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## On the Hardness of Approximating Multicut and Sparsest-Cut (2005)

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Venue: | In Proceedings of the 20th Annual IEEE Conference on Computational Complexity |

Citations: | 99 - 5 self |

### Citations

1257 | Approximation Algorithms, - Vazirani - 2001 |

793 | Proof verification and the hardness of approximation problems.
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context .... Proving the conjecture using current techniques appears quite hard. In particular, the asserted NP-hardness is much stronger than what we can obtain via standard constructions using the PCP theorem =-=[5, 3]-=- and the parallel repetition theorem [28], two deep results in computational complexity. Although the conjecture seems difficult to prove in general, some special cases are well-understood. In particu... |

748 | Some optimal inapproximability results.
- Hastad
- 2001
(Show Context)
Citation Context ...nd that the value of c is not specified therein, but it is certainly much smaller than 2. The MIN-2CNF≡ DELETION problem is also known to be APX-hard, as follows, e.g., from linear equations modulo 2 =-=[17]-=-. Assuming the Unique Games Conjecture, Khot [19, Theorem 3] essentially obtained an arbitrarily large constantfactor hardness for MIN-2CNF≡ DELETION, and this implies, using the aforementioned reduct... |

521 | The geometry of graphs and some of its algorithmic applications
- Linial, London, et al.
- 1995
(Show Context)
Citation Context ...atorial problems, with connections to multicommodity flow, edge expansion, and metric embeddings. Both problems can be approximated to within an O(log k) factor through linear programming relaxations =-=[25, 16, 6, 26]-=-. These bounds match the lower bounds on the integrality gaps up to constant factors [25, 16]. MIN-2CNF≡ DELETION can also be approximated to within an O(log n) factor, as implied by the results of Kl... |

412 | Probabilistic checking of proofs: a new characterization of NP.
- Arora, Safra
- 1998
(Show Context)
Citation Context .... Proving the conjecture using current techniques appears quite hard. In particular, the asserted NP-hardness is much stronger than what we can obtain via standard constructions using the PCP theorem =-=[5, 3]-=- and the parallel repetition theorem [28], two deep results in computational complexity. Although the conjecture seems difficult to prove in general, some special cases are well-understood. In particu... |

362 | A parallel repetition theorem.
- Raz
- 1998
(Show Context)
Citation Context ...niques appears quite hard. In particular, the asserted NP-hardness is much stronger than what we can obtain via standard constructions using the PCP theorem [5, 3] and the parallel repetition theorem =-=[28]-=-, two deep results in computational complexity. Although the conjecture seems difficult to prove in general, some special cases are well-understood. In particular, if at all the Unique Games Conjectur... |

322 | Correlation clustering.
- Bansal, Blum, et al.
- 2004
(Show Context)
Citation Context ...iterals, and the goal is to find a Boolean assignment to the variables minimizing the total weight of unsatisfied clauses. 2 Our results immediately extend also to the Correlation Clustering problem (=-=Bansal et al. 2004-=-; Charikar et al. 2003; Demaine & Immorlica 2003; Emanuel & Fiat 2003) of minimizing disagreements in a weighted graph, since the approximability of this problem is known to be equivalent to within co... |

311 | On the power of unique 2-prover 1-round games
- Khot
(Show Context)
Citation Context ...and the goal is to find a multicut M whose total cost c(M) = � e∈M c(e) is minimal. This problem is known to be APX-hard [12]. We prove that if a strong version of the Unique Games Conjecture of Khot =-=[19]-=- is true, then MULTICUT is NP-hard to approximate to within a factor of Ω(log log n). Under the original version of this conjecture, our reduction shows that for every constant L > 0, it is NP-hard to... |

309 | Expander flows, geometric embeddings and graph partitioning.
- Arora, Rao, et al.
- 2009
(Show Context)
Citation Context ...s of Klein et al. [23], who give an approximation-preserving reduction from this problem to MULTICUT. Recently, starting with the groundbreaking O( √ log n)-approximation for the uniform demands case =-=[4]-=-, improved approximation algorithms have 1 In general, the demand pairs may have positive weights (demands), but for our purpose of inapproximability results, it clearly suffices to consider the more ... |

293 | The influence of variables on boolean functions
- Kahn, Kalai, et al.
- 1988
(Show Context)
Citation Context ...ast β fraction of the antipodal pairs in C. Then for all x > 0, � a∈[d] I M a ≤ βx ⇒ max a∈[d] IM a ≥ 2 −6x /27. To prove this, we will make use of the following lemma, due to Kahn, Kalai, and Linial =-=[18]-=- (see also [29, Section 1.5]). Lemma 2.4 (Kahn, Kalai, and Linial [18]). Let f be a Boolean function defined on a hypercube, and suppose the fraction of inputs x for which f(x) = 1 is p ≤ 1/2. Then fo... |

244 | An approximate maxflow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms.
- Leighton, Rao
- 1988
(Show Context)
Citation Context ...atorial problems, with connections to multicommodity flow, edge expansion, and metric embeddings. Both problems can be approximated to within an O(log k) factor through linear programming relaxations =-=[25, 16, 6, 26]-=-. These bounds match the lower bounds on the integrality gaps up to constant factors [25, 16]. MIN-2CNF≡ DELETION can also be approximated to within an O(log n) factor, as implied by the results of Kl... |

223 | Optimal inapproximability results for Max-Cut and other 2-variable CSPs
- Khot, Kindler, et al.
(Show Context)
Citation Context ... particular for fixed d. Plausibility of the conjecture and its stronger version. The Unique Games Conjecture has been used to show optimal inapproximability results for VERTEX COVER [21] and MAX-CUT =-=[20, 27]-=-. Proving the conjecture using current techniques appears quite hard. In particular, the asserted NP-hardness is much stronger than what we can obtain via standard constructions using the PCP theorem ... |

212 | Free bits, PCPs, and nonapproximability – towards tight results.
- Bellare, Goldreich, et al.
- 1998
(Show Context)
Citation Context ...he hythat contains u. The set D of demand pairs then percube C p i 3 This is a standard technique in PCP constructions for graph optimization problems. A hypercube can be interpreted as a “long code” =-=[8]-=-, and a dimension cut is the encoding of an answer in the 2-prover game. contains every pair of antipodal vertices in G, and hence k = |D| = n2 d−1 . Note that every vertex of G belong to exactly one ... |

192 | The complexity of multiterminal cuts.
- Dahlhaus, Johnson, et al.
- 1994
(Show Context)
Citation Context ...his problem, the input also specifies a positive cost c(e) for each edge e ∈ E and the goal is to find a multicut M whose total cost c(M) = � e∈M c(e) is minimal. This problem is known to be APX-hard =-=[12]-=-. We prove that if a strong version of the Unique Games Conjecture of Khot [19] is true, then MULTICUT is NP-hard to approximate to within a factor of Ω(log log n). Under the original version of this ... |

170 | The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into l1
- Khot, Vishnoi
(Show Context)
Citation Context ...letion are restricted to equality (and effectively non-equality) constraints. 2sFor Sparsest-Cut, no hardness of approximation result was previously known. Independently of our work, Khot and Vishnoi =-=[KV05]-=- have recently used a different construction to show an arbitrarily large constant factor hardness for Sparsest-Cut assuming the Unique Games Conjecture. Their hardness factor could, in principle, be ... |

157 | Approximate max-flow min-(multi)cut theorems and their applications.
- Garg, Vazirani, et al.
- 1996
(Show Context)
Citation Context ...atorial problems, with connections to multicommodity flow, edge expansion, and metric embeddings. Both problems can be approximated to within an O(log k) factor through linear programming relaxations =-=[25, 16, 6, 26]-=-. These bounds match the lower bounds on the integrality gaps up to constant factors [25, 16]. MIN-2CNF≡ DELETION can also be approximated to within an O(log n) factor, as implied by the results of Kl... |

150 | Vertex cover might be hard to approximate to within 2−ε,
- Khot, Regev
- 2008
(Show Context)
Citation Context ... O(log n), and in particular for fixed d. Plausibility of the conjecture and its stronger version. The Unique Games Conjecture has been used to show optimal inapproximability results for VERTEX COVER =-=[21]-=- and MAX-CUT [20, 27]. Proving the conjecture using current techniques appears quite hard. In particular, the asserted NP-hardness is much stronger than what we can obtain via standard constructions u... |

128 | An O(log k) approximate min-cut max-flow theorem and approximation algorithm
- Aumann, Rabani
- 1998
(Show Context)
Citation Context |

126 | Noise stability of functions with low influences: Invariance and optimality.
- Mossel, O’Donnell, et al.
- 2010
(Show Context)
Citation Context ... particular for fixed d. Plausibility of the conjecture and its stronger version. The Unique Games Conjecture has been used to show optimal inapproximability results for VERTEX COVER [21] and MAX-CUT =-=[20, 27]-=-. Proving the conjecture using current techniques appears quite hard. In particular, the asserted NP-hardness is much stronger than what we can obtain via standard constructions using the PCP theorem ... |

117 | A.: Clustering with qualitative information
- Charikar, Guruswami, et al.
- 2005
(Show Context)
Citation Context ...x and y are literals, and the goal is to find a Boolean assignment to the variables minimizing the total weight of unsatisfied clauses. 2 Our results also extend to the CORRELATION CLUSTERING problem =-=[7, 10, 13, 14]-=- of minimizing disagreements in a weighted graph, because the approximability of this problem is known to be equivalent to that of MULTICUT in weighted graphs [10, 14]. 1.1. Known results on MULTICUT,... |

113 | Euclidean distortion and the sparsest cut
- Arora, Lee, et al.
- 2005
(Show Context)
Citation Context ...at the constraints in MIN-2CNF≡ DELETION are restricted to equality (and effectively non-equality) constraints.sbeen developed for the SPARSEST-CUT problem using a semidefinite programming relaxation =-=[4, 11, 2]-=-. The best approximation factor currently known for general demands is O( √ log k log log k) [2]. The obvious modification of the semidefinite program used for SPARSEST-CUT to solve MULTICUT was recen... |

103 |
Free bits, PCPs, and non-approximability—towards tight results.
- Bellare, Goldreich, et al.
- 1995
(Show Context)
Citation Context ... if v is in containing the all-ones vector 1. For every v ∈ C p i 3 This is a standard technique in PCP constructions for graph optimization problems. A hypercube can be interpreted as a “long code” (=-=Bellare et al. 1998-=-), and a dimension cut is the encoding of an answer in the 2-prover game.104 Chawla et al. cc 15 (2006) the same side as 0, and f(v) = 1 otherwise. This is exactly the A p i in v, i.e., f(v) = v p Ai... |

100 |
Collective coin flipping
- Ben-Or, Linial
- 1990
(Show Context)
Citation Context .... The main bottleneck to improving the hardness factor lies in Lemma 2.3, which in turn crucially depends on Lemma 2.4, due to [18]. These bounds are tight in general, as shown by the tribes function =-=[9]-=-. However, in our context, in the reduction to the (non-bicriteria) MULTICUT problem, one may additionally assume that f is odd, that is, f(u) �= f(u) for all inputs u (because a multicut should separ... |

90 | Boolean functions with low average sensitivity depend on few coordinates
- Friedgut
- 1998
(Show Context)
Citation Context ... on η was c1(log η(nc2) ). Interestingly, the two versions employ exactly the same reduction, but the analysis is different (and perhaps simpler), as the current version uses Friedgut’s Junta Theorem =-=[Fri98]-=- rather than a theorem of Kahn, Kalai, and Linial [KKL88]. This improvement was also motivated, in part, by the integrality ratio of [KV05] for Unique Games, which suggests a significant asymmetry bet... |

54 |
O(√logn) approximation algorithms for min uncut, min 2CNF deletion, and directed cut problems
- Agarwal, Charikar, et al.
(Show Context)
Citation Context ... general demands is O( √ log k log log k) [2]. The obvious modification of the semidefinite program used for SPARSEST-CUT to solve MULTICUT was recently shown to have an integrality ratio of Ω(log k) =-=[1]-=-, which matches, up to constant factors, the approximation factor and integrality gap of previously analyzed linear programming relaxations for this problem. On the hardness side, it is known that MUL... |

53 | Correlation Clustering with Partial Information
- Demaine, Emanuel, et al.
- 2003
(Show Context)
Citation Context ...x and y are literals, and the goal is to find a Boolean assignment to the variables minimizing the total weight of unsatisfied clauses. 2 Our results also extend to the CORRELATION CLUSTERING problem =-=[7, 10, 13, 14]-=- of minimizing disagreements in a weighted graph, because the approximability of this problem is known to be equivalent to that of MULTICUT in weighted graphs [10, 14]. 1.1. Known results on MULTICUT,... |

47 |
Approximation through multicommodity flow
- Klein, Agarwal, et al.
- 1990
(Show Context)
Citation Context ...nds match the lower bounds on the integrality gaps up to constant factors [25, 16]. MIN-2CNF≡ DELETION can also be approximated to within an O(log n) factor, as implied by the results of Klein et al. =-=[23]-=-, who give an approximation-preserving reduction from this problem to MULTICUT. Recently, starting with the groundbreaking O( √ log n)-approximation for the uniform demands case [4], improved approxim... |

46 | Near-optimal algorithms for unique games
- Charikar, Makarychev, et al.
(Show Context)
Citation Context ... [Kho02]). These algorithms imply that a stronger version of the Unique Games Conjecture can only be true if η(n) = Ω(1/ log n) (assuming P �= NP). Very recently, Charikar, Makarychev, and Makarychev =-=[CMM06]-=- improved upon the rounding algorithm of Khot [Kho02] to obtain better approximation algorithms for Unique Games. Their results imply that for Unique Games Conjecture to be true, we must have η ≥ Ω(1/... |

46 |
N.K.: Integrality gaps for sparsest cut and minimum linear arrangement problems
- Devanur, Khot, et al.
- 2006
(Show Context)
Citation Context ... 2-prover system has a low-cost balanced cut, then the corresponding graph on hypercubes would have a low-cost balanced cut regardless of the value of the 2-prover game. Very recently, Devanur et al. =-=[DKSV06]-=- have shown a lower bound of Ω(log log n) on the integrality ratio of the natural SDP relaxation for this problem. Alternatively, of course, one might improve the approximation algorithms for any of t... |

44 | Approximation algorithms for unique games
- Trevisan
- 2005
(Show Context)
Citation Context ... result falls short of the Unique Games Conjecture in that Lδ is bounded away from 1. 3sStronger versions of the conjecture in which d, η, and δ are functions of n have also been considered. Trevisan =-=[Tre05]-=- and Gupta and Talwar [GT06] recently developed approximation algorithms for solving instances of Unique Games where η(n) is a sufficiently small function of n (based on an SDP and an LP relaxation re... |

39 | Improved lower bounds for embeddings into L1
- Krauthgamer, Rabani
(Show Context)
Citation Context ...4−o(1) on the integrality ratio of the semidefinite programming relaxations used in the recent approximation algorithms for Sparsest-Cut. This last lower bound was further improved to Ω(log log n) in =-=[KR06]-=-, using in part ideas from the current paper. 1.2 The Unique Games Conjecture Unique 2-prover game is the following problem. The input is a bipartite graph GQ = (Q, EQ), where each side p = 1, 2 conta... |

23 | Approximating unique games
- Gupta, Talwar
- 2006
(Show Context)
Citation Context ...ique Games Conjecture in that Lδ is bounded away from 1. 3sStronger versions of the conjecture in which d, η, and δ are functions of n have also been considered. Trevisan [Tre05] and Gupta and Talwar =-=[GT06]-=- recently developed approximation algorithms for solving instances of Unique Games where η(n) is a sufficiently small function of n (based on an SDP and an LP relaxation respectively, different from t... |

22 | Correlation clustering — minimizing disagreements on arbitrary weighted graphs
- Emanuel, Fiat
- 2003
(Show Context)
Citation Context ...x and y are literals, and the goal is to find a Boolean assignment to the variables minimizing the total weight of unsatisfied clauses. 2 Our results also extend to the CORRELATION CLUSTERING problem =-=[7, 10, 13, 14]-=- of minimizing disagreements in a weighted graph, because the approximability of this problem is known to be equivalent to that of MULTICUT in weighted graphs [10, 14]. 1.1. Known results on MULTICUT,... |

17 |
An improved approximation to sparsest cut
- Chawla, Gupta, et al.
- 2005
(Show Context)
Citation Context ...at the constraints in MIN-2CNF≡ DELETION are restricted to equality (and effectively non-equality) constraints.sbeen developed for the SPARSEST-CUT problem using a semidefinite programming relaxation =-=[4, 11, 2]-=-. The best approximation factor currently known for general demands is O( √ log k log log k) [2]. The obvious modification of the semidefinite program used for SPARSEST-CUT to solve MULTICUT was recen... |

15 |
On systems of linear equations with two variables per equation
- Feige, Reichman
- 2004
(Show Context)
Citation Context ...log n) hardness result (see Corollary 1.4 below) requires the existence of a constant c > 0, such that max{η, δ} ≤ 1/(log n) c and d ≤ O(log n), which is notsexcluded by the above. Feige and Reichman =-=[15]-=- recently showed that for every constant L > 0 there exists a constant δ > 0, such that it is NP-hard to distinguish whether a unique 2-prover game (with d = d(L, δ)) has value at least Lδ or at most ... |

12 | Fourier Transform in Computer Science - Stefankovic - 2000 |

10 |
Correlation clustering–minimizing disagreements on arbitrary weighted graphs
- Emanuel, Fiat
- 2003
(Show Context)
Citation Context ...es minimizing the total weight of unsatisfied clauses. 2 Our results immediately extend also to the Correlation Clustering problem (Bansal et al. 2004; Charikar et al. 2003; Demaine & Immorlica 2003; =-=Emanuel & Fiat 2003-=-) of minimizing disagreements in a weighted graph, since the approximability of this problem is known to be equivalent to within constant factors to that of Multicut in weighted graphs (Charikar et al... |

2 |
On embeddability of negative type metrics into ℓ1
- Khot, Vishnoi
- 2004
(Show Context)
Citation Context ...hypercube), and thus may prove useful in further investigation of such questions. For SPARSEST-CUT, no hardness of approximation result was previously known. Independent of our work, Khot and Vishnoi =-=[22]-=- have recently used a different construction to show an arbitrarily large constant factor hardness for SPARSEST-CUT assuming the Unique Games Conjecture; their hardness factor could, in principle, be ... |

2 | Vishnoi (2005) The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1 - Khot, K - 1997 |

2 | Vazirani [2001] Approximation Algorithms - V |

1 | Räcke (2005a). Improved approximations to sparsest cut - Chawla, Gupta, et al. |

1 | Rabani & D. Sivakumar (2005b). On the hardness of approximating multicut and sparsest-cut - Chawla, Krauthgamer, et al. |

1 | Oleszkiewicz (2005). Noise stability of functions with low influences: Invariance and optimality - Mossel, O’Donnell, et al. |

1 | Manuscript received 19 September 2005 - Pittsburgh |