Title: Accelerating Matroid Optimization through Fast Imprecise Oracles
Venue: NeurIPS 2024
Accepted: yes
Keywords: matroid optimization

weak-strong models

learning-augmented algorithms

algorithms with predictions

query minimization

robustness
Abstract: Querying complex models for precise information (e.g. traffic models, database systems, large ML models) often entails intense computations and results in long response times. Thus, weaker models which give imprecise results quickly can be advantageous, provided inaccuracies can be resolved using few queries to a stronger model. In the fundamental problem of computing a maximum-weight basis of a matroid, a well-known generalization of many combinatorial optimization problems, algorithms have access to a clean oracle to query matroid information. We additionally equip algorithms with a fast but dirty oracle. We design and analyze practical algorithms which only use few clean queries w.r.t. the quality of the dirty oracle, while maintaining robustness against arbitrarily poor dirty oracles, approaching the performance of classic algorithms for the given problem. Notably, we prove that our algorithms are, in many respects, best-possible. Further, we outline extensions to other matroid oracle types, non-free dirty oracles and other matroid problems.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the problem of finding a maximum-weight basis of a matroid in a learning-augmented setting, where there are two different oracles that the algorithm can query to check whether a set is independent: An exact oracle that always gives correct answers (but may be slow), and a dirty oracle that may give incorrect answers. Formally, the answers of the dirty oracle correspond to some other \"dirty\" matroid (or more generally, downward closed set system) that ideally is close to the true matroid. In the setting considered, queries to the dirty oracle are free, and the goal is to minimize the number of queries to the exact oracle to find a maximum weight basis.\n\nThe quality of the dirty oracle is parametrized by two error measures, corresponding roughly to the numbers of elements that would need to be added/removed to move from a max-weight basis for the dirty matroid to a max-weight basis for the true matroid. The main result is an algorithm whose number of clean queries interpolates (as a function of the error parameters) between n-r+k (where n is the number of items in the ground set and r the size of any basis, and $k\\ge 1$ is an algorithm parameter that can be chosen) and $n(1+1/k)$. One should note that without dirty queries, the problem requires n exact queries. The dependence on the error in the interpolation is shown to be asymptotically optimal.\n\nIf the algorithm has access to rank queries rather than just independence queries (rank queries return the rank of a set, i.e., the size of the maximal independent subset) the guarantees can be improved to a quantity interpolating beetween 2 and n+1 exact queries depending on the input.\n\nAn extension of the main results includes an application to the matrix intersection problem.\n\nSTRENGTHS:\n- Essentially tight bounds.\n- Non-trivial algorithms. They are not overly complicated, but getting everything right requires some care.\n- Clear presentation. The simpler algorithm for the unweighted setting is helpful before diving into the unweighted setting.\n\nWEAKNESSES:\n- In order for the improvement of the algorithm for independence queries to be a super-constant factor requires n-r is sub-linear in n. It is not clear how realistic this is. The authors hint at graphic networks in sparse graphs having high rank -- does this mean $r=\\Omega(n)$ or even $r = n - o(n)$? The result in the case of rank queries avoids this issue though.\n- Significant work is required for the robustness result, i.e., to ensure the number of clean queries is simultaneously bounded by a function of the error parameters and by a quantity slightly larger than n. However, the trivial approach of running two algorithms in parallel would achieve a similar result, losing only a factor of 2, so the improvement over this trivial approach is relatively small.\n- The extension to matroid intersection seems relatively weak (strong assumptions and an error measure that can be exponentially large in n in the first setting). Since this is only an extension rather than a main result, this does not affect the main contribution though.\n\nQUESTIONS:\nPlease elaborate on the size of n-r in typical settings, in relation to the first possible weakness above.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper aims to solve fundamental matroid optimization problems, specifically, computing a maximum-weight basis of a matroid, a complex combinatorial optimization problem, To this end, the author proposes a two-oracle model, which uses fast but dirty oracle to reduce the time to call clean oracles. Then, the author proposes the algorithm to compute the maximum-weight basis and gives the upper bound of the oracles calls needed. Finally, the author also discusses advanced settings like different kinds of oracles and other matroid optimization problem.\n\nSTRENGTHS:\n1. The theoretical technique of this paper seems strong, the paper gives a theorem and detailed proof of the bound of the cost to call oracles.\n\n2. The two-model design is superior to traditional methods which require n+1 clean calls, this seems to be a huge increase.\n\n3. The extended result in the Section 4 is interesting, it helps to show that the algorithm is generalizable and can be suitable for different settings.\n\nWEAKNESSES:\n1. The organization of this paper is very confusing, for example, the author includes a large part of the preliminary in its introduction rather than an independent section. Moreover, the author includes many extended results and propositions in its final section 4, but doesn't include a conclusion.\n\n2. The paper studies combinatorial optimization problems which can be hard to comprehend for readers with not much background knowledge, while the paper is not self-contained enough. The preliminary in Section 1.3 alone is not enough to grasp an intuitive concept of that.\n\n3. This paper doesn't include any empirical experiment.\n\nQUESTIONS:\nPerhaps it will be beneficial to include more figures as illustrations? For example, you can give an example of the algorithm walkthrough as a toy example. Also, it can also be helpful for the readers to understand the new proposed concept like k-safe and modification example.\n\nLIMITATIONS:\nI think the author has stated the limitations as he/she clearly stated the necessary assumptions in the respective theorem.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper mainly studies the problem of finding a maximum weight basis in a matroid $\\mathcal{M}=(E,\\mathcal{I})$ using two types of independence oracles \"clean\" and \"dirty\". The clean oracle determines whether a set $S \\subseteq E$ is an independent set in $\\mathcal{M}$, and the dirty oracle determines independence according to another matroid $\\mathcal{M}_d=(E,\\mathcal{I}_d)$. The dirty oracle is free but might be imprecise for $\\mathcal{M}$. For measuring the error of $\\mathcal{M}_d$ with respect to $\\mathcal{M}$, the parameters $\\eta_A$ and $\\eta_R$ are defined, which intuitively are the number of elements that have to be added to/removed from a maximum-weight basis of $\\mathcal{M}_d$ to reach a maximum-weight basis of $\\mathcal{M}$.\nThe main result of the paper is an algorithm that computes a maximum-weight basis of $\\mathcal{M}$ using at most $\\min (n-r+k+\\eta_A \\cdot (k+1)+ \\eta_R \\cdot (k+1) \\lceil \\log_2 r_d \\rceil, (1+1/k)n )$ calls to the clean oracle, where $n=|E|$, $k$ is a positive integer, and $r$ and $r_d$ are the ranks of $\\mathcal{M}$ and $\\mathcal{M}_d$, respectively. The authors also prove lower bounds that show any deterministic algorithm should have dependencies with respect to $n$, $r$, $\\eta_A$, and $\\eta_B$ that are similar to those of the proposed algorithm.\n\nSTRENGTHS:\n* The authors provide interesting and nontrivial upper and lower bounds for a fundamental problem. \n* The two-oracle model considered, which can be viewed as a learning-augmented model, is theoretically interesting.\n* The paper is well-written, and the algorithms are presented in a logical sequence that is easy to follow. The warm-up algorithms introduced in section 2 are especially helpful in facilitating understanding of the techniques and challenges.\n\nWEAKNESSES:\n* The parameter $k$ in the main result of the paper is intuitively used to determine how much we want to trust the dirty oracle. When \n$k$ is set close to 0 to heavily favor the dirty oracle, if the errors $\\eta_A$ and $\\eta_R$ turn out to be high, the algorithm might use $\\Omega(n \\log n)$ calls to the clean oracle. This significantly exceeds the calls required by the optimal worst-case algorithm without predictions, which uses $n$ calls to the clean oracle. In this sense, the algorithm is not robust.\n* The results lack empirical verification. Even basic proof-of-concept experiments would be valuable to assess whether this model could be practically applicable.\n\nQUESTIONS:\n* Minor remarks:\n  * In lines 71-73 and 86-87, the wording was initially confusing to me. I think the way these results are stated later as the minimum or maximum of two values is easier to read.\n  * The error measures defined in lines 375-380 can be exponential in $n$.\n\nLIMITATIONS:\nThe authors state their theoretical results formally, describing all assumptions.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper explores the concept of 2-oracle algorithms for matroid optimisation problems. The underlying idea is to equip algorithms with a second, somewhat \"weaker\" oracle. The second oracle is also permits the algorithm to query matroid information (similar to the first oracle) but only gets imprecise answer (different from the first oracle). Therefore the second oracle is also called \"fast\" as the assumption is that its use is cheaper than the use of the first oracle. In this context the problem of computing a maximum-weight basis for a matroid is investigated. The main result obtained is the existence of an algorithm for computing a maximum-weight basis with a prescribed number of oracle calls. Some related tools and aspects like error-depdent and robust algorithms are also considered.\n\nSTRENGTHS:\n- The main results are established in form of formal theorems that come with a proofs; I did not check the proofs in detail but they look solid (the proof techniques look standard and adequate for this purpose); thus the soundness of the material appears good\n\n- While this is not the first paper that studies the power of 2-oracle algorithms its study in the context of matroid optimisation appears to be original; thus the novelty of the paper is good\n\nWEAKNESSES:\n- The paper is not easy to read; at the end of the introduction (i.e. Section 1) there should be an outline of the organisation of the paper to give the reader orientation\n\n- The organisation of the paper is poor: the 9 pages text consist of 4 pages introduction (Section 1), 1.5 pages warm-up (computing an \"unweighted\" basis) (Section 2), 2 pages computing a max-weighted basis (Section 3) and 1.5 pages future work (Section 4); the way how the material is organised is really unfriendly for the reader\n\n- The main results require a long list of lemmas (which are given in the paper or its appendix); so this might be tedious for the reader to follow the line of proof\n\n- The presentation of the material in the paper is at most fair; changing this may require quite some rewriting of the paper\n\nQUESTIONS:\n- While matroid optimisation is closely related to combinatorial optimisation it is not clear why this paper is submitted to NeurIPS as the connection to neural computing is not that obvious; so this should be justified better\n\n- The paper speaks of \"robust\" algorithms and of \"robustifying\" error-dependent algorithms in a sense of dealing with errors from the second oracle; but robust optimisation has been studied widely in the combinatorial optimisation community is usually concerned with uncertainty in the costs or times given as input; so these two meaning of \"robust\" may easily be mixed up and may be confusing for the reader\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}