Title: Tight Bounds for Maximum Weight Matroid Independent Set and Matching in the Zero Communication Model
Venue: NeurIPS 2025
Accepted: yes
Keywords: matroids; matching; distributed algorithms; parallel algorithms; lower bounds
Abstract: Recent years have revealed an unprecedented demand for AI-based technology, leading to a common setting where immense data is distributed across multiple locations. This creates a communication bottleneck among the storage facilities, often aiming to jointly solve tasks of small solution size $k$ from input of astronomically large size $n$. Motivated by federated and distributed machine learning applications, we study two fundamental optimization problems, maximum weight matroid independent set (MW-IS) and maximum weight matching (MWM), in a zero communication computational model. In this model, the data is dispersed between $m$ servers. Without any communication, each server has to send a message to a central server, which is required to compute an optimal solution for the original (large) instance. The goal is to minimize the size of the maximum message sent. For this natural restrictive model, we obtain deterministic algorithms that use $k$-data per server for MW-IS and $O(k^2)$-data per server for MWM, where $k$ is the solution size. We complement these results with tight lower bounds -- ruling out any asymptotic improvement even if randomization is allowed. Our algorithms are simple and run in nearly linear time. Interestingly, we show how our zero communication algorithms yield deterministic parallel algorithms with running times $O\left(\sqrt{k} \cdot \log n\right)$ and $O\left(k^4 \cdot \log n\right)$ for MW-IS and MWM, respectively.


Peer reviews:
comments: SUMMARY:
This paper investigates the problem of computing exact solutions for the Maximum Weight Matroid Independent Set (MW-IS) and the Maximum Weight Matching (MWM) problems under a strict zero-communication mode. Each of the m distributed servers processes part of the data and then transmits a message to the central coordinator without any inter-server coordination. The authors provide deterministic algorithms that solve this problem with tight bounds: O(k) per server data for the MW-IS problem and O(k^2) for the MWM problem where k is the solution size. This paper also presents matching lower bounds. The techniques proposed in this paper are further leveraged to derive deterministic parallel algorithms for both of the discussed problems (MW-IS and MWM).

STRENGTHS AND WEAKNESSES:
Strengths:
- Good theoretical contributions with rigorous proofs and tight bounds for both the algorithm results and the lower bounds. 
- Relevant exploration of a severely constrained communication model.
- Demonstrates applicability in parallel computing by linking distributed computation to efficient parallelism. 

Weaknesses:
- There weren't any experimental results provided but this is a soft weakness since the authors may think it's not necessary. 
- The initial setting of this paper (zero-communication exact algorithms) is a fairly specialized setting so applicability to the real-world is only lightly motivated. Perhaps it could be further motivated.

QUESTIONS:
1. What happens in the presence of noisy or faulty server inputs? Do the bounds proposed in this paper still hold?
2. Have the authors tested any of these methods in practice? If so, could they comment on how these deterministic algorithms compare in runtime to the randomized ones discussed in the paper?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper is interested in the problem of selecting a maximum weight feasible set (independent set in a matroid, matching, etc) using $m$ servers which may only communicate through a central coordinator. The motivation is that perhaps the dataset is huge and needs to be partitioned onto $m$ servers -- what should the servers send the central coordinator to guarantee that the central coordinator can output the right answer? How much data do the servers have to send? How much computation do they individually have to do? The paper gives upper and lower bounds to these questions.\n\nSTRENGTHS AND WEAKNESSES:\nWeaknesses: \n\nSection 3's proof is a little easy. But there's sort of a subculture within theory to de-value papers just because the proofs seem easy, so I don't think that's good grounds to reject a paper. Still, you devote almost two pages of valuable space to a proof that really just needs a few paragraphs. I'm much more interested in the proofs of Section 4's results (which are not in the main body, but the appendix). Section 4 proofs should be taking the pages you've devoted to Section 3. For example, I would really like to understand more intuition in Definition 4.1. I see how properties (1) and (3) follow from the algorithm in lines 331 - 337. Can you say a few lines about why property (2) holds (this should be in the main body, not appendices)? Also, include a paragraph as to why the union of strong sets must contain a maximum weight matching. \n\nMy main worry is that Section 4's results seem the most interesting to me, but not enough details of that proof have been added to the main body to convince me the result is true. I think this is critical in a theory focused paper (even though there's only 9 pages the reviewers will be reading). _Note: if you can write more about why your more non-trivial results are true and how you went about proving it, I would be more likely to bump my assessment to a 5_. \n\nThis is a nice theoretical result, and I'm not super sure what it's doing here in NeurIPS over some other theory conference. The application is quite nice. There's no evidence / results about empirical performance though, which I usually like to see in a theory focused paper in a venue like this. \n\nStrengths: \n\nThis is a great model. The proofs I read are clean and correct. The writing is wonderful and pleasant to read. If NeurIPS is interested in accepting more theory like papers, this is a pretty good candidate. Nice job!\n\nQUESTIONS:\nCan your results with matroids extend to matroid intersection in any way? That might imply some (bipartite) matching results. I would be interested in knowing that. \n\nYou should cite coresets in your paper. You're basically creating coresets. This is a rich body of work, which may have implications about some of your questions / results. \n\nWhat's the difference between $d$-data and the definition of bandwidth?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses two related problems : Maximum-Weight Matroid Independent Set (MW-IS) and Maximum Weight Matching (MWM) under a 0 communication model, the original problem is partitioned among smaller servers and each server must send a message to a central server such that the union of all messages sent \"contains\" the optimal solution for the overall problem.  The goal is to minimize the size of the largest message sent. \n\nMore specifically, this paper investigates a specialized form of the above problem, parameterized by $k$, which is the \"solution size\". For example, in the case of MWM, this paper is interested in computing the \"k-MWM\", that is, the maximum-cost matching of cardinality $k$. Parameterizing based on this $k$, the authors give simple algorithms for finding $k$-MW-IS and $k$-MWM with maximum message sizes of $k$ and $k^2$ respectively. They also provide lower bounds that show these results are optimal.\n\nAs a direct implication, the authors also obtain bounds for parallel versions of $O(\\sqrt{k}\\log{n})$ for $k$-MW-IS and $O(k^4\\log{n})$ $k$-MWM.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths : The paper is generally well written, especially with regards to notation definitions, aside from a few things highlighted below. (There is one major concern in the parallel section for MWM that needs addressing). \n\nThe inclusion of lower bounds strengthens the theoretical contribution of this paper, especially because it is not clear whether this exact parameterization (based on $k$) has been investigated before under this model.\n\nWeaknesses : \nDue to being less familiar with the literature on MW-IS, my feedback here is mainly related to $k$-MWM. \n\n- Your algorithm only makes sense when k is very small. However, why should we expect k to be small? I.e., for what practical applications would k ever be small? \n- You mention several applications of, for example, MWM, in a generic sense, and I agree MWM is a very important problem, and solving MWM in a zero communication model is an interesting problem. However, you are solving k-MWM, which is a totally different (probably much easier to solve) problem. As a result, you have to properly motivate *that* specific problem, which I see no discussion of; you only motivate MWM in a somewhat vague / general sense. This is both a practical concern and a theoretical concern, since I don’t think any prior work on the k-MWM variant was cited, and the value of a purely theoretical result is in some sense rooted in its comparison to prior theoretical work.  \nIn particular:\n(1)\tWhat are some examples of practical scenarios (for a NeurIPS audience) that motivate the *k*-MWM variant you propose?\n(2)\tWhy should someone expect that this k-MWM variant is a difficult problem to begin with?\n(3)\tHas the *k*-MWM problem been proposed and studied *outside* of your zero communication scenario? If so, then that would provide a good motivation for the problem to begin with. If not, then why solve it for the zero-communication setting if it’s not interesting in general?\nI would like to elaborate by saying that I am aware of the importance of the “partial p-Wasserstein” distance problem, related also to the “partial optimal transport” problem, which are generalizations of (the minimization variant of) your k-MWM problem, and have applications to a machine learning audience. Optimal cardinality-k matchings are important. My concern is that, for these applications, I don’t think k << n. I would normally expect k to be more like Theta(n), or, at least Omega(\\sqrt{n}) in most practical cases. \nI appreciate the theoretical nature of this work, but I also think that the problem needs stronger practical motivation that relates to a machine learning audience, specifically for motivating that “very small k”.\n\nI think that the literature review is somewhat lacking. The nature of the citations in the related work section seems somewhat disconnected from the nature of this work. The cited related work mentions several results for various communication models, and mainly gives approximation results. However, so far as I can tell, not a single citation in this document addresses the *k*-MWM problem, which is computationally very different from the MWM problem. This absence of prior work cited makes it difficult to assess the theoretical contribution of this work relative to prior work. If nothing else, the Hungarian algorithm is missing, which can compute an optimal cardinality $k$ matching in $O((m + n\\log{n}) * k)$ time.\n\nFinally, I don’t find the parallel result for MWM to be particularly surprising, and I don’t think it is at all tight – this appears as more of a direct by-product of your (optimal) 0-communication model, without any attempt at optimizing it for the parallel setting. I believe that prior (old) work for MWM is much stronger for the parallel setting. For example, see:\n\nOrlin, J. B., & Stein, C. (1993). Parallel algorithms for the assignment and minimum-cost flow problems. Operations research letters, 14(4), 181-186.\n\nThis also cites some work even prior to that. In this line of work, some bounds are given with parallel running times for the perfect matching problem that have running times of polylogarithmic in $n$ (with perhaps some dependency on the largest edge cost as well). Of course, this is a slightly different problem, since it computes a perfect matching, but in my experience, finding perfect matchings is usually more difficult that finding cardinality $k$ matchings for most combinatorial algorithms.\n\nIt is also not clear how your parallel bound for MWM works exactly, as stated. It seems as if you are assuming that the central server ends up with $k^2$ edges, but there are multiple servers sending a message of size $k^2$, right? So shouldn’t the central server end up with a solution much larger than $k^2$, due to accounting for the potentially very large number of servers? Regardless, the way you achieve your parallel bound is collecting a set of edges on the central server and running a (mysteriously unspecified) sequential algorithm. Why run a sequential algorithm instead of some existing parallel algorithm, such as:\n\nGabow, H., & Tarjan, R. (1988, January). Almost-optimum speed-ups of algorithms for bipartite matching and related problems. In Proceedings of the twentieth annual ACM symposium on Theory of computing (pp. 514-527).\nIn general, it seems to me like this parallel section for MWM is not thoroughly treated, and this section is perhaps the weakest part of the paper.\n\nCorrections / Improvements : \n55 : There are multiple types of randomization. One option is that randomization affects the accuracy, and another is that it affects the efficiency (Monte Carlo or Las Vegas). My point being, it’s possible to have an algorithm that is guaranteed optimal with probability 1, but that terminates within the desired time bound, with, say, probability 1/2. Most would still consider that “randomized”, but it is outside of the definition you’ve specified. This is not a major concern, but please add clarification to your model description to clarify the differences between these probability models, and also double check your proofs (especially the lower bounds) to ensure that they continue to apply under the more general notion of “randomized algorithms” if needed. If you are able to use a more general definition of probability, you should do so. Otherwise, you should clearly emphasize the limitation.\n\nIntroduction : Whenever you mention that the edges sent to the coordinator *contain* the optimal solution, be a bit more formal: i.e., say that there exists an optimal solution such that it’s a subset of the provided edges. The reason for this recommendation is that, as written, the reader might wrongly assume that the optimal consists of the optimal rather than being a superset of it. There is nothing incorrect here; simply a matter of readability.\n\n108 : “running time of server i can be computed in “. I don’t think you mean “computing the *running time*”. Reword this.\n\n180 : “In this work, we *showed* tight bounds”  : Due to the placement, this wording seems odd, more like it would be a conclusion. Same again on next line. Showed -> show.\nIntroduction : It is mentioned in multiple places (line 84 and 86) that k is the “solution size”, which seems to imply the “size of the optimal solution”. However, you really mean that k is a “constraint on the set of feasible solutions”. I.e., in your official problem statements, you are considering the maximum weight solution restricted to subsets of size k. I think some parts of the introduction need to be reworded to clarify this distinction, as I found it confusing. (It was correct and complete, but rewording these occurrences will help readability).\n\nLines 377 – 378 : Question about your k^4 bound: Where do you get k^4? Which sequential algorithm are you applying? Are you trying to apply the Hungarian algorithm, and assuming it runs in O(m^2) time? There should be a trivial improvement to this: The Hungarian algorithm takes only $O((m + n\\log{n})*k)$ running time when computing an optimal cardinality k matching. (Just run the first k iterations of it, using Dijkstra’s algorithm as a Hungarian search). So, your running time should simply be roughly ~O(k^3) I think if there are indeed only $k^2$ many edges. I know this to be true for the minimization problem where all costs are non-negative, but it seems like some small modifications to the edge costs can make the two problems equivalent (maximization for all matchings of size at most k). Please double check on this.\n\nRegardless, you have to state what sequential algorithm you are using. Make it explicit!\n\nQUESTIONS:\nI have highlighted several concerns in my discussion above, but, to streamline the response, please specifically address the following, referring to my discussion above for more details:\n(1) Please provide practical justification for why one should expect $k$ to be small for MWM, noting that, when $k \\geq \\sqrt{n}$, your algorithm effectively has no benefit for MWM under either model.\n(2) Please provide clarification regarding your parallel MWM algorithm, in light of my discussion above. In partciular:\n- Why are there $k^2$ total edges on the central server? (How do you account for the total message size from all servers?).\n- What sequential algorithm are you using?\n- Can the Hungarian algorithm (or any other faster sequential algorithm, or even another parallel algorithm) yield better results here?\n\nLIMITATIONS:\nNothing to note here.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors consider a low-round distributed algorithm for matroid independent set and maximum weighted matching. Their model of computation is called \"zero-communication\": the input is distributed across machines, the machines report a small message to a central server, and the central server can extract a solution from these small messages. Each message is bounded by size k, where k is the size of the solution. This is very similar to 2-round parallel algorithms, though the second step is more restricted in that it requires a single centralized computation.\n\nThese problems are highly interesting, and have been studied in related models of computation (e.g., 2-player 1-round communication, message passing, etc.), but not this model exactly. It could be said that this model is a tad niche, but these problems are also highly fundamental that improvements in more standard models are exceedingly rare. Also of note is that these algorithms are perfect algorithms, not approximations.\n\nTheir first result is a k-cardinality maximum weight matroid independent set. It is quite simple: find a solution on each machine, send the solution to a central machine, and then the central machine finds the best solution on what it's given. The interesting aspect of this work is the analysis, which carefully examines the properties of independent sets and cycles in matroids to show that this yields a perfect solution. The final algorithm uses O(sqrt(|E|)) deterministic parallel time where |E| is the size of the edge set on a machine by leveraging existing algorithms on their individual machines. The messages passed contain only k elements each.\n\nThe next result uses a similar method for maximum weight matching, though it is insufficient to simply solve the problem on each machine and pass them to a central machine. Instead, they extract what are called \"strong sets\" of size at most 10k^2 + 1 to pass along to the central machine, and the machine can extract the solution from this. Now the messages are slightly larger, but this is also a more difficult problem to solve. The parallel runtime is O(|E|).\n\nThey also show how to adapt these to parallel algorithms - both generally for any subset selection problem and specifically for the two studied problems.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper studies very interesting problems and is generally clear though quite dense at times. The model used is a bit niche (and possibly not studied before), but these are extremely fundamental problems, so progress in more standard models is more difficult. For that reason, I think this work is interesting and important.\n\nMy overall rating comes mostly from gaps in my understanding or explanations in the paper itself. If the authors clear these up, I am happy to increase my score.\n\nQUESTIONS:\n1. Theorem 1.1 says that the running time of each i can be computed in ... time. Do you mean that the runtime IS this value? Seems weird to be measuring the runtime of a computation of the runtime.\n\n2. Is Theorem 1.3 supposed to be for all k in N?\n\n3. In the proof of Claim 3.5, you only need to cite Claim 3.4 the first time. The second time you use it, you get B_i' is independent because you took B_i \\cup {e} and removed an element e'' in its only cycle, therefore leaving an independent set. Therefore you're not really using it a second time as implied by how you wrote it. This is correct, right?\n\n4. In the proof of Claim 3.5, once you have B_i' is an independent set, why is it a basis? Is it true that any independent set of a matroid with the same cardinality as a basis must be a basis (due to the second property of matroids), and that means B_i' is a basis?\n\n5. Has this zero-communication model been studied before, or are you introducing it? I think it's well-motivated either way, but I don't know if \"zero-communication\" is a misleading term term, since there is certainly communication, even if it's only from machines to a central server. It's extremely similar to a 2-round parallel algorithm, but instead you have the second round is a single centralized computation instead of more parallel computation. Do you think there's a more appropriate name for this?\n\n6. Your explanation at the beginning of section 4 is a bit lacking. MWM is not a matroid problem, so you can't necessarily transfer the results from section 3. However, the process you describe does sound like a zero communication algorithm. I'm assuming the issue is, if you distribute the graph across machines and find the k-MWM on each machine, you can't then derive an MWM from the set of solutions on across machines. Is this correct? Or otherwise, why is what you've described not a zero communication algorithm?\n\n7. Can you add some intuition of why the strong set helps you determine the maximum weight matching?\n\n8. Can you describe at a high level what the algorithm for 5.1 looks like?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the cardinality-constrained version of the maximum-weight independent set problem of matroids and the maximum-weight graph matching problem under the \"zero communication\" model. In this model, the elements and their weights of the ground set (or the edge set for the matching problem) are partitioned and stored over a multiple number of servers. Every element is in exactly one server. Each server processes its data, and sends a small message to the coordinator, which must use these messages to produce the final output.\n\nFor the maximum-weight independent set problem, the answer is intuitive: each server finds the solution for the matroid restricted to the elements the server has, and sends this solution to the coordinator. Since the problem is cardinality-constrained, each server will send at most $k$ items to the coordinator. The coordinator then finds the solution for the matroid restricted to, this time, the union of the solutions received from the servers. Its correctness is shown using an exchange argument on bases.\n\nFor the maximum-weight matching problem, the message is larger than before: each server sends $O(k^2)$ edges to the coordinator. The intuition is that a server needs to send to the coordinator only those edges that are valuable enough to be one of the k edges chosen in the final solution. As such, an edge is \"useful\" only if every cardinality-$(2k+1)$ matching contains the edge and it is one of the $(2k+1)$ most valuable edges incident with each endpoint. The number of these useful edges is $O(k^2)$, as can be easily seen from an argument that a high-degree graph with many edges must have a large matching.\n\nBased on these two zero-communication results, the paper also gives a parallel algorithm. The algorithm partitions the ground/edge set into small pieces to achieve $\\mathrm{poly}(k)\\cdot \\log(n)$ running time. The paper also presents lower bound results that complement the main results. Finally in the appendix, it also shows that the shortest path problem cannot be solved without sending the entire set of edges to the coordinator.\n\nSTRENGTHS AND WEAKNESSES:\n### Strength\n\n- The zero communication model is an interesting model and the paper proposes to consider fundamental problems of combinatorial optimization under this model.\n\n- Presentation is simple and clear.\n\n### Weakness\n\n- (See questions.) The algorithm requires a cardinality constraint to keep the messages small. This seems somewhat artificial.\n\n- As was shown by the paper itself, the model is too restrictive to solve even simple problems like the shortest path problem. It would be nice to see some more arguments why the zero communication model should be studied (e.g. as a means to obtain good parallel algorithms).\n\n- A suggestion: the analysis of the two algorithms, especially the one for the independent set problem, uses techniques that are quite standard. Can we omit some of these arguments from the first nine pages and instead bring some of the matrials in the appendix?\n\nQUESTIONS:\n- Although the paper states on Page 3 that the version with the cardinality constraint is more general, using the reduction may blow up the size of the message. It seems that messages with size $O(r)$ is sufficient for the independent set problem, where $r$ denotes the rank of the matroid, but what can we do for the matching problem?\n\n- The model assumes that the matroid structure is known by all servers (at least when restricted to its elements) and the coordinator. How practical is this assumption?\n\nLet me write a few minor comments here too:\n\n- The abstract does not make it clear that k is given as input.\n\n- Line 55, (probability 1): even if an algorithm succeeds with probability 1, it may still be randomized. \n\n- \"pseudo-polynomial\" sounds somewhat misleading. How about saying \"polylogarithmic in the size of the ground set\"?\n\nLIMITATIONS:\nYes. While the paper clearly states assumptions it makes for its theoretical results, I discussed some additional concerns above (see Questions).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}