729 research outputs found
The Stable Roommates problem with short lists
We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists (sri) that are degree constrained, i.e., preference lists are of bounded length. The first variant, egald-sri, involves finding an egalitarian stable matching in solvable instances of sri with preference lists of length at most d. We show that this problem is NP-hard even if d = 3. On the positive side we give a 2d+372d+37-approximation algorithm for d ∈{3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called d-srti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-srti admits a stable matching is NP-complete even if d = 3. We also consider the “most stable” version of this problem and prove a strong inapproximability bound for the d = 3 case. However for d = 2 we show that the latter problem can be solved in polynomial time
Stable marriage and roommates problems with restricted edges: complexity and approximability
In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs.
Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs.
Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs
The Stable Roommates problem with short lists
We consider two variants of the classical Stable Roommates problem with
Incomplete (but strictly ordered) preference lists SRI that are degree
constrained, i.e., preference lists are of bounded length. The first variant,
EGAL d-SRI, involves finding an egalitarian stable matching in solvable
instances of SRI with preference lists of length at most d. We show that this
problem is NP-hard even if d=3. On the positive side we give a
(2d+3)/7-approximation algorithm for d={3,4,5} which improves on the known
bound of 2 for the unbounded preference list case. In the second variant of
SRI, called d-SRTI, preference lists can include ties and are of length at most
d. We show that the problem of deciding whether an instance of d-SRTI admits a
stable matching is NP-complete even if d=3. We also consider the "most stable"
version of this problem and prove a strong inapproximability bound for the d=3
case. However for d=2 we show that the latter problem can be solved in
polynomial time.Comment: short version appeared at SAGT 201
Matching measure, Benjamini-Schramm convergence and the monomer-dimer free energy
We define the matching measure of a lattice L as the spectral measure of the
tree of self-avoiding walks in L. We connect this invariant to the
monomer-dimer partition function of a sequence of finite graphs converging to
L.
This allows us to express the monomer-dimer free energy of L in terms of the
measure. Exploiting an analytic advantage of the matching measure over the
Mayer series then leads to new, rigorous bounds on the monomer-dimer free
energies of various Euclidean lattices. While our estimates use only the
computational data given in previous papers, they improve the known bounds
significantly.Comment: 18 pages, 3 figure
The weighted stable matching problem
We study the stable matching problem in non-bipartite graphs with incomplete
but strict preference lists, where the edges have weights and the goal is to
compute a stable matching of minimum or maximum weight. This problem is known
to be NP-hard in general. Our contribution is two fold: a polyhedral
characterization and an approximation algorithm. Previously Chen et al. have
shown that the stable matching polytope is integral if and only if the subgraph
obtained after running phase one of Irving's algorithm is bipartite. We improve
upon this result by showing that there are instances where this subgraph might
not be bipartite but one can further eliminate some edges and arrive at a
bipartite subgraph. Our elimination procedure ensures that the set of stable
matchings remains the same, and thus the stable matching polytope of the final
subgraph contains the incidence vectors of all stable matchings of our original
graph. This allows us to characterize a larger class of instances for which the
weighted stable matching problem is polynomial-time solvable. We also show that
our edge elimination procedure is best possible, meaning that if the subgraph
we arrive at is not bipartite, then there is no bipartite subgraph that has the
same set of stable matchings as the original graph. We complement these results
with a -approximation algorithm for the minimum weight stable matching
problem for instances where each agent has at most two possible partners in any
stable matching. This is the first approximation result for any class of
instances with general weights.Comment: This is an extended version of a paper to appear at the The Fourth
International Workshop on Matching Under Preferences (MATCH-UP 2017
Approximability of Connected Factors
Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by
Tutte's reduction to the matching problem. By the same reduction, it is easy to
find a minimal or maximal d-factor of a graph. However, if we require that the
d-factor is connected, these problems become NP-hard - finding a minimal
connected 2-factor is just the traveling salesman problem (TSP).
Given a complete graph with edge weights that satisfy the triangle
inequality, we consider the problem of finding a minimal connected -factor.
We give a 3-approximation for all and improve this to an
(r+1)-approximation for even d, where r is the approximation ratio of the TSP.
This yields a 2.5-approximation for even d. The same algorithm yields an
(r+1)-approximation for the directed version of the problem, where r is the
approximation ratio of the asymmetric TSP. We also show that none of these
minimization problems can be approximated better than the corresponding TSP.
Finally, for the decision problem of deciding whether a given graph contains
a connected d-factor, we extend known hardness results.Comment: To appear in the proceedings of WAOA 201
Optimal Approximation Algorithms for Multi-agent Combinatorial Problems with Discounted Price Functions
Submodular functions are an important class of functions in combinatorial
optimization which satisfy the natural properties of decreasing marginal costs.
The study of these functions has led to strong structural properties with
applications in many areas. Recently, there has been significant interest in
extending the theory of algorithms for optimizing combinatorial problems (such
as network design problem of spanning tree) over submodular functions.
Unfortunately, the lower bounds under the general class of submodular functions
are known to be very high for many of the classical problems.
In this paper, we introduce and study an important subclass of submodular
functions, which we call discounted price functions. These functions are
succinctly representable and generalize linear cost functions. In this paper we
study the following fundamental combinatorial optimization problems: Edge
Cover, Spanning Tree, Perfect Matching and Shortest Path, and obtain tight
upper and lower bounds for these problems.
The main technical contribution of this paper is designing novel adaptive
greedy algorithms for the above problems. These algorithms greedily build the
solution whist rectifying mistakes made in the previous steps
The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable
Monadic second order logic can be used to express many classical notions of
sets of vertices of a graph as for instance: dominating sets, induced
matchings, perfect codes, independent sets or irredundant sets. Bounds on the
number of sets of any such family of sets are interesting from a combinatorial
point of view and have algorithmic applications. Many such bounds on different
families of sets over different classes of graphs are already provided in the
literature. In particular, Rote recently showed that the number of minimal
dominating sets in trees of order is at most and that
this bound is asymptotically sharp up to a multiplicative constant. We build on
his work to show that what he did for minimal dominating sets can be done for
any family of sets definable by a monadic second order formula.
We first show that, for any monadic second order formula over graphs that
characterizes a given kind of subset of its vertices, the maximal number of
such sets in a tree can be expressed as the \textit{growth rate of a bilinear
system}. This mostly relies on well known links between monadic second order
logic over trees and tree automata and basic tree automata manipulations. Then
we show that this "growth rate" of a bilinear system can be approximated from
above.We then use our implementation of this result to provide bounds on the
number of independent dominating sets, total perfect dominating sets, induced
matchings, maximal induced matchings, minimal perfect dominating sets, perfect
codes and maximal irredundant sets on trees. We also solve a question from D.
Y. Kang et al. regarding -matchings and improve a bound from G\'orska and
Skupie\'n on the number of maximal matchings on trees. Remark that this
approach is easily generalizable to graphs of bounded tree width or clique
width (or any similar class of graphs where tree automata are meaningful)
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