Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems

Lewin, M. R.; Livnat, Dror; Zwick, Uri

doi:10.1007/3-540-47867-1_6

Cited by 106 publications

(115 citation statements)

References 10 publications

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“…As the problem of Max-2CSP(2) has an approximation algorithm with a guarantee of 0.874 [18], this implies an approximation algorithm for Max-EGI 2 with the same guarantee and since Max-2Lin-2 is hard to approximate beyond 0.878 under UGC [16], we have almost matching upper and lower bounds for Max-EGI 2 under UGC.…”

Section: Lemma 10mentioning

confidence: 81%

Approximate Graph Isomorphism

Arvind

Köbler

Kuhnert

et al. 2012

Mathematical Foundations of Computer Science 2012

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Abstract. We study optimization versions of Graph Isomorphism. Given two graphs G1, G2, we are interested in finding a bijection π from V (G1) to V (G2) that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an n O(log n) time approximation scheme that for any constant factor α < 1, computes an α-approximation. We prove this by combining the n O(log n) time additive error approximation algorithm of Arora et al. [Math. Program., 92, 2002] with a simple averaging algorithm. We also consider the corresponding minimization problem (of mismatches) and prove that it is NP-hard to α-approximate for any constant factor α. Further, we show that it is also NP-hard to approximate the maximum number of edges mapped to edges beyond a factor of 0.94. We also explore these optimization problems for bounded color class graphs which is a well studied tractable special case of Graph Isomorphism. Surprisingly, the bounded color class case turns out to be harder than the uncolored case in the approximate setting.

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Section: Lemma 10mentioning

confidence: 81%

Approximate Graph Isomorphism

Arvind

Köbler

Kuhnert

et al. 2012

Mathematical Foundations of Computer Science 2012

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“…, n}. For k = 2, which is Max-DiCut, the semidefinite programming (SDP) relaxation is shown to yield a ≈ 1.144-approximation in [14], improving upon previous analyses of [15], [21], and [5]. As mentioned above, RMAS is a generalization of Max-k-Ordering, and a 2 √ 2-approximation for it based on linear programming (LP) rounding was shown recently by Grandoni et al [7] which is also the best approximation for Max-k-Ordering for k = 3.…”

Section: Related Workmentioning

confidence: 99%

On the Approximability of Digraph Ordering

Kenkre¹,

Pandit²,

Purohit

et al. 2016

Algorithmica

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Abstract. Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling ℓ : V → [k] maximizing the number of forward edges, i.e. edges (u, v) such that ℓ(u) < ℓ(v). For different values of k, this reduces to maximum acyclic subgraph (k = n), and Max-DiCut (k = 2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k = {2, . . . , n}, improving on the known 2k/(k − 1) -approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k = {2, . . . , n} and constant ε > 0, Max-k-Ordering has an LP integrality gap of 2 − ε for n Ω( 1/log log k ) rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels Sv ⊆ Z + . We prove an LP rounding based 4 √ 2 √ 2 + 1 ≈ 2.344 approximation for it, improving on the 2 √ 2 ≈ 2.828 approximation recently given by Grandoni et al. [7]. In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any k ∈ [n]. A vertex deletion version was studied earlier by Paik et al. [17], and Svensson [18].

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“…It gives a 1 4 approximation to Max 2-AND and a 1 2 approximation to Max CUT. There are algorithms that use semidefinite programming to achieve about 0.874 approximation to Max 2-AND [17] and about 0.878 approximation to Max CUT [11]. Assum-ing the Unique Games Conjecture, these algorithms are optimal for Max CUT [15,19], and nearly optimal for Max 2-AND (which under this assumption is hard to approximate within 0.87435 [5]).…”

Section: Related Workmentioning

confidence: 99%

Oblivious Algorithms for the Maximum Directed Cut Problem

Feige

Jozeph

2013

Algorithmica

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This paper introduces a special family of randomized algorithms for Max DICUT that we call oblivious algorithms. Let the bias of a vertex be the ratio between the total weight of its outgoing edges and the total weight of all its edges. An oblivious algorithm selects at random in which side of the cut to place a vertex v, with probability that only depends on the bias of v, independently of other vertices. The reader may observe that the algorithm that ignores the bias and chooses each side with probability 1/2 has an approximation ratio of 1/4, whereas no oblivious algorithm can have an approximation ratio better than 1/2 (with an even directed cycle serving as a negative example). We attempt to characterize the best approximation ratio achievable by oblivious algorithms, and present results that are nearly tight. The paper also discusses natural extensions of the notion of oblivious algorithms, and extensions to the more general problem of Max 2-AND.

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Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems

Cited by 106 publications

References 10 publications

Approximate Graph Isomorphism

Approximate Graph Isomorphism

On the Approximability of Digraph Ordering

Oblivious Algorithms for the Maximum Directed Cut Problem

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