Size Of Maximum Matching Research Articles

Improved approximation results for the stable marriage problem

The stable marriage problem has recently been studied in its general setting, where both ties and incomplete lists are allowed. It is NP-hard to find a stable matching of maximum size, while any stable matching is a maximal matching and thus trivially we can obtain a 2-approximation algorithm. In this article, we give the first nontrivial result for approximation of factor less than two. Our algorithm achieves an approximation ratio of 2/(1 + L −2 ) for instances in which only men have ties of length at most L . When both men and women are allowed to have ties but the lengths are limited to two, then we show a ratio of 13/7(<1.858). We also improve the lower bound on the approximation ratio to 21/19(>1.1052).

Degree-sequenced matching algorithms for input-queued switches

This paper presents a class of algorithms for scheduling packets in input-queued switches. As opposed to previously known algorithms that focus only on achieving high throughput, these algorithms seek to achieve low average delay without compromising the throughput achieved. Packet scheduling in input-queued switches based on the virtual-output-queued architecture is a bipartite graph matching problem wherein ports are represented by vertices and the traffic flows by the edges. The set of matched edges determine the packets that are to be transferred from the input ports to the output ports. Current matching algorithms implicitly prioritize high-degree vertices, i.e., ports with a large number of flows, causing longer delays at ports with a smaller number of flows. Motivated by this observation, we present three matching algorithms based on explicitly prioritizing low-degree vertices and the edges through them. Using both real gateway traffic traces as well as synthetically generated traffic, we present simulation results showing that this class of algorithms achieves a low average delay as compared to other scheduling algorithms of equivalent complexity while still achieving similar throughput. We also show that these algorithms determine the maximum size matching in almost all cases.

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = maxi,j|λi − λj|, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U*(n,k), and the graph with the largest spread in U(n,k).

Analytical evaluation of PHM convergence

The parallel hierarchical matching (PHM) algorithm is a distributed maximal size matching scheduler for virtual output-queued switches. In a previous letter, we formulated an upper bound on the maximum number of iterations PHM requires to achieve a maximal size matching in any traffic scenario. In this letter, we follow an analytical approach to find the average number of iterations for PHM to achieve a maximal size matching under diverse traffic models. The estimated number of iterations is O(log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> N), as in the case of iSLIP-like algorithms

Decoupled parallel hierarchical matching schedulers

AbstractThe load balanced Birkhoff–von Neumann switch is an elegant VOQ architecture with two outstanding characteristics: (i) it has a computational cost of O(1) iterations and (ii) input controllers do not exchange information (as a result, it allows decoupled implementations with a low power density). The load balancing stage guarantees stability under a broad class of traffic patterns. It may alter packet sequence, but this can be solved with appropriate packet selection strategies.The average packet delay caused by previous maximal size matching algorithms, such as iSLIP, RDSRR, or PHM is noticeably lower than that of a Birkhoff–von Neumann switch, especially for low and medium loads. However, they need tightly coupled VOQ controllers, which implies higher power density. For example, this makes difficult to apply those algorithms to optical switching architectures. Moreover, they require O(log2 N) iterations to converge, and this computational cost may be unacceptable for the slot lengths in optical packet switches.In this paper, we propose a family of decoupled Parallel Hierarchical Matching (PHM) VOQ controllers (DPHM). They outperform the Birkhoff–von Neumann scheduler, which can be viewed as a member of the family (in fact, the simplest one). DPHM schedulers have a computational cost of O(1) iterations and, unlike last generation maximal size matching algorithms, they allow a low input controller interconnection complexity (low power density switch implementation). Copyright © 2006 John Wiley &amp; Sons, Ltd.

Stability of a frame-based maximal weight matching algorithm with transfer speedup

It has been shown that maximal size matching algorithms for input-queued switches are stable under any admissible traffic conditions with a scheduling speedup of 2. However, as link speeds increase, the computational complexity of these algorithms limits their applicability in high port-density switches and routers. In this letter we describe a Frame-Based Maximal Weight Matching (FMWM) algorithm in which a new scheduling decision is issued once every several cell times. Between scheduling decisions, the configuration of the crossbar switch remains unchanged. We prove that the FMWM algorithm is stable with an internal buffer transfer speedup of 2, thereby significantly relaxing the timing constraints on the scheduling process. Simulation results illustrate the impact of the algorithm on the average cell delay for different traffic scenarios.

Stability analysis of input-buffered packet switches with maximal size matching

We analyze the stability of input-buffered packet switches with virtual output queue (VOQ), maximal size matching (MSM) algorithm and speedup. We first introduce a queueing system for the stability analysis. Using supplementary variables to represent the packet switching mode and remaining packet service time, we describe the behavior of the queueing system. Then, we propose a Lyapunov function to prove speedup of 2 is sufficient to achieve 100 % throughput in input-buffered packet switches with MSM algorithm.

Tight bounds on maximal and maximum matchings

In this paper, we study lower bounds on the size of maximal and maximum matchings in 3-connected planar graphs and graphs with bounded maximum degree. For each class, we give a lower bound on the size of matchings, and show that the bound is tight for some graph within the class.

Maximum size matching is unstable for any packet switch

Input-queued packet switches use a matching algorithm to configure a nonblocking switch fabric (e.g., a crossbar). Ideally, the matching algorithm will guarantee 100% throughput for a broad class of traffic, so long as the switch is not oversubscribed. An intuitive choice is the maximum size matching (MSM) algorithm, which maximizes the instantaneous throughput. It was shown (McKeown et al. (1999)) that with MSM the throughput can be less than 100% when N /spl ges/ 3, even with Terms-Instability,benign Bernoulli i.i.d. arrivals. In this letter, we extend this result to N /spl ges/ 2, and hence show it to be true for switches of any size.

On achieving throughput in an input-queued switch

We establish some lower bounds on the speedup required to achieve throughput for some classes of switching algorithms in a input-queued switch with virtual output queues (VOQs). We use a weak notion of throughput, which will only strengthen the results, since an algorithm that cannot achieve weak throughput cannot achieve stronger notions of throughput. We focus on priority switching algorithms, i.e., algorithms that assign priorities to VOQs and forward packets of high priority first. We show a lower bound on the speedup for two fairly general classes of priority switching algorithms: input priority switching algorithms and output priority switching algorithms. An input priority scheme prioritizes the VOQs based on the state of the input queues, while an output priority scheme prioritizes the VOQs based on their output ports. We first show that, for output priority switching algorithms, a speedup S/spl ges/2 is required to achieve weak throughput. From this, we deduce that both maximal and maximum size matching switching algorithms do not imply weak throughput unless S/spl ges/2. The bound of S/spl ges/2 is tight in all cases above, based on a result in Dai et al. Finally, we show that a speedup S/spl ges/3/2 is required for the class of input priority switching algorithms to achieve weak throughput.

Bounds on delays and queue lengths in input-queued cell switches

In this article, we develop a general methodology, mainly based upon Lyapunov functions, to derive bounds on average delays, and on averages and variances of queue lengths in complex systems of queues. We apply this methodology to cell-based switches and routers, considering first output-queued (OQ) architectures, in order to provide a simple example of our methodology, and then both input-queued (IQ), and combined input/output queued (CIOQ) architectures. These latter switching architectures require a scheduling algorithm to select at each slot a subset of input-buffered cells that can be transferred toward output ports. Although the stability properties (i.e., the limit throughput) of IQ and CIOQ cell-based switches were already studied for several classes of scheduling algorithms, very few analytical results concerning cell delays or queue lengths are available in the technical literature. We concentrate on Maximum Weight Matching (MWM) and Maximal Size Matching (mSM) scheduling algorithms; while the former was proved to maximize throughput, the latter allows simpler implementation. The derived bounds are shown to be rather tight when compared to simulation results.

On the behavior of phm distributed schedulers for input buffered packet switches

iSLIP and parallel hierarchical matching (PHM) are distributed maximal size matching schedulers for input-buffered switches. Previous research has analyzed the hardware cost of those schedulers and their performance after a small number of iterations. In this paper, we formulate an upper bound for the number of iterations required by PHM to converge. Then, we compare the number of iterations required by iSLIP and PHM to achieve a maximal throughput under uniform Bernoulli traffic, by means of simulation. Finally, we obtain the corresponding delay performances, which are similar. The results suggest that PHM has both the advantages of previous hierarchical matching algorithms (low hardware complexity) and iSLIP (low number of iterations).

Finding Maximum Induced Matchings in Subclasses of Claw-Free and P 5-Free Graphs, and in Graphs with Matching and Induced Matching of Equal Maximum Size

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P5,Dm)-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.

Hard variants of stable marriage

The Stable Marriage Problem and its many variants have been widely studied in the literature (Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989; Roth and Sotomayor, Two-sided matching: a study in game-theoretic modeling and analysis, Econometric Society Monographs, vol. 18, Cambridge University Press, Cambridge, 1990; Knuth, Stable Marriage and its Relation to Other Combinatorial Problems, CRM Proceedings and Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 1997), partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program (Roth, J. Political Economy 92(6) (1984) 991) and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stable—even if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an ‘egalitarian’ and a ‘minimum regret’ stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes.

A concentration inequality for maximum matching size in random graphs1

Consider a function f defined on the set of graphs on a fixed set of n vertices. Assume that f satisfies the following “continuity” condition: (*) |f(G) - f(G′)| ≦ 1 whenever G and G′ differ by at most one edge. (An example of such a function f is the maximum matching of the graph G) Then we prove that in either of the classical models and G n,p of random graphsf is very concentrated around its expectation. (The concentration bounds we obtain are of optimal order.).

On the maximum number of independent edges in cubic graphs

Let the reals be extended to include oo with o~ > r for every real nuraber r. Given an extended real number r, a property P(r) of graphs is super-hereditary if, whenever graph G has property P(r) and H is a subgraph of G, then H has property P(s) with s I> r. Notice that girth is a super-hereditary property if a forest has girth oo. In this paper, we prove that, given a super-hereditary proper~y P of graphs, if g is the smallest order of a graph with property P and having one vertex of degree 2 and all others of degree 3, and if G is a cubic graph with n vertices and property P, then G contains a matching with at least 1⁄2n(3g1)/(3g-~1) edges. In the case that P is the property of having girth r we describe all graphs having exactly the maximum matching size named. Our result has the corollary that every cubic graph with n vertices has a matching containing at least ~ n edges. Let L be a graph with super-hereditary property P in which one vertex has degree two and all others have degree 3 and such that the the order g of L is as s~nall as possible. For example, if P is 'having girth 3', L is /(4 with one edge subdivided and g = 5. If P is 'having girth 4', then L i,; K3.3 with one edge subdivided, and g = 7. The existence of L in the case that P is 'having girth r' ~.s shown by observing that an r-cage with one edge subdivided will serve (see [3, Theorem 8.82], for a proof that r-cages exist for all r>~3). Bollob~is and Eldridge [2], by the same method used in this paper, obtained the greatest lower bound for the size of a maximum matching in a k-connected (or Aedge-connected) graph with a given number of vertices anzl given minimal and maximal degrees. In this paper we find the greatest lower bound for the size of a maximum matching in a cubic graph with n vertices and with sup,~r-hereditary property P(r). In particular:

Algorithms for Loop Matchings

A simplified (two-base) version of the problem of planar folding of long chains (e.g., RNA and DNA biomolecules) is formulated as a matching problem. The chain is prescribed as a loop or circular sequence of letters A and B, n units long. A matching here means a set of A-B base pairings or matches obeying a planarity condition: no two matches may cross each other if drawn on the interior of the loop. Also, no two adjacent letters may be matched. We present a dynamic programming algorithm requiring $O( {n^3 } )$ steps and $O( {n^2 } )$ storage which computes the size of the maximum for the given A-B base sequence and which also allows reconstructing a particular folded form of the original string which realizes the maximum matching size. The algorithm can be adapted to deal with sequences with larger alphabets and with weighted matchings.An algorithm is also presented for a modified problem closer to the biochemical problem of interest: We demand that every match must be adjacent to another match, forcing ...