Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

David R. Karger, Philip Klein, Cliff Stein, Mikkel Thorup and Neal E. Young
Mathematics of Operations Research
Vol. 29, No. 3 (Aug., 2004), pp. 436-461
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/30035660
Page Count: 26
  • Download ($30.00)
  • Cite this Item
Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut
Preview not available

Abstract

Given an undirected graph with edge costs and a subset of k ≥ 3 nodes called terminals, a multiway, or k-way, cut is a subset of the edges whose removal disconnects each terminal from the others. The multiway cut problem is to find a minimum-cost multiway cut. This problem is Max-SNP hard. Recently, Calinescu et al. (Calinescu, G., H. Karloff, Y. Rabani. 2000. An improved approximation algorithm for MULTIWAY CUT. J. Comput. System Sci. 60(3) 564-574) gave a novel geometric relaxation of the problem and a rounding scheme that produced a (3/2-1/k)-approximation algorithm. In this paper, we study their geometric relaxation. In particular, we study the worst-case ratio between the value of the relaxation and the value of the minimum multicut (the so-called integrality gap of the relaxation). For k = 3, we show the integrality gap is 12/11, giving tight upper and lower bounds. That is, we exhibit a family of graphs with integrality gaps arbitrarily close to 12/11 and give an algorithm that finds a cut of value 12/11 times the relaxation value. Our lower bound shows that this is the best possible performance guarantee for any algorithm based purely on the value of the relaxation. Our upper bound meets the lower bound and improves the factor of 7/6 shown by Calinescu et al. For all k, we show that there exists a rounding scheme with performance ratio equal to the integrality gap, and we give explicit constructions of polynomial-time rounding schemes that lead to improved upper bounds. For k = 4 and 5, our best upper bounds are based on computer-constructed rounding schemes (with computer proofs of correctness). For general k we give an algorithm with performance ratio $1.3438-\varepsilon_{k}$. Our results were discovered with the help of computational experiments that we also describe here.

Page Thumbnails

  • Thumbnail: Page 
436
    436
  • Thumbnail: Page 
437
    437
  • Thumbnail: Page 
438
    438
  • Thumbnail: Page 
439
    439
  • Thumbnail: Page 
440
    440
  • Thumbnail: Page 
441
    441
  • Thumbnail: Page 
442
    442
  • Thumbnail: Page 
443
    443
  • Thumbnail: Page 
444
    444
  • Thumbnail: Page 
445
    445
  • Thumbnail: Page 
446
    446
  • Thumbnail: Page 
447
    447
  • Thumbnail: Page 
448
    448
  • Thumbnail: Page 
449
    449
  • Thumbnail: Page 
450
    450
  • Thumbnail: Page 
451
    451
  • Thumbnail: Page 
452
    452
  • Thumbnail: Page 
453
    453
  • Thumbnail: Page 
454
    454
  • Thumbnail: Page 
455
    455
  • Thumbnail: Page 
456
    456
  • Thumbnail: Page 
457
    457
  • Thumbnail: Page 
458
    458
  • Thumbnail: Page 
459
    459
  • Thumbnail: Page 
460
    460
  • Thumbnail: Page 
461
    461