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176
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
- Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 1211 (13 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...
Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 223 (32 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [25]. This implies that if the Unique Games
Approximate Graph Coloring by Semidefinite Programming.
- In Proceedings of 35th Annual IEEE Symposium on Foundations of Computer Science,
, 1994
"... Abstract. We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on n vertices with min{O(⌬ 1/3 log 1/2 ⌬ log n), O(n 1/4 log 1/2 n)} colors where ⌬ is the maximum degree of any vertex ..."
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Cited by 210 (7 self)
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Abstract. We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on n vertices with min{O(⌬ 1/3 log 1/2 ⌬ log n), O(n 1/4 log 1/2 n)} colors where ⌬ is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first nontrivial approximation result as a function of the maximum degree ⌬. This result can be generalized to k-colorable graphs to obtain a coloring using min{O(⌬ 1Ϫ2/k log 1/2 ⌬ log n), O(n 1Ϫ3/(kϩ1) log 1/2 n)} colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász -function. We show lower bounds on the gap between the Permission to make digital / hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery (ACM), Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and / or a fee. © 1998 ACM 0004-5411/98/0300-0246 $05.00 Journal of the ACM, Vol. 45, No. 2, March 1998, pp. 246 -265. optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the -function.
The Dense k-Subgraph Problem
- Algorithmica
, 1999
"... This paper considers the problem of computing the dense k-vertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense k-subgraph (D ..."
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Cited by 199 (11 self)
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This paper considers the problem of computing the dense k-vertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense k-subgraph (DkS) maximization problem, of computing the dense k- vertex subgraph of a given graph. That is, on input a graph G and a parameter k, we are interested in finding a set of k vertices with maximum average degree in the subgraph induced by this set. As this problem is NP-hard (say, by reduction from Clique), we consider approximation algorithms for this problem. We obtain a polynomial time algorithm that on any input (G; k) returns a subgraph of size k whose average degree is within a factor of at most n ffi from the optimum solution, where n is the number of vertices in the input graph G, and ffi ! 1=3 is some universal constant. Unfortunately, we are unable to present a complementary negati...
Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT
- IN PROCEEDINGS OF THE THIRD ISRAEL SYMPOSIUM ON THEORY OF COMPUTING AND SYSTEMS
, 1995
"... It is well known that two prover proof systems are a convenient tool for establishing hardness of approximation results. In this paper, we show that two prover proof systems are also convenient starting points for establishing easiness of approximation results. Our approach combines the Feage-Lovdsz ..."
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Cited by 141 (10 self)
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It is well known that two prover proof systems are a convenient tool for establishing hardness of approximation results. In this paper, we show that two prover proof systems are also convenient starting points for establishing easiness of approximation results. Our approach combines the Feage-Lovdsz (STOC92) semidefinite programming relaxation of one-round two-prover proof systems, together with rounding techniques for the solutions of semidefinite progmms, as introduced by Goemans and Williamson (STO C94). As a consequence of our approach, we present improved approximation algorithms for MAX 2SAT and MAX DICUT. The algorithms are guamnteed to deliver solutions within a factor of 0.931 of the optimum for MAX 2SAT and within a factor of 0.859 for MAX DICUT, improving upon the guarantees of 0.878 and 0.796 of Goemans and Williamson.
Some Applications of Laplace Eigenvalues of Graphs
- GRAPH SYMMETRY: ALGEBRAIC METHODS AND APPLICATIONS, VOLUME 497 OF NATO ASI SERIES C
, 1997
"... In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of ..."
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Cited by 129 (0 self)
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In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.
Clustering with qualitative information
- In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
, 2003
"... We consider the problem of clustering a collection of el-ements based on pairwise judgments of similarity and dis-similarity. Bansal, Blum and Chawla [1] cast the problem thus: given a graph G whose edges are labeled “+ ” (sim-ilar) or “− ” (dissimilar), partition the vertices into clus-ters so that ..."
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Cited by 122 (9 self)
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We consider the problem of clustering a collection of el-ements based on pairwise judgments of similarity and dis-similarity. Bansal, Blum and Chawla [1] cast the problem thus: given a graph G whose edges are labeled “+ ” (sim-ilar) or “− ” (dissimilar), partition the vertices into clus-ters so that the number of pairs correctly (resp. incorrectly) classified with respect to the input labeling is maximized (resp. minimized). Complete graphs, where the classifier la-bels every edge, and general graphs, where some edges are not labeled, are both worth studying. We answer several questions left open in [1] and provide a sound overview of clustering with qualitative information. We give a factor 4 approximation for minimization on complete graphs, and a factor O(log n) approximation for general graphs. For the maximization version, a PTAS for complete graphs is shown in [1]; we give a factor 0.7664 approximation for general graphs, noting that a PTAS is unlikely by proving APX-hardness. We also prove the APX-hardness of minimization on complete graphs. 1.
Minimum-interference channel assignment in multi-radio wireless mesh networks
- IN SECON
, 2006
"... In this paper, we consider multi-hop wireless mesh networks, where each router node is equipped with multiple radio interfaces and multiple channels are available for communication. We address the problem of assigning channels to communication links in the network with the objective of minimizing ov ..."
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Cited by 107 (2 self)
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In this paper, we consider multi-hop wireless mesh networks, where each router node is equipped with multiple radio interfaces and multiple channels are available for communication. We address the problem of assigning channels to communication links in the network with the objective of minimizing overall network interference. Since the number of radios on any node can be less than the number of available channels, the channel assignment must obey the constraint that the number of different channels assigned to the links incident on any node is atmost the number of radio interfaces on that node. The above optimization problem is known to be NP-hard. We design centralized and distributed algorithms for the above channel assignment problem. To evaluate the quality of the solutions obtained by our algorithms, we develop a semidefinite program formulation of our optimization problem to obtain a lower bound on overall network interference. Empirical evaluations on randomly generated network graphs show that our algorithms perform close to the above established lower bound, with the difference diminishing rapidly with increase in number of radios. Also, detailed ns-2 simulation studies demonstrate the performance potential of our channel assignment algorithms in 802.11-based multi-radio mesh networks.
Semidefinite Programming and Combinatorial Optimization
- DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 106 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Spectral Partitioning: The More Eigenvectors, the Better
- PROC. ACM/IEEE DESIGN AUTOMATION CONF
, 1995
"... The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) w ..."
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Cited by 75 (3 self)
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The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy isincontrast to that of the widely-used spectral bipartitioning (SB) heuristic (which uses a single eigenvector to construct a 2-way partitioning) and several previous multiway partitioning heuristics [7][10][16][26][37] (which usek eigenvectors to construct a k-way partitioning). Our result motivates a simple ordering heuristic that is a multiple-eigenvector extension of SB. This heuristic not only signi cantly outperforms SB, but can also yield excellent multi-way VLSI circuit partitionings as compared to [1] [10]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective heuristics.
