S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica 20:393-415, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....within 7 5 #, for any # 0. We also show that the Matrix To Line# problem cannot be approximated within 2 # in polynomial time, unless 3 colorable graphs can be colored with colors in polynomial time. Khanna et al. showed that it is NP hard to find a 4 coloring of a 3 colorable graph [KLS93] The problem of k coloring a 3 colorable graph is not known to be NP hard for k 5. However, it is a very well studied problem, and despite this there is currently no polynomial time algorithm that colors a 3 colorable graph with less than O( V polylog( V ) colors, which is accomplished ....

Sanjeev Khanna, Nathan Linial, and Shmuel Safra. On the hardness of approximating the chromatic number. In ISTCS93, 1993.

....for any constant f and some arbitrarily small constant # 0. 3.2 Connection Between PCPs and Min Chromatic Number Lund and Yannakakis [19] reduced Min Chromatic Number to Max Clique. This reduction, which did not preserve the ratio of approximation, was improved by Khanna, Linial, and Safra [17] and Bellare and Sudan [5] As a final improvement, Furer [12] constructed a randomized reduction showing that if Max Clique cannot be approximated within n 1 (1 f) then Min Chromatic Number cannot be approximated within n min 1 2,1 (1 2f) o(1) In the reduction used above to prove strong ....

Sanjeev Khanna, Nathan Linial, and Shmuel Safra. On the hardness of approximating the chromatic number. In Proceedings of 2nd Israel Symposium on Theory of Computing and Systems, pages 250--260, Natanya, Israel, 1993. IEEE Computer Society.

....3 uniform hypergraphs requires O(n colors [20] and the best coloring algorithm for 3 colorable graphs, requires O(n ) colors [5] On the lower bound side, not much is known. For graphs, the best hardness result states that using 4 colors to color a 3 colorable graph is NP hard [17, 13]. It would already be a signi cant step to prove that coloring a 3 colorable graph with O(1) colors is NP hard. The property of being 2 colorable is well studied in combinatorics and is also referred to as property B (see [14] for further references) Nevertheless, prior to this work no ....

Sanjeev Khanna, Nathan Linial, and Shmuel Safra. On the hardness of approximating the chromatic number. Combinatorica, 20(3):393-415, 2000.

....to within a factor n for some # 0 would imply P = RP. There is still some room for improvement of the results of Karger et al. for fixed values of k the best related hardness result states that all 3 colourable graphs cannot be coloured in polynomial time using 4 colours, unless P = NP [20]. The best known (exponential) algorithm for exact 3 colouring runs in O(1.3446 ) time [4] The graph colouring problem for k colourable graphs can be seen as a special case of the unweighted MAX k CUT problem. Approximation algorithms for the MAX k CUT problem assign a colour from a set of ....

S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica, 20:393--415, 2000.

....to Section 2.4.2 for the definitions of the problems. The conclusion for Max Clique follows, of course, from the FGLSS reduction and the first proof system listed above. The conclusion for the Chromatic Number follows from a recent reduction of Furer [Fu] which in turn builds on reductions in [LuYa, KLS, BeSu]. Furer s work and ours are contemporaneous and thus we view the N hardness result as jointly due to both papers. The improvements for the MaxSNP problems are perhaps more significant than the Max Clique one: We see hardness results for MaxSNP problems that are comparable to the factors ....

....ALMSS] this implies the chromatic number is hard to approximate within N for some ffi 0. But, again, ffi is very small. Improvements to ffi were derived both by improvements to ffl and improvements to the function h used by the reduction. A subsequent reduction of Khanna, Linial and Safra [KLS] is simpler but in fact slightly less efficient, having h(ffl) ffl= 5 ffl) A more efficient reduction is given by [BeSu] they present a reduction obtaining h(ffl) ffl= 3 Gamma 2ffl) Our N hardness for Clique would yield, via this, a N 1=7 hardness for the chromatic number. But more ....

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S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. Proceedings of the Second Israel Symposium on Theory and Computing Systems, IEEE, 1993, pp. 250--260.

....to this, we show that the Matrix To Line problem cannot be approximated within a factor 4 3 unless P=NP. We also show that the MatrixTo Line problem cannot be approximated within 2 # in polynomial time, colors in polynomial time. It is NP hard to find a 4 coloring of a 3 colorable graph [9]. The problem of k coloring a 3 colorable graph is not known to be NP hard for k 5. However, it is a very well studied problem and despite this there is currently no polynomial time algorithm that colors a 3 colorable graph with less than ) colors [4] Su#cient conditions and ....

....that if Matrix To Line is approximable within 2 # in polynomial time, then every 3 colorable graph can be #4 ## colored in polynomial time. The problem of k coloring a 3 colorable graph is a well studied problem. The problem is not known to be NP hard. In fact, the best result so far is from [9] where they show that it is NP hard to find a 4 coloring of a 3 colorable graph. However, the best approximation algorithm known for the corresponding optimization problem Minimum Graph Coloring for 3 colorable graphs, has performance ratio ) 4] Our lower bound of 4 3 is obtained by a ....

Sanjeev Khanna, Nathan Linial, and Shmuel Safra. On the hardness of approximating the chromatic number. In ISTCS93, 1993.

....(2; n ; n 1 ) approximation problem is NP hard under randomized reductions. However, for graphs whose chromatic number is a small constant, the known hardness results are much weaker. For example, for 3 colorable graphs the best known hardness result only rules out coloring using 4 colors [20, 16]. This paper is motivated by the quest for strong (super constant) inapproximability for coloring graphs whose chromatic number is a small constant. We do not get such results for graph coloring, but do get such inapproximability results for hypergraph coloring and in particular for coloring ....

....3 colorable graphs, a natural question arises: Are PCPs really necessary to show such hardness results, or would something weaker suce To date there are no reasons showing PCPs are necessary. And while the rst result showing the intractability of coloring 3 colorable graphs with 4 colors [20] did use the PCP technique, 16] show that PCPs are not needed in this result. The starting point of our work is the observation that covering PCPs are indeed necessary for showing strong hardness results for graph coloring. Speci cally, in Proposition 2.1, we show that if the (2; c; ....

S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. Combinatorica, 20(3) (2000), pp. 393-415.

....a number of linear programs obtained from a gadget construction. We also show that the Matrix To Line problem cannot be approximated within 2 in polynomial time, unless 3 colorable graphs can be colored with d4=e colors in polynomial time. It is NP hard to find a 4coloring of a 3 colorable graph [9]. The problem of k coloring a 3 colorable graph is not known to be NP hard for k 5. However, it is a very well studied problem, and despite this there is currently no polynomial time algorithm that colors a 3 colorable graph with less than Sufficient conditions and non polynomial time ....

....we show that if Matrix To Line is approximable within 2 in polynomial time, then every 3 colorable graph can be d4=e colored in polynomial time. The problem of k coloring a 3 colorable graph is a well studied problem. The problem is not known to be NP hard. The best result so far is from [9] where they show that it is NP hard to find a 4 coloring of a 3 colorable graph. However, the best approximation algorithm known for the corresponding optimization problem Minimum Graph Coloring for 3 colorable graphs, has performance ratio ) 4] i.e. O(n log k (n) for some constant ....

Sanjeev Khanna, Nathan Linial, and Shmuel Safra. On the hardness of approximating the chromatic number. In Proceedings of the 2nd Israel Symposium on Theory and Computing Systems, ISTCS, pages 250--260. IEEE Computer Society Press, 1993.

....very large independent sets. The extension of the Alon and Kahale result may be of independent interest. 1 Introduction Finding a 3 coloring of a given 3 colorable graph is a well known NP hard problem. Finding a 4 coloring of such a graph is also known to be NP hard (Khanna, Linial and Safra [KLS00] and Guruswami and Khanna [GK00] Karger, Motwani and Sudan [KMS98] show, on the other hand, using semidefinite programming, that a 3 colorable graph on n vertices with maximum degree Delta can be colored, in polynomial time, using School of Computer Science, Tel Aviv University, Tel Aviv 69978, ....

.... the chromatic number of general graphs cannot be approximated, in polynomial time, to within a ratio of n 1 Gammaffl , for every ffl 0, unless NP = RP (Feige and Killian [FK96] It is only known, however, that coloring 3 colorable graphs using 4 colors in NP hard (Khanna, Linial and Safra [KLS00] and Guruswami and Khanna [GK00] Obtaining improved hardness results for coloring 3colorable graphs is a challenging open problem. Another interesting problem is the following: how large can the chromatic number of vector 3 colorable (or vector k colorable) graphs be See [KMS98] for a ....

S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica, 20:393--415, 2000.

....there is no polynomial time algorithm to approximate the Chromatic Number within N o(1) for = 1= a bf ) Thus the problem is to design (a; b) chromatic number reductions with a; b as small as possible. The reduction of [20] achieves a = 1 and b = 5. A simple reduction supplied later by [17] is slightly less ecient, achieving a = 6 and b = 5. Applying this and Theorem 1.4 we can conclude that approximating Chromatic Number within N 1 16 o(1) is hard, which is already better than the best previous hardness factor of N 1=71 due to [13] However, we have the following improvement ....

....is already better than the best previous hardness factor of N 1=71 due to [13] However, we have the following improvement in the reduction. Theorem 1.5 There is a (a; b) chromatic number reduction achieving a = 1 and b = 3. Theorem 1.2 follows. Our reduction is an extension of the one of [17], staying within the same framework but using linear algebra and coding theory techniques to implement shifts in a di erent way. See Section 4. 1.3 History and explanations Non approximability results based on PCPs begin with [12] They have since been proved under many di erent assumptions. ....

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S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. ISTCS 93.

....is at least 1=s. The class cPCP arises naturally in the study of certain minimization problems, and in particular in the study of the approximability of graph coloring. Traditionally, however the class has not been focussed on explicitly. Instead all previous (PCP based) results on graph coloring [20, 17, 10] have implicitly relied on the obvious containment PCP 1;s [r; q] cPCP 1;s [r; q] Thus it sufficed to prove strong containments of NP in PCP to get hardness result for graph coloring. This approach was quite successful in proving strong (and in fact essentially tight) inapproximability of ....

....essentially tight) inapproximability of graph coloring for general graphs [10] but for graphs whose chromatic number is a small constant, however, the known hardness results are much weaker. For example, for 3 colorable graphs the best known hardness result only rules out coloring using 4 colors [17, 14]. This paper is motivated by the quest for strong (superconstant) inapproximability for coloring graphs whose chromatic number is a small constant, and the kind of PCP constructions that this question motivates. A necessary (but not sufficient condition) for such a result is a containment of NP in ....

S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. In Proceedings of the 2nd Israel Symposium on Theory and Computing Systems, ISTCS, pp. 250-260, IEEE Computer Society Press, 1993.

....is at least 1=s. The class cPCP arises naturally in the study of certain minimization problems, and in particular in the study of the approximability of graph coloring. Traditionally, however the class has not been focussed on explicitly. Instead all previous (PCP based) results on graph coloring [21, 18, 10] have implicitly relied on the obvious containment PCP 1;s [r; q] cPCP 1;s [r; q] Thus it sufficed to prove strong containments of NP in PCP to get hardness result for graph coloring. This approach was quite successful in proving strong (and in fact essentially tight) inapproximability of ....

....essentially tight) inapproximability of graph coloring for general graphs [10] but for graphs whose chromatic number is a small constant, however, the known hardness results are much weaker. For example, for 3 colorable graphs the best known hardness result only rules out coloring using 4 colors [18, 14]. This paper is motivated by the quest for strong (super constant) inapproximability for coloring graphs whose chromatic number is a small constant, and the kind of PCP constructions that this question motivates. A necessary (but not sufficient condition) for such a result is a containment of NP ....

S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. In Proceedings of the 2nd Israel Symposium on Theory and Computing Systems, ISTCS, pp. 250-260, IEEE Computer Society Press, 1993.

....N 1 3 . Theorem 1.3 For every 0, NP FPCP[ log; 2 ] Thus the Max Clique size of an N vertex graph is NP hard (under randomized Karp reductions) to approximate within N 1=3 . Combined with a recent reduction by Furer [Fu] which in turn builds upon the reductions presented in [LuYa, KLS, BeSu], we get: Theorem 1.4 For any 0, it is NP Hard (under randomized Karp reductions) to approximate the chromatic number of an N vertex graph to within N 1 5 . Previous improvements in the efficiency of proof systems [BFL, BFLS, FGLSS, ArSa, ALMSS, BGLR, FeKi, BeSu] had required ....

S. KHANNA, N. LINIAL AND S. SAFRA. On the hardness of approximating the chromatic number. ISTCS, 1993.

....to 1=10 unless NQP 6= co RQP, and to 1=13 unless NP = co RP. However, none of these hardness results apply to the special case of the problem where the input graph is guaranteed to be k colorable for some small k. The best hardness result in this direction is due to Khanna, Linial, and Safra [26] who show that it is not possible to color a 3 colorable graph with 4 colors in polynomial time unless P = NP. In this paper we present improvements on the result of Blum. In particular, we provide a randomized polynomial time algorithm which colors a 3 colorable graph of maximum degree with ....

S. Khanna, N. Linial, and S. Safra. On the Hardness of Approximating the Chromatic Number. In Proceedings 2nd Israeli Symposium on Theory and Computing Systems, pp. 250--260, 1992.

....Lund and Yannakakis [79] show hardness results for approximation versions of a large set of maximum subgraph problems. These problems involve nding the largest subgraph that satis es a a property , where is a nontrivial graph property closed under vertex deletion. Khanna, Linial and Safra [71] study the hardness of coloring 3 colorable graph. They show that coloring a 3 colorable graph with 4 colors is NP hard. Arora, Babai, Stern, and Sweedyk [3] prove hardness results for a collection of problems involving integral lattices, codes, or linear equations inequations. These include ....

S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Proceedings of the Second Israel Symposium on Theory and Computing Systems, 1993.

....[9] that, for any xed 0, the chromatic number of graphs on n vertices is not approximable within a factor of n 1 unless NP = ZPP . If a graph on n vertices is 3 colorable, then one can color it using O(n 3=14 log O(1) n) colors [5] but it is NP hard to color it using four colors [17]. As for approximating the independence number of a graph, Boppana and Halld orsson presented an algorithm ( 6] with approximation ratio O(n= log 2 n) for graphs on n vertices, based on the so called Local Ratio Approach, to be discussed later in this paper. If a graph contains an independent ....

S. Khanna, N. Linial and M. Safra, On the hardness of approximating the chromatic number, Proc. 2 nd Israeli Symposium on Theor. Comp. Sci., IEEE (1992), 250-260.

....the Hardness of 4 coloring a 3 colorable Graph Venkatesan Guruswami Sanjeev Khanna Abstract We give a new proof showing that it is NP hard to color a 3 colorable graph using just four colors. This result is already known [19], but our proof is novel as it does not rely on the PCP theorem, while the one in [19] does. This highlights a qualitative difference between the known hardness result for coloring 3 colorable graphs and the factor n hardness for approximating the chromatic number of general graphs, as the ....

....4 coloring a 3 colorable Graph Venkatesan Guruswami Sanjeev Khanna Abstract We give a new proof showing that it is NP hard to color a 3 colorable graph using just four colors. This result is already known [19] but our proof is novel as it does not rely on the PCP theorem, while the one in [19] does. This highlights a qualitative difference between the known hardness result for coloring 3 colorable graphs and the factor n hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [3] Another aspect in ....

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S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. In Proceedings of the 2nd Israel Symposium on Theory and Computing Systems, ISTCS, pp. 250-260, IEEE Computer Society Press, 1993. 16

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S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica 20:393-415, 2000.

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Sanjeev Khanna, Nathan Linial, and Shmuel Safra. On the hardness of approximating the chromatic number. Combinatorica, 20(3):393--415, 2000.

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S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. Proceedings of the Second Israel Symposium on Theory and Computing Systems, 1993.

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S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica, 20:393-415, 2000.

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S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. Proceedings of the Second Israel Symposium on Theory and Computing Systems, 1993.

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S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica, 20:393-415, 2000. 11

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S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. Proceedings of the Second Israel Symposium on Theory and Computing Systems, 1993.

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S. Khanna, N. Linial and S. Safra. On the hardness of approximating the chromatic number. Proceedings of the Second Israel Symposium on Theory and Computing Systems, 1993.

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