8 research outputs found
Kombinatorikus OptimalizĂĄlĂĄs: Algoritmusok, StrukturĂĄk, AlkalmazĂĄsok = Combinatorial optimization: algorithms, structures, applications
Mint azt az OTKA-pĂĄlyĂĄzat munkaterve tartalmazza, a pĂĄlyĂĄzatban rĂ©sztvevĆ kutatĂłk alkotjĂĄk a tĂ©mavezetĆ irĂĄnyĂtĂĄsĂĄval mƱködĆ EgervĂĄry JenĆ Kombinatorikus OptimalizĂĄlĂĄsi KutatĂłcsoportot. A csoport a kutatĂĄsi tervben szereplĆ több tĂ©mĂĄban jelentĆs eredmĂ©nyeket Ă©rt el az elmĂșlt 4 Ă©vben, ezekrĆl a pĂĄlyĂĄzat rĂ©sztvevĆinek több mint 50 folyĂłiratcikke jelent meg, Ă©s szĂĄmos rangos nemzetközi konferenciĂĄn ismertetĂ©sre kerĂŒltek. NĂ©hĂĄny kiemelendĆ eredmĂ©ny: sikerĂŒlt polinomiĂĄlis kombinatorikus algoritmust adni irĂĄnyĂtott grĂĄf pont-összefĂŒggĆsĂ©gĂ©nek növelĂ©sĂ©re; jelentĆs elĆrelĂ©pĂ©s törtĂ©nt a hĂĄromdimenziĂłs tĂ©rben merev grĂĄfok jellemzĂ©sĂ©vel Ă©s a molekulĂĄris sejtĂ©ssel kapcsolatban; 2 dimenziĂłban sikerĂŒlt bizonyĂtani Hendrickson sejtĂ©sĂ©t; a pĂĄrosĂtĂĄselmĂ©letben egy ĂșjdonsĂĄgnak szĂĄmĂtĂł mĂłdszerrel szĂĄmos Ășj algoritmikus eredmĂ©ny szĂŒletett; több, grĂĄfok Ă©lösszefĂŒggĆsĂ©gĂ©t jellemzĆ tĂ©telt sikerĂŒlt hipergrĂĄfokra ĂĄltalĂĄnosĂtani. | As the research plan indicates, the researchers participating in the project are the members of the EgervĂĄry Research Group, led by the coordinator. The group has made important progress in the past 4 years in the research topics declared in the research plan. The results have been published in more than 50 journal papers, and have been presented at several prestigious international conferences. The most significant results are the following: a polynomial algorithm has been found for the node-connectivity augmentation problem of directed graphs; considerable progress has been made towards the characterization of 3-dimensional rigid graphs and towards the proof of the molecular conjecture; Hendrickson's conjecture has been proved in 2 dimensions; several new algorithmic results were obtained in matching theory using a novel approach; several theorems characterizing connectivity properties of graphs have been generalized to hypergraphs
Many-to-one matchings with lower quotas : algorithms and complexity
We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = (AUP,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-toone matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-toone matching with lower and upper quotas (wmlq), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of wmlq from the viewpoints of classic polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umax as basis, and we prove that this dependence is necessary unless FPT = W[1]. Finally, we also present an approximation algorithm for the general case with performance guarantee umax+1, which is asymptotically best possible unless P = NP
Packing non-returning A-paths algorithmically
In this paper we present an algorithmic approach to packing A-paths. It is regarded as a generalization of Edmonds' matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the so-called 3-way lemma, which either provides augmentation, or a dual, or a subgraph which can be used for contraction. The method works in the general setting of packing non-returning A-paths. It also implies an ear-decomposition of criticals, as a generalization of the odd ear-decomposition of factor-critical graph
Representative set statements for delta-matroids and the Mader delta-matroid
We present representative sets-style statements for linear delta-matroids,
which are set systems that generalize matroids, with important connections to
matching theory and graph embeddings. Furthermore, our proof uses a new
approach of sieving polynomial families, which generalizes the linear algebra
approach of the representative sets lemma to a setting of bounded-degree
polynomials. The representative sets statements for linear delta-matroids then
follow by analyzing the Pfaffian of the skew-symmetric matrix representing the
delta-matroid. Applying the same framework to the determinant instead of the
Pfaffian recovers the representative sets lemma for linear matroids.
Altogether, this significantly extends the toolbox available for kernelization.
As an application, we show an exact sparsification result for Mader networks:
Let be a graph and a partition of a set of terminals , . A -path in is a path with endpoints
in distinct parts of and internal vertices disjoint from . In
polynomial time, we can derive a graph with ,
such that for every subset there is a packing of
-paths with endpoints in if and only if there is one in
, and . This generalizes the (undirected version of the)
cut-covering lemma, which corresponds to the case that contains
only two blocks.
To prove the Mader network sparsification result, we furthermore define the
class of Mader delta-matroids, and show that they have linear representations.
This should be of independent interest
Packing T-paths
Katedra aplikované matematikyDepartment of Applied MathematicsFaculty of Mathematics and PhysicsMatematicko-fyzikålnà fakult
矀ă©ăă«ä»ăă°ă©ăă«ăăăç”ćăæé©ć
ćŠäœăźçšźć„: èȘČçšć棫毩æ»ć§ćĄäŒć§ćĄ : ïŒäž»æ»ïŒæ±äșŹć€§ćŠææ ćČ©ç° èŠ, æ±äșŹć€§ćŠææ ćźć
Œ éŠćœŠ, æ±äșŹć€§ćŠææ ä»äș 攩, ćœç«æ
ć ±ćŠç 究æææ æČłćæ ć„äž, æ±äșŹć€§ćŠćææ ćčłäș ćșćżUniversity of Tokyo(æ±äșŹć€§ćŠ
The Path-Packing Structure of Graphs
We prove Edmonds-Gallai type structure theorems for Mader's edge- and vertex-disjoint paths including also capacitated variants, and state a conjecture generalizing Mader's minimax theorems on path packings and Cunningham and Geelen's path-matching theorem