 34rd ACM STOC, 2002.

....the Boolean functions, in which every pair of functions disagree on at least the distance d of the functions plausible inputs. The Hadamard code then consists of all multiplicative functions, that is, all characters. The long code, which has been used in numerous hardness of approximation results [BGS98, Has97, Has99, DS02, Kho02], would comprise all increasing dictatorship (all functions f(x) x i ) When using the long code to prove hardness of approximation, one is typically required to show that a Boolean function that satisfies a certain condition, must have a short list decoding as a code word. The su#cient ....

....condition above, for a function to be close to a junta, implies exactly that. The code word associated with any variable i [n] cannot have a nonnegligible correlation with f unless it is one of the variables in the junta determining f. The biased distribution. We note that for some results (e.g. [DS02]) the correlation is weighted according to the p biased measure on , for which Theorem 1 does not extend (as shown below) In the p biased measure over the binary cube, each variable is independently set to 1 with probability p and to 1 with probability 1 p, namely p (x) p x (1 ....

I. Dinur and S. Safra. On the importance of being biased. In Proc. 34th ACM Symp. on Theory of Computing, 2002.

....and instead of Fourier analysis uses some properties concerning intersecting families of nite sets. The authors also give a more complicated reduction that shows a factor (k 3 ) hardness for Ek Vertex Cover. The crucial impetus for that work came from the recent result of Dinur and Safra [7] on the hardness of approximating vertex cover (on graphs) and as in [7] the notion of biased long codes and some extremal combinatorics relating to intersecting families of sets play an important role. In addition to ideas from [7] the factor (k 3 ) hardness result also exploits the notion of ....

....families of nite sets. The authors also give a more complicated reduction that shows a factor (k 3 ) hardness for Ek Vertex Cover. The crucial impetus for that work came from the recent result of Dinur and Safra [7] on the hardness of approximating vertex cover (on graphs) and as in [7] the notion of biased long codes and some extremal combinatorics relating to intersecting families of sets play an important role. In addition to ideas from [7] the factor (k 3 ) hardness result also exploits the notion of covering complexity introduced by Guruswami, H astad and Sudan [11] Both ....

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I. Dinur and S. Safra. The importance of being biased. Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 33-42, May 2002.

....of parts increases, the number of colors required to color an unsatis able instance increases as well. The Long Code and the Kneser Graph. The Long Code [4] of a domain R is a powerful tool in numerous hardness of approximation results. We adopt a completely combinatorial viewpoint, following [7], and discover a new connection between the Long Code and a well known combinatorial object called the Kneser graph [19] The Long Code of a domain R consists of all possible subsets of R. The Long Code Graph, explicitly de ned in [7] is the graph whose 2 vertices are the subsets of R, ....

....We adopt a completely combinatorial viewpoint, following [7] and discover a new connection between the Long Code and a well known combinatorial object called the Kneser graph [19] The Long Code of a domain R consists of all possible subsets of R. The Long Code Graph, explicitly de ned in [7], is the graph whose 2 vertices are the subsets of R, and whose edges connect disjoint subsets. Let us consider the following encoding of an element a 2 R: color all subsets containing a red, and the rest orange. This coloring corresponds to the legal encoding of a 2 R via the Long Code. It is ....

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I. Dinur and S. Safra. On the importance of being biased. In Proc. 34th ACM Symp. on Theory of Computing, 2002.

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I. Dinur and S. Safra. On the importance of being biased. Proceedings of the 34rd ACM Symposium on Theory of Computing, 2002.

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I. Dinur and S. Safra. The importance of being biased. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Montr eal, Canada, pages 33--42, 2002.

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I. Dinur and S. Safra. The importance of being biased. In Proceedings of STOC'02, pages 33--42, 2002.

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Irit Dinur and Shmuel Safra. The importance of being biased. In Proceedings of the 34th Symposium on Theory of Computing, pages 33--42. ACM Press, 2002.

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I. Dinur and S. Safra. On the importance of being biased. Proceedings of the 34rd ACM Symposium on Theory of Computing, 2002.

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Irit Dinur and Shmuel Safra. The importance of being biased. In Annual ACM Symposium on Theory of Computing (Montreal, 2002). To appear. 20

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Irit Dinur and Shmuel Safra. The importance of being biased. Manuscript, October 01.

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I. Dinur and S. Safra. The importance of being biased. In Proc. 34th Ann. ACM Symp. on the Theory of Computing, pages 33--42, 2002.

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I. Dinur and S. Safra. The importance of being biased. In Proc. 34th ACM Symp. on the Theory of Computing (STOC 2002.

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I. Dinur, S. Safra, The importance of being biased, in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, Montral, Quebc, Canada, 2002, pp. 3342.

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Irit Dinur and Shmuel Safra. The importance of being biased. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 3342. Montral, Qubec, Canada, 1921 May 2002. 143

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I. Dinur, S. Safra, The importance of being biased, in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, Montral, Quebc, Canada, 2002, pp. 3342.

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I. Dinur and S. Safra. The importance of being biased. In Proc. 34th Ann. ACM Symp. on the Theory of Computing, pages 33--42, 2002.

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Irit Dinur and Shmuel Safra. The importance of being biased. In Annual ACM Symposium on Theory of Computing (Montral, 2002). To appear.

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