--- _id: '10904' abstract: - lang: eng text: Multi-dimensional mean-payoff and energy games provide the mathematical foundation for the quantitative study of reactive systems, and play a central role in the emerging quantitative theory of verification and synthesis. In this work, we study the strategy synthesis problem for games with such multi-dimensional objectives along with a parity condition, a canonical way to express ω-regular conditions. While in general, the winning strategies in such games may require infinite memory, for synthesis the most relevant problem is the construction of a finite-memory winning strategy (if one exists). Our main contributions are as follows. First, we show a tight exponential bound (matching upper and lower bounds) on the memory required for finite-memory winning strategies in both multi-dimensional mean-payoff and energy games along with parity objectives. This significantly improves the triple exponential upper bound for multi energy games (without parity) that could be derived from results in literature for games on VASS (vector addition systems with states). Second, we present an optimal symbolic and incremental algorithm to compute a finite-memory winning strategy (if one exists) in such games. Finally, we give a complete characterization of when finite memory of strategies can be traded off for randomness. In particular, we show that for one-dimension mean-payoff parity games, randomized memoryless strategies are as powerful as their pure finite-memory counterparts. acknowledgement: 'Author supported by Austrian Science Fund (FWF) Grant No P 23499-N23, FWF NFN Grant No S11407 (RiSE), ERC Start Grant (279307: Graph Games), Microsoft faculty fellowship.' alternative_title: - LNCS article_processing_charge: No author: - first_name: Krishnendu full_name: Chatterjee, Krishnendu id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87 last_name: Chatterjee orcid: 0000-0002-4561-241X - first_name: Mickael full_name: Randour, Mickael last_name: Randour - first_name: Jean-François full_name: Raskin, Jean-François last_name: Raskin citation: ama: 'Chatterjee K, Randour M, Raskin J-F. Strategy synthesis for multi-dimensional quantitative objectives. In: Koutny M, Ulidowski I, eds. CONCUR 2012 - Concurrency Theory. Vol 7454. Berlin, Heidelberg: Springer; 2012:115-131. doi:10.1007/978-3-642-32940-1_10' apa: 'Chatterjee, K., Randour, M., & Raskin, J.-F. (2012). Strategy synthesis for multi-dimensional quantitative objectives. In M. Koutny & I. Ulidowski (Eds.), CONCUR 2012 - Concurrency Theory (Vol. 7454, pp. 115–131). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-642-32940-1_10' chicago: 'Chatterjee, Krishnendu, Mickael Randour, and Jean-François Raskin. “Strategy Synthesis for Multi-Dimensional Quantitative Objectives.” In CONCUR 2012 - Concurrency Theory, edited by Maciej Koutny and Irek Ulidowski, 7454:115–31. Berlin, Heidelberg: Springer, 2012. https://doi.org/10.1007/978-3-642-32940-1_10.' ieee: K. Chatterjee, M. Randour, and J.-F. Raskin, “Strategy synthesis for multi-dimensional quantitative objectives,” in CONCUR 2012 - Concurrency Theory, Newcastle upon Tyne, United Kingdom, 2012, vol. 7454, pp. 115–131. ista: 'Chatterjee K, Randour M, Raskin J-F. 2012. Strategy synthesis for multi-dimensional quantitative objectives. CONCUR 2012 - Concurrency Theory. CONCUR: Conference on Concurrency Theory, LNCS, vol. 7454, 115–131.' mla: Chatterjee, Krishnendu, et al. “Strategy Synthesis for Multi-Dimensional Quantitative Objectives.” CONCUR 2012 - Concurrency Theory, edited by Maciej Koutny and Irek Ulidowski, vol. 7454, Springer, 2012, pp. 115–31, doi:10.1007/978-3-642-32940-1_10. short: K. Chatterjee, M. Randour, J.-F. Raskin, in:, M. Koutny, I. Ulidowski (Eds.), CONCUR 2012 - Concurrency Theory, Springer, Berlin, Heidelberg, 2012, pp. 115–131. conference: end_date: 2012-09-07 location: Newcastle upon Tyne, United Kingdom name: 'CONCUR: Conference on Concurrency Theory' start_date: 2012-09-04 date_created: 2022-03-21T08:00:21Z date_published: 2012-09-15T00:00:00Z date_updated: 2023-02-23T10:55:06Z day: '15' department: - _id: KrCh doi: 10.1007/978-3-642-32940-1_10 ec_funded: 1 editor: - first_name: Maciej full_name: Koutny, Maciej last_name: Koutny - first_name: Irek full_name: Ulidowski, Irek last_name: Ulidowski external_id: arxiv: - '1201.5073' intvolume: ' 7454' language: - iso: eng month: '09' oa_version: Preprint page: 115-131 place: Berlin, Heidelberg project: - _id: 2584A770-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: P 23499-N23 name: Modern Graph Algorithmic Techniques in Formal Verification - _id: 25863FF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: S11407 name: Game Theory - _id: 2581B60A-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '279307' name: 'Quantitative Graph Games: Theory and Applications' - _id: 2587B514-B435-11E9-9278-68D0E5697425 name: Microsoft Research Faculty Fellowship publication: CONCUR 2012 - Concurrency Theory publication_identifier: eisbn: - '9783642329401' isbn: - '9783642329395' issn: - 0302-9743 - 1611-3349 publication_status: published publisher: Springer quality_controlled: '1' related_material: record: - id: '2716' relation: later_version status: public scopus_import: '1' status: public title: Strategy synthesis for multi-dimensional quantitative objectives type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 7454 year: '2012' ...