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New approaches to multi-objective optimization

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Abstract

A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Many classical optimization problems, such as maximum spanning tree and forest, shortest path, maximum weight (perfect) matching, maximum weight independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. Not much is known however about the case of two or more budgets: filling this gap, at least partially, is the main goal of this paper. In more detail, we obtain the following main results: Using iterative rounding for the first time in multi-objective optimization, we obtain multi-criteria PTASs (which slightly violate the budget constraints) for spanning tree, matroid basis, and bipartite matching with \(k=O(1)\) budget constraints. We present a simple mechanism to transform multi-criteria approximation schemes into pure approximation schemes for problems whose feasible solutions define an independence system. This gives improved algorithms for several problems. In particular, this mechanism can be applied to the above bipartite matching algorithm, hence obtaining a pure PTAS. We show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for \(k\)-budgeted matroid independent set. We present a deterministic approximation scheme for \(k\)-budgeted matching (in general graphs), where \(k=O(1)\). Interestingly, to show that our procedure works, we rely on a non-constructive result by Stromquist and Woodall, which is based on the Ham Sandwich Theorem.

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Notes

  1. The assumption that \(k\) is a constant is crucial in this paper, since many of the presented algorithms will have a running time that is exponential in \(k\), but polynomial for constant \(k\).

  2. We recall that \(E\) is a finite ground set and \(\mathcal{I }\subseteq 2^{E}\) is a nonempty family of subsets of \(E\) (independent sets) which have to satisfy the following two conditions: (i) \(I\in \mathcal{I },\;J\subseteq I\;\Rightarrow \;J\in \mathcal{I }\) and (ii) \(I,J\in \mathcal{I },|I|>|J|\;\Rightarrow \;\exists z\in I{\setminus } J:\;J\cup \{z\}\in \mathcal{I }\). A basis is a maximal independent set. For all matroids used in this paper we make the usual assumption that independence of a set can be checked in polynomial time. For additional information on matroids, see e.g. [38, Volume B].

  3. A matroid \(M=(E,\mathcal{I })\) is representable if its ground set \(E\) can be mapped in a bijective way to the columns of a matrix over some field, and \(I\subseteq E\) is independent in \(M\) iff the corresponding columns are linearly independent.

  4. For some given matroid \(M=(E,\mathcal{I })\), the corresponding matroid polytope \(P_\mathcal{I }\) is the convex hull of the incidence vectors of all independent sets.

  5. Notice that it suffices to assume that \(\fancyscript{F}\) is closed under contractions, since a restriction can be emulated by setting the weights of the elements to be removed to zero.

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Acknowledgments

We are grateful to Christian Reitwießner and Maximilian Witek for pointing us to the paper of Stromquist and Woodall [42]. Furthermore, we are thankful to the two anonymous referees, whose comments and suggestions considerably improved the presentation of this paper. F. Grandoni: Partially supported by the ERC Starting Grant NEWNET 279352. R. Ravi: Supported in part by NSF grant CCF-0728841. R. Zenklusen: Supported by Swiss National Science Foundation Grant PBEZP2-129524, by NSF Grants CCF-1115849 and CCF-0829878, and by ONR Grants N00014-11-1-0053 and N00014-09-1-0326.

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Authors and Affiliations

  1. IDSIA, University of Italian Switzerland, Manno, Switzerland

    Fabrizio Grandoni

  2. Tepper School of Business, Carnegie Mellon University, Pittsburgh, USA

    R. Ravi

  3. School of Computer Science, McGill University, Montreal, Canada

    Mohit Singh

  4. Institute for Operations Research (IFOR), Department of Mathematics, ETH, Zurich, Switzerland

    Rico Zenklusen

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  1. Fabrizio Grandoni
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  2. R. Ravi
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  3. Mohit Singh
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  4. Rico Zenklusen
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Correspondence to Rico Zenklusen.

Additional information

A preliminary version of many results in this paper appeared in ESA’09 [17] and ESA’10 [18].

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Grandoni, F., Ravi, R., Singh, M. et al. New approaches to multi-objective optimization. Math. Program. 146, 525–554 (2014). https://doi.org/10.1007/s10107-013-0703-7

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