2026-07-26T04:50:19Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:18068 2024-03-06T11:00:55Z ddc:004 open_access

Approximation Algorithms for Envy-Free Cake Division with Connected Pieces Barman, Siddharth Kulkarni, Pooja Fair Division Envy-Freeness Envy-Cycle Elimination Cake cutting is a classic model for studying fair division of a heterogeneous, divisible resource among agents with individual preferences. Addressing cake division under a typical requirement that each agent must receive a connected piece of the cake, we develop approximation algorithms for finding envy-free (fair) cake divisions. In particular, this work improves the state-of-the-art additive approximation bound for this fundamental problem. Our results hold for general cake division instances in which the agents' valuations satisfy basic assumptions and are normalized (to have value 1 for the cake). Furthermore, the developed algorithms execute in polynomial time under the standard Robertson-Webb query model. Prior work has shown that one can efficiently compute a cake division (with connected pieces) in which the additive envy of any agent is at most 1/3. An efficient algorithm is also known for finding connected cake divisions that are (almost) 1/2-multiplicatively envy-free. Improving the additive approximation guarantee and maintaining the multiplicative one, we develop a polynomial-time algorithm that computes a connected cake division that is both (1/4 +o(1))-additively envy-free and (1/2 - o(1))-multiplicatively envy-free. Our algorithm is based on the ideas of interval growing and envy-cycle elimination. In addition, we study cake division instances in which the number of distinct valuations across the agents is parametrically bounded. We show that such cake division instances admit a fully polynomial-time approximation scheme for connected envy-free cake division. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Siddharth Barman and Pooja Kulkarni 2023 Is Part Of LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2023.16 urn:nbn:de:0030-drops-180685 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.16 eng https://creativecommons.org/licenses/by/4.0/legalcode