340 research outputs found
Cycle packing
In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every
graph on n vertices can be partitioned into O(n) cycles and edges. They
observed that one can easily get an O(n log n) upper bound by repeatedly
removing the edges of the longest cycle. We make the first progress on this
problem, showing that O(n log log n) cycles and edges suffice. We also prove
the Erd\H{o}s-Gallai conjecture for random graphs and for graphs with linear
minimum degree.Comment: 18 page
Kernelization of Cycle Packing with Relaxed Disjointness Constraints
A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP subseteq coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of "constraints" that separate "kernelizable" from "non-kernelizable" variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a "global" relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a "local" relaxation parameter and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t >= 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k/c, where c could be a function of k, we obtain a kernel of size O(2^{c^{2}}*k^{7+c}*log^3(k)) whenever c in o(sqrt(k))). Thus the kernel size varies from being sub-exponential when c in o(sqrt(k)), to quasipolynomial when c in o(log^l(k)), l in R_+, and polynomial when c in O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t in O(k^{epsilon}), for any 0 <= epsilon < 1
Maximum weight cycle packing in directed graphs, with application to kidney exchange programs
Centralized matching programs have been established in several countries to organize kidney exchanges between incompatible patient-donor pairs. At the heart of these programs are algorithms to solve kidney exchange problems, which can be modelled as cycle packing problems in a directed graph, involving cycles of length 2, 3, or even longer. Usually, the goal is to maximize the number of transplants, but sometimes the total benefit is maximized by considering the differences between suitable kidneys. These problems correspond to computing cycle packings of maximum size or maximum weight in directed graphs. Here we prove the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles. We also present an approximation algorithm and an exact algorithm for the problem of finding a maximum weight exchange involving cycles of bounded length. The exact algorithm has been used to provide optimal solutions to real kidney exchange problems arising from the National Matching Scheme for Paired Donation run by NHS Blood and Transplant, and we describe practical experience based on this collaboration
Chordless Cycle Packing Is Fixed-Parameter Tractable
A chordless cycle or hole in a graph G is an induced cycle of length at least 4. In the Hole Packing problem, a graph G and an integer k is given, and the task is to find (if exists) a set of k pairwise vertex-disjoint chordless cycles. Our main result is showing that Hole Packing is fixed-parameter tractable (FPT), that is, can be solved in time f(k)n^O(1) for some function f depending only on k
Packing cycles faster than Erdos-Posa
The Cycle Packing problem asks whether a given undirected graph contains vertex-disjoint cycles. Since the publication of the classic ErdĂśs--PĂłsa theorem in 1965, this problem received significant attention in the fields of graph theory and algorithm design. In particular, this problem is one of the first problems studied in the framework of parameterized complexity. The nonuniform fixed-parameter tractability of Cycle Packing follows from the Robertson--Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time using exponential space. In the case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a -time (deterministic) algorithm using exponential space, which is a consequence of the ErdĂśs--PĂłsa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on parameterized complexity. Yet, no algorithm that runs in time , beating the bound , has been found. In light of this, it seems natural to ask whetherthe bound is essentially optimal. In this paper, we answer this question negatively by developing a -time (deterministic) algorithm for Cycle Packing. In the case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound , our algorithm runs in time linear in , and its space complexity is polynomial in the input size.publishedVersio
Improved Approximation Algorithms for Cycle and Path Packings
Given an edge-weighted (metric/general) complete graph with vertices, the
maximum weight (metric/general) -cycle/path packing problem is to find a set
of vertex-disjoint -cycles/paths such that the total weight is
maximized. In this paper, we consider approximation algorithms. For metric
-cycle packing, we improve the previous approximation ratio from to
for , and from for to
for constant odd and to for even . For metric -path packing, we
improve the approximation ratio from to
for even . For the case of
, we improve the approximation ratio from to for metric
4-cycle packing, from to for general 4-cycle packing, and from
to for metric 4-path packing.Comment: To appear in WALCOM 202
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