Room Allocation With Capacity Diversity and Budget Constraints

With the development of the room rental market, many room rental websites have been created, e.g., SpareRoom and EasyRoommate. On these websites, people find not only rooms for rent but also suitable roommates. Inspired by the rental mode in practice, a benchmark room allocation model was introduced by Chan <italic>et al.</italic>, in which <inline-formula> <tex-math notation="LaTeX">$2{n}$ </tex-math></inline-formula> agents must be allocated to <inline-formula> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> rooms that have the same capacity and each agent can be allocated to any room. However, in practice, rooms may differ in terms of capacity, e.g., college dorms or apartments may contain both two-bed rooms and four-bed rooms. Moreover, an agent can only be allocated to a room of which the rent does not exceed the agent’s budget. In this scenario, we must consider not only the agents’ preferences but also the capacity diversity of the rooms and the budget constraints while allocating the rooms. Therefore, this paper investigates the room allocation problem with capacity diversity and budget constraints. We mainly focus on finding an allocation that maximizes social welfare. First, this paper demonstrates that finding an allocation that maximizes the social welfare is NP-hard (i.e., non-deterministic polynomial-time hard), even if only one room’s capacity is larger than 1 and the other rooms’ capacities are all 1. Second, this paper presents a (<inline-formula> <tex-math notation="LaTeX">${c} ^{*}$ </tex-math></inline-formula> + 2)/2 + <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>-factor approximation algorithm (with <inline-formula> <tex-math notation="LaTeX">$\varepsilon > 0$ </tex-math></inline-formula>) for the case in which the capacity of each room does not exceed a constant <inline-formula> <tex-math notation="LaTeX">${c} ^{*}$ </tex-math></inline-formula>. Third, this paper proposes a heuristic algorithm based on the local search for the general case in which the capacity of each room is not bounded by a constant. The experimental results demonstrate that the proposed algorithm can produce near-optimal solutions. Finally, this paper investigates how to find a roommate stable or room envy-free allocation with a social welfare guarantee.

[1]  Robert W. Irving,et al.  The Stable marriage problem - structure and algorithms , 1989, Foundations of computing series.

[2]  Mihalis Yannakakis,et al.  The Complexity of Multiterminal Cuts , 1994, SIAM J. Comput..

[3]  Rolf Niedermeier,et al.  Stable roommates with narcissistic, single-peaked, and single-crossing preferences , 2019, Autonomous Agents and Multi-Agent Systems.

[4]  Mohammed Abdulrahman Alqahtani,et al.  Grid Resource Allocation for Real-Time Data-Intensive Tasks , 2017, IEEE Access.

[5]  Eytan Ronn,et al.  NP-Complete Stable Matching Problems , 1990, J. Algorithms.

[6]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[7]  Kurt Mehlhorn,et al.  Popular matchings , 2005, SODA '05.

[8]  David Manlove,et al.  The exchange-stable marriage problem , 2005, Discret. Appl. Math..

[9]  Katarzyna E. Paluch Capacitated Rank-Maximal Matchings , 2013, CIAC.

[10]  Xin Huang,et al.  Online Roommate Allocation Problem , 2017, IJCAI.

[11]  David Manlove,et al.  Popular matchings in the weighted capacitated house allocation problem , 2010, J. Discrete Algorithms.

[12]  J. Drèze,et al.  HEDONIC COALITIONS: OPTIMALITY AND STABILITY , 1980 .

[13]  Konstantin Andreev,et al.  Balanced Graph Partitioning , 2004, SPAA '04.

[14]  Min Sheng,et al.  Energy-Efficient Resource Allocation for Heterogeneous Wireless Network With Multi-Homed User Equipments , 2018, IEEE Access.

[15]  Piotr Berman,et al.  A d/2 Approximation for Maximum Weight Independent Set in d-Claw Free Graphs , 2000, Nord. J. Comput..

[16]  Gerhard J. Woeginger,et al.  Group Activity Selection Problem , 2012, WINE.

[17]  Felix Brandt,et al.  Pareto Optimality in Coalition Formation , 2011, SAGT.

[18]  Vittorio Bilò,et al.  Nash Stable Outcomes in Fractional Hedonic Games: Existence, Efficiency and Computation , 2018, J. Artif. Intell. Res..

[19]  Felix Brandt,et al.  Stable partitions in additively separable hedonic games , 2011, AAMAS.

[20]  Mohammad Mahdian,et al.  Random popular matchings , 2006, EC '06.

[21]  Li Liu,et al.  Resource Allocation Optimization Based on Mixed Integer Linear Programming in the Multi-Cloudlet Environment , 2018, IEEE Access.

[22]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[23]  Tayfun Sönmez,et al.  Core in a simple coalition formation game , 2001, Soc. Choice Welf..

[24]  Michael A. Trick,et al.  Stable matching with preferences derived from a psychological model , 1986 .

[25]  Mikkel Thorup,et al.  Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut , 2004, Math. Oper. Res..

[26]  Haris Aziz Stable marriage and roommate problems with individual-based stability , 2013, AAMAS.

[27]  Robert W. Irving An Efficient Algorithm for the "Stable Roommates" Problem , 1985, J. Algorithms.

[28]  Katarína Cechlárová,et al.  On the complexity of exchange-stable roommates , 2002, Discret. Appl. Math..

[29]  Pasin Manurangsi,et al.  An Improved Integrality Gap for the Călinescu-Karloff-Rabani Relaxation for Multiway Cut , 2017, IPCO.

[30]  Roy Schwartz,et al.  Simplex Partitioning via Exponential Clocks and the Multiway-Cut Problem , 2018, SIAM J. Comput..

[31]  Barun Chandra,et al.  Greedy local improvement and weighted set packing approximation , 2001, SODA '99.

[32]  Roger Wattenhofer,et al.  Optimizing file availability in a secure serverless distributed file system , 2001, Proceedings 20th IEEE Symposium on Reliable Distributed Systems.

[33]  José Alcalde Exchange-proofness or divorce-proofness? Stability in one-sided matching markets , 1994 .

[34]  Nimrod Megiddo,et al.  A sublinear parallel algorithm for stable matching , 1994, SODA '94.

[35]  Howard J. Karloff,et al.  A lower bound of 8/(7+(1/k)-1) on the integrality ratio of the Calinescu-Karloff-Rabani relaxation for multiway cut , 2000, Inf. Process. Lett..

[36]  Chi-Kit Lam,et al.  A (1 + 1/e)-Approximation Algorithm for Maximum Stable Matching with One-Sided Ties and Incomplete Lists , 2019, SODA.

[37]  Julián Mestre,et al.  Weighted popular matchings , 2006, TALG.

[38]  Boris G. Pittel,et al.  On Likely Solutions of the Stable Matching Problem with Unequal Numbers of Men and Women , 2017, Math. Oper. Res..

[39]  Marcin Peski,et al.  Large roommate problem with non-transferable random utility , 2017, J. Econ. Theory.

[40]  Paul Harrenstein,et al.  Fractional Hedonic Games , 2014, AAMAS.

[41]  Yuval Rabani,et al.  An improved approximation algorithm for multiway cut , 1998, STOC '98.

[42]  David Manlove,et al.  The Stable Roommates Problem with Ties , 2002, J. Algorithms.

[43]  Felix Brandt,et al.  Optimal Partitions in Additively Separable Hedonic Games , 2011, IJCAI.

[44]  Jan Vondrák,et al.  Multiway cut, pairwise realizable distributions, and descending thresholds , 2014, STOC.

[45]  Joachim Gudmundsson,et al.  Welfare Maximization in Fractional Hedonic Games , 2015, IJCAI.

[46]  Matthew O. Jackson,et al.  The Stability of Hedonic Coalition Structures , 2002, Games Econ. Behav..

[47]  Roy Schwartz,et al.  Simplex Transformations and the Multiway Cut Problem , 2017, SODA.

[48]  Xin Huang,et al.  Assignment and Pricing in Roommate Market , 2016, AAAI.

[49]  L. S. Shapley,et al.  College Admissions and the Stability of Marriage , 2013, Am. Math. Mon..

[50]  Katarína Cechlárová,et al.  Stability in coalition formation games , 2000, Int. J. Game Theory.

[51]  David Manlove,et al.  Popular Matchings in the Capacitated House Allocation Problem , 2006, ESA.

[52]  David Manlove,et al.  Popular Matchings in the Marriage and Roommates Problems , 2010, CIAC.

[53]  Vasileios Pappas,et al.  Improving the Scalability of Data Center Networks with Traffic-aware Virtual Machine Placement , 2010, 2010 Proceedings IEEE INFOCOM.

[54]  Edith Elkind,et al.  Group Activity Selection on Social Networks , 2017, AAAI.