6.889: Algorithms for Planar Graphs and Beyond
November 30, 2011
Problem Set 11
This problem set is due Wednesday, December 7 at noon.
1. Recall the cycle xing procedure in the multiple-source multiple-sink maximum ow algorithm
which nds a ow which elimin
6.889: Algorithms for Planar Graphs and Beyond
Problem Set 4 - Solutions
1. Solution:
(a) Let (T, (Bt )tV (T ) ) be a tree decomposition of G. We show that there exists a bag Bz such that
u
C Bz . For any edge uv of T , dene Tuv as the connected component
6.889: Algorithms for Planar Graphs and Beyond
October 5, 2011
Problem Set 4
This problem set is due Wednesday, October 12 at noon.
1. (a) Show that if C is the vertex set of a clique in a graph G, then the treewidth of G
is at least |C | 1.
(b) Show that
6.889: Algorithms for Planar Graphs and Beyond
September 28, 2011
Problem Set 3
This problem set is due Wednesday, October 5 at noon.
1. Let P := cfw_p1 , . . . , pk be points in the plane and cfw_Q1 , . . . , Qt be a partition of P into
t sets. Argue (
6.889: Algorithms for Planar Graphs and Beyond
September 28, 2011
Problem Set 3
This problem set is due Wednesday, October 5 at noon.
1. Let P := cfw_p1 , . . . , pk be points in the plane and cfw_Q1 , . . . , Qt be a partition of P into
t sets. Argue (
6.889: Algorithms for Planar Graphs and Beyond
September 21, 2011
Problem Set 2
This problem set is due Wednesday, September 28 at noon.
1. Prove that any undirected planar graph G with non-negative edge weights can be
transformed into an undirected plana
6.889: Algorithms for Planar Graphs and Beyond
September 21, 2011
Problem Set 2
This problem set is due Wednesday, September 28 at noon.
1. Prove that any undirected planar graph G with non-negative edge weights can be
transformed into an undirected plana
6.889: Algorithms for Planar Graphs and Beyond
Problem Set 1 - Solutions
1. Solution: Since every face has size at least three, and each edge is in exactly two faces,
3f 2m (here, f is the number of faces). Substituting into Eulers formula we get 2 =
n m
6.889: Algorithms for Planar Graphs and Beyond
September 14, 2011
Problem Set 1
This problem set is due Thursday, September 22 at noon.
1. Sparsity Lemma: Prove that for a planar a embedded graph in which every face has
size at least three, m 3n 6, where
Algorithmica manuscript No.
(will be inserted by the editor)
Diameter and Treewidth in Minor-Closed Graph
Families, Revisited
Erik D. Demaine, MohammadTaghi Hajiaghayi
MIT Computer Science and Articial Intelligence Laboratory, 32 Vassar St., Cambridge,
MA
6.889: Algorithms for Planar Graphs and Beyond
October 12, 2011
Problem Set 5
This problem set is due Wednesday, October 19 at noon.
1. For m > 2k , let G be an m m grid, and G be the central (m 2k ) (m 2k ) subgrid
of G. Let N be a set of at least k 4 ve
6.889: Algorithms for Planar Graphs and Beyond
October 19, 2011
Problem Set 6
This problem set is due Wednesday, October 26 at noon.
For this problem set it is important to know that the separation property is dened somewhat
dierently for contraction-bidi
6.889: Algorithms for Planar Graphs and Beyond
October 19, 2011
Problem Set 6
This problem set is due Wednesday, October 26 at noon.
For this problem set it is important to know that the separation property is dened somewhat
dierently for contraction-bidi
6.889: Algorithms for Planar Graphs and Beyond
November 30, 2011
Problem Set 11
This problem set is due Wednesday, December 7 at noon.
1. Recall the cycle xing procedure in the multiple-source multiple-sink maximum ow algorithm
which nds a ow which elimin
6.889: Algorithms for Planar Graphs and Beyond
Problem Set 10 - Solutions
In this problem set you will develop an algorithm for canceling ow cycles in a given ow assignment.
In general graphs this can be done in O(m log n) time using Sleators and Tarjans
6.889: Algorithms for Planar Graphs and Beyond
November 16, 2011
Problem Set 10
This problem set is due Wednesday, November 23 at noon.
In this problem set you will develop an algorithm for canceling ow cycles in a given ow assignment.
In general graphs t
6.889: Algorithms for Planar Graphs and Beyond
November 9, 2011
Problem Set 9
This problem set is due Wednesday, November 16 at noon.
1. Prove that the Steiner tree spanner as constructed in the lecture actually contains
a Steiner tree of the given termin
6.889: Algorithms for Planar Graphs and Beyond
Problem Set 8 - Solutions
1. Solution:
(a) The Monge property cannot be satised because shortest paths need not cross.
(b) This is essentially the bipartite case treated in class. The Monge property holds. Ch
Hint for problem set 8:
Try nding two paths P1 , P2 with the property that for every i C1 and j C2 ,
there exists a shortest i-to-j path in G that does not cross both P1 and P2 .
1
6.889: Algorithms for Planar Graphs and Beyond
November 2, 2011
Problem Set 8
This problem set is due Wednesday, November 9 at noon.
1. Recall that in lecture 14 we represented the edges of the dense distance graph in a matrix Ai . We saw
that performing
6.889: Algorithms for Planar Graphs and Beyond
October 26, 2011
Problem Set 7
This problem set is due Wednesday, 11/2/2011 at noon.
Problem: Give an O(n log n)time algorithm to compute an rdivision (O(n/r) pieces of
size O(r) and boundary O( r) with the a
6.889: Algorithms for Planar Graphs and Beyond
October 26, 2011
Problem Set 7
This problem set is due Wednesday, 11/2/2011 at noon.
Problem: Give an O(n log n)time algorithm to compute an rdivision (O(n/r) pieces of
size O(r) and boundary O( r) with the a
6.889 Lecture 25: Single-Source Shortest Paths
with Negative Lengths in Minor-Free Graphs
Christian Sommer csom@mit.edu
December 12, 2011
Setting directed graph G = (V, A), underlying undirected graph is H minor-free, arbitrary real arc lengths
: A R or i
References
[BDT09] Glencora Borradaile, Erik D. Demaine, and Siamak Tazari.
Polynomial-time approximation schemes for subset-connectivity
problems in bounded-genus graphs. In STACS 09: Proceedings
of the 26th Symposium on Theoretical Aspects of Computer S
6.889 Lecture 11: Multiple-Source Shortest Paths
Christian Sommer csom@mit.edu
(gures by Philip Klein)
October 19, 2011
Single-Source Shortest Path (SSSP) Problem: given a graph G = (V, E ) and a source vertex s V , compute
shortest-path distance dG (s, v
References
[DFHT05] Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free
graphs. J. ACM, 52(6):866893, 2005.
[DH05a]
Erik D. Demaine and Moh
References
[DH04]
Erik D. Demaine and MohammadTaghi Hajiaghayi. Equivalence
of local treewidth and linear local treewidth and its algorithmic applications. In SODA 04: Proceedings of the 15th annual
ACM-SIAM Symposium on Discrete Algorithms, pages 840849,
References
[AST90] Noga Alon, Paul Seymour, and Robin Thomas. A separator theorem for nonplanar graphs. Journal of the American Mathematical
Society, 3(4):801808, 1990.
[RW09] Bruce Reed and David R. Wood. A linear-time algorithm to nd
a separator in a gr
References
[HKRS97] Monika R. Henzinger, Philip N. Klein, Satish Rao, and Sairam
Subramanian. Faster shortest-path algorithms for planar graphs.
Journal of Computer and System Sciences, 55(1):323, 1997. Previously appeared STOC 94: Proceedings of the 26th