On a k-matching algorithm and finding k-factors in random graphs with minimum degree k+1 in linear time

Anastos, Michael

doi:10.48550/arxiv.2107.03523

Cited by 2 publications

(14 citation statements)

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“…Query the entries of L v , one by one, in which of the sets V 1 , ..., V 4 they belong to until either you have identified for each j ′ ∈ [4] a set of 4000 entries that belong to V j ′ or you have queried min{q, |L v |} entries. If the later occurs then remove v from ∪ i∈ [4] V i and add it to V 7 . 5: end for 6: If more than 0.8 • 10 5 n queries have been made in total so far then return F AILU RE0.…”

Section: Phase 1: Partitioning the Vertex Setmentioning

confidence: 99%

“…We apply Algorithm 2 to each of the pairs (V 1 , V 2 ), (V 2 , V 3 ), (V 3 , V 4 ) and (U 1 , U 4 ) to find a perfect matching in the underlying bipartite graph. Before applying the algorithm we order the vertices in V i ⊂ [n] in an increasing order for i ∈ [4]. For i ∈ {1, 4} we also order the vertices in U i as follows.…”

Section: Phase 2: Finding Perfect Matchings Between the Partitionsmentioning

confidence: 99%

“…13: Let P = ∪{P C : C is a component of F }. 14: For P ∈ P remove V (P ) from ∪ i∈ [4] V i , add one of its endpoints to V 5 , the other to V 6 and all of its interior vertices to V 7 . 15:…”

mentioning

confidence: 99%

“…If no such vertices exists return FAILURE4. 21: end for 22: For each path P created remove V (P ) from ∪ i∈ [4] V i , add one of its endpoints to V 5 , the other to V 6 and all of its interior vertices to V 7 .…”

mentioning

confidence: 99%

“…Lemma 2. (d) For every v ∈ V the number of vertices in ∪ i∈ [4] I i within distance at most 10 from v is at most 300.…”

mentioning

confidence: 99%

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Fast algorithms for solving the Hamilton Cycle problem with high probability

Anastos¹

2021

Preprint

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We study the Hamilton cycle problem with input a random graph G ∼ G(n, p) in two settings. In the first one, G is given to us in the form of randomly ordered adjacency lists while in the second one we are given the adjacency matrix of G. In each of the settings we give a deterministic algorithm that w.h.p. either it finds a Hamilton cycle or it returns a certificate that such a cycle does not exists, for p ≥ 0. The running times of our algorithms are w.h.p. O(n) and O( n p ) respectively each being best possible in its own setting.Let u ∈ E ′ (v). Query the edges from u ∪ N M (u) to V ′ \ F orbidden.

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Section: Phase 1: Partitioning the Vertex Setmentioning

confidence: 99%

Section: Phase 2: Finding Perfect Matchings Between the Partitionsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Lemma 2. (d) For every v ∈ V the number of vertices in ∪ i∈ [4] I i within distance at most 10 from v is at most 300.…”

mentioning

confidence: 99%