24 research outputs found
A push–relabel approximation algorithm for approximating the minimum-degree MST problem and its generalization to matroids
AbstractIn the minimum-degree minimum spanning tree (MDMST) problem, we are given a graph G, and the goal is to find a minimum spanning tree (MST) T, such that the maximum degree of T is as small as possible. This problem is NP-hard and generalizes the Hamiltonian path problem. We give an algorithm that outputs an MST of degree at most 2Δopt (G)+o(Δopt (G)), where Δopt (G) denotes the degree of the optimal tree. This result improves on a previous result of Fischer [T. Fischer, Optimizing the degree of minimum weight spanning trees. Technical Report 14853, Dept. of Computer Science, Cornell University, Ithaca, NY, 1993] that finds an MST of degree at most bΔopt (G)+logbn, for any b>1.The MDMST problem is a special case of the following problem: given a k-ary hypergraph G=(V,E) and weighted matroid M with E as its ground set, find a minimum-cost basis (MCB) T of M such that the degree of T in G is as small as possible. Our algorithm immediately generalizes to this problem, finding an MCB of degree at most k2Δopt (G,M)+O(kkΔopt (G,M)).We use the push–relabel framework developed by Goldberg [A. V. Goldberg, A new max-flow algorithm, Technical Report MIT/LCS/TM-291, Massachusetts Institute of Technology, 1985 (Technical Report)] for the maximum-flow problem. To our knowledge, this is the first use of the push–relabel technique in an approximation algorithm for an NP-hard problem.The MDMST problem is closely connected to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph G and degree bound B on its nodes, the BDMST problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. Previous algorithms for this problem by Könemann and Ravi [J. Könemann, R. Ravi, A matter of degree: Improved approximation algorithms for degree-bounded minimum spanning trees, SIAM Journal on Computing 31(6) (2002) 1783–1793; J. Könemann, R. Ravi, Primal-dual meets local search: Approximating MST’s with nonuniform degree bounds, in: Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, 2003, pp. 389–395] and by Chaudhuri et al. [K. Chaudhuri, S. Rao, S. Riesenfeld, K. Talwar, What would Edmonds do? Augmenting paths and witnesses for bounded degree MSTs, in: Proceedings of APPROX/RANDOM, 2005, pp. 26–39] incur a near-logarithmic additive error in the degree. We give the first BDMST algorithm that approximates both the degree and the cost to within a constant factor of the optimum. These results generalize to the case of nonuniform degree bounds
Matroidal Degree-Bounded Minimum Spanning Trees
We consider the minimum spanning tree (MST) problem under the restriction
that for every vertex v, the edges of the tree that are adjacent to v satisfy a
given family of constraints. A famous example thereof is the classical
degree-constrained MST problem, where for every vertex v, a simple upper bound
on the degree is imposed. Iterative rounding/relaxation algorithms became the
tool of choice for degree-bounded network design problems. A cornerstone for
this development was the work of Singh and Lau, who showed for the
degree-bounded MST problem how to find a spanning tree violating each degree
bound by at most one unit and with cost at most the cost of an optimal solution
that respects the degree bounds.
However, current iterative rounding approaches face several limits when
dealing with more general degree constraints. In particular, when several
constraints are imposed on the edges adjacent to a vertex v, as for example
when a partition of the edges adjacent to v is given and only a fixed number of
elements can be chosen out of each set of the partition, current approaches
might violate each of the constraints by a constant, instead of violating all
constraints together by at most a constant number of edges. Furthermore, it is
also not clear how previous iterative rounding approaches can be used for
degree constraints where some edges are in a super-constant number of
constraints.
We extend iterative rounding/relaxation approaches both on a conceptual level
as well as aspects involving their analysis to address these limitations. This
leads to an efficient algorithm for the degree-constrained MST problem where
for every vertex v, the edges adjacent to v have to be independent in a given
matroid. The algorithm returns a spanning tree T of cost at most OPT, such that
for every vertex v, it suffices to remove at most 8 edges from T to satisfy the
matroidal degree constraint at v
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal
In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G=(V,E) with a degree upper bound Bv on each vertex v∈V, and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this problem. In this paper, we present a polynomial time algorithm which returns a spanning tree T of cost at most OPT and dT(v)≤Bv+1 for all v, where dT(v) denotes the degree of v in T. This generalizes a result of Fürer and Raghavachari [1994] to weighted graphs, and settles a conjecture of Goemans [2006] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound Av and a degree upper bound Bv, and returns a spanning tree with cost at most OPT and Av−1≤dT(v) ≤ Bv+1 for all v ∈ V. This is essentially the best possible. The main technique used is an extension of the iterative rounding method introduced by Jain [2001] for the design of approximation algorithms
Degree-bounded generalized polymatroids and approximating the metric many-visits TSP
In the Bounded Degree Matroid Basis Problem, we are given a matroid and a
hypergraph on the same ground set, together with costs for the elements of that
set as well as lower and upper bounds and for
each hyperedge . The objective is to find a minimum-cost basis
such that for
each hyperedge . Kir\'aly et al. (Combinatorica, 2012) provided an
algorithm that finds a basis of cost at most the optimum value which violates
the lower and upper bounds by at most , where is the
maximum degree of the hypergraph. When only lower or only upper bounds are
present for each hyperedge, this additive error is decreased to .
We consider an extension of the matroid basis problem to generalized
polymatroids, or g-polymatroids, and additionally allow element multiplicities.
The Bounded Degree g-polymatroid Element Problem with Multiplicities takes as
input a g-polymatroid instead of a matroid, and besides the lower and
upper bounds, each hyperedge has element multiplicities
. Building on the approach of Kir\'aly et al., we provide an
algorithm for finding a solution of cost at most the optimum value, having the
same additive approximation guarantee.
As an application, we develop a -approximation for the metric
Many-Visits TSP, where the goal is to find a minimum-cost tour that visits each
city a positive number of times. Our approach combines our algorithm
for the Bounded Degree g-polymatroid Element Problem with Multiplicities with
the principle of Christofides' algorithm from 1976 for the (single-visit)
metric TSP, whose approximation guarantee it matches.Comment: 17 page
A 3/2-Approximation for the Metric Many-visits Path TSP
In the Many-visits Path TSP, we are given a set of cities along with
their pairwise distances (or cost) , and moreover each city comes
with an associated positive integer request .
The goal is to find a minimum-cost path, starting at city and ending at
city , that visits each city exactly times.
We present a -approximation algorithm for the metric Many-visits
Path TSP, that runs in time polynomial in and poly-logarithmic in the
requests .
Our algorithm can be seen as a far-reaching generalization of the
-approximation algorithm for Path TSP by Zenklusen (SODA 2019), which
answered a long-standing open problem by providing an efficient algorithm which
matches the approximation guarantee of Christofides' algorithm from 1976 for
metric TSP.
One of the key components of our approach is a polynomial-time algorithm to
compute a connected, degree bounded multigraph of minimum cost.
We tackle this problem by generalizing a fundamental result of Kir\'aly, Lau
and Singh (Combinatorica, 2012) on the Minimum Bounded Degree Matroid Basis
problem, and devise such an algorithm for general polymatroids, even allowing
element multiplicities.
Our result directly yields a -approximation to the metric
Many-visits TSP, as well as a -approximation for the problem of
scheduling classes of jobs with sequence-dependent setup times on a single
machine so as to minimize the makespan.Comment: arXiv admin note: text overlap with arXiv:1911.0989
A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and Beyond
Short spanning trees subject to additional constraints are important building
blocks in various approximation algorithms. Especially in the context of the
Traveling Salesman Problem (TSP), new techniques for finding spanning trees
with well-defined properties have been crucial in recent progress. We consider
the problem of finding a spanning tree subject to constraints on the edges in
cuts forming a laminar family of small width. Our main contribution is a new
dynamic programming approach where the value of a table entry does not only
depend on the values of previous table entries, as it is usually the case, but
also on a specific representative solution saved together with each table
entry. This allows for handling a broad range of constraint types.
In combination with other techniques -- including negatively correlated
rounding and a polyhedral approach that, in the problems we consider, allows
for avoiding potential losses in the objective through the randomized rounding
-- we obtain several new results. We first present a quasi-polynomial time
algorithm for the Minimum Chain-Constrained Spanning Tree Problem with an
essentially optimal guarantee. More precisely, each chain constraint is
violated by a factor of at most , and the cost is no larger than
that of an optimal solution not violating any chain constraint. The best
previous procedure is a bicriteria approximation violating each chain
constraint by up to a constant factor and losing another factor in the
objective. Moreover, our approach can naturally handle lower bounds on the
chain constraints, and it can be extended to constraints on cuts forming a
laminar family of constant width.
Furthermore, we show how our approach can also handle parity constraints (or,
more precisely, a proxy thereof) as used in the context of (Path) TSP and one
of its generalizations, and discuss implications in this context.Comment: A short version of this work appeared in the proceedings of the 30th
annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum