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Yuta Nozaki

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4

SODA Conference 2022 Conference Paper

Monotone edge flips to an orientation of maximum edge-connectivity à la Nash-Williams

  • Takehiro Ito
  • Yuni Iwamasa
  • Naonori Kakimura
  • Naoyuki Kamiyama
  • Yusuke Kobayashi 0001
  • Shun-ichi Maezawa
  • Yuta Nozaki
  • Yoshio Okamoto

We initiate the study of k -edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2 k -edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge-connectivity, and the final orientation is k -edge-connected. This yields an “edge-flip based” new proof of Nash-Williams' theorem: an undirected graph G has a k -edge-connected orientation if and only if G is 2 k -edge-connected. As another consequence of the theorem, we prove that the edge-flip graph of k -edge-connected orientations of an undirected graph G is connected if G is (2 k + 2)-edge-connected. This has been known to be true only when k = 1.

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AAAI Conference 2022 Conference Paper

Reforming an Envy-Free Matching

  • Takehiro Ito
  • Yuni Iwamasa
  • Naonori Kakimura
  • Naoyuki Kamiyama
  • Yusuke Kobayashi
  • Yuta Nozaki
  • Yoshio Okamoto
  • Kenta Ozeki

We consider the problem of reforming an envy-free matching when each agent is assigned a single item. Given an envy-free matching, we consider an operation to exchange the item of an agent with an unassigned item preferred by the agent that results in another envy-free matching. We repeat this operation as long as we can. We prove that the resulting envy-free matching is uniquely determined up to the choice of an initial envy-free matching, and can be found in polynomial time. We call the resulting matching a reformist envy-free matching, and then we study a shortest sequence to obtain the reformist envy-free matching from an initial envy-free matching. We prove that a shortest sequence is computationally hard to obtain even when each agent accepts at most four items and each item is accepted by at most three agents. On the other hand, we give polynomial-time algorithms when each agent accepts at most three items or each item is accepted by at most two agents. Inapproximability and fixed-parameter (in)tractability are also discussed.

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