4,495 research outputs found
Algorithms for finding K-best perfect matchings
AbstractIn the K-best perfect matching problem (KM) one wants to find K pairwise different, perfect matchings M1,…,Mk such that w(M1) ≥ w(M2) ≥ ⋯ ≥ w(Mk) ≥ w(M), ∀M ≠M1, M2,…, Mk. The procedure discussed in this paper is based on a binary partitioning of the matching solution space. We survey different algorithms to perform this partitioning. The best complexity bound of the resulting algorithms discussed is O(Kn3), where n is the number of nodes in the graph
Exact Covers via Determinants
Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such
that every hyperedge includes one vertex from each part, the k-dimensional
matching problem asks whether there is a disjoint collection of the hyperedges
which covers all vertices. We show it can be solved by a randomized polynomial
space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors
polynomial in n and k.
When we drop the partition constraint and permit arbitrary hyperedges of
cardinality k, we obtain the exact cover by k-sets problem. We show it can be
solved by a randomized polynomial space algorithm in time O*(c_k^n), where
c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k.
Both results substantially improve on the previous best algorithms for these
problems, especially for small k, and follow from the new observation that
Lovasz' perfect matching detection via determinants (1979) admits an embedding
in the recently proposed inclusion-exclusion counting scheme for set covers,
despite its inability to count the perfect matchings
Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
We consider the general problem of finding the minimum weight \bm-matching
on arbitrary graphs. We prove that, whenever the linear programming (LP)
relaxation of the problem has no fractional solutions, then the belief
propagation (BP) algorithm converges to the correct solution. We also show that
when the LP relaxation has a fractional solution then the BP algorithm can be
used to solve the LP relaxation. Our proof is based on the notion of graph
covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara
2007}.
These results are notable in the following regards: (1) It is one of a very
small number of proofs showing correctness of BP without any constraint on the
graph structure. (2) Variants of the proof work for both synchronous and
asynchronous BP; it is the first proof of convergence and correctness of an
asynchronous BP algorithm for a combinatorial optimization problem.Comment: 28 pages, 2 figures. Submitted to SIAM journal on Discrete
Mathematics on March 19, 2009; accepted for publication (in revised form)
August 30, 2010; published electronically July 1, 201
Space Saving by Dynamic Algebraization
Dynamic programming is widely used for exact computations based on tree
decompositions of graphs. However, the space complexity is usually exponential
in the treewidth. We study the problem of designing efficient dynamic
programming algorithm based on tree decompositions in polynomial space. We show
how to construct a tree decomposition and extend the algebraic techniques of
Lokshtanov and Nederlof such that the dynamic programming algorithm runs in
time , where is the maximum number of vertices in the union of
bags on the root to leaf paths on a given tree decomposition, which is a
parameter closely related to the tree-depth of a graph. We apply our algorithm
to the problem of counting perfect matchings on grids and show that it
outperforms other polynomial-space solutions. We also apply the algorithm to
other set covering and partitioning problems.Comment: 14 pages, 1 figur
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
- …