{"id":"f98cf9ce-f3fe-48bf-8b58-55bcaa1a4ff5","arxiv_id":"2411.18292","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Mader's S-path packing admits a deterministic O(mnk) time algorithm, improving the previous best deterministic bound O(mn^omega).","lead":"This paper gives a faster deterministic algorithm for Mader's S-path packing, a graph problem that generalizes matching and disjoint paths. The new method runs in O(mnk) time and avoids expensive matrix operations inside the augmenting path framework.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2's unproved adaptation of Gabow-Stallmann from dummy singleton copies to terminal singletons is the load-bearing gap; if it fails, both correctness and the O(mnk) bound collapse.","rationale":"I read the paper in good faith. The reduction via Schrijver and the structural lemmas (Lemmas 1-4) are standard and, apart from a sign typo in Eq. (16) that does not affect the conclusions, the exposition is coherent. Algorithm 3's traversal updates are internally consistent: with the orientation convention mu_e(u) = -mu_e(v), the recurrence v^o = u^o + mu_e(v)e^o and the two-pass terminal formula correctly recover representations; the O(mn) complexity accounting is plausible because each tree component is traversed at most twice. The genuinely load-bearing weakness is Proposition 2: it imports the Gabow-Stallmann oracle into a setting with terminal singletons and no proof that the original exchange arguments survive the replacement. The paper's own one-paragraph justification is too weak, and Proposition 3 is asserted rather than proved. If this adaptation fails, both correctness and the O(mnk) bound collapse. This does not mean the theorem is false; it means the current manuscript does not establish it. The reader's CONDITIONAL verdict is appropriate, and my read does not change it; I only partially agree with the reader because I see the Gabow-Stallmann adaptation as the more fundamental concern, while Algorithm 3 itself appears fixable with a formal invariant.","tokens_in":11404,"tokens_out":40390,"duration_ms":372708,"concrete_test":"Independently re-derive the key exchange-graph lemma of Gabow-Stallmann [11, Lemma 3.1] for the terminal-singleton setting: prove that if B is a feasible base and |E[B]| is not maximum, the dependence graph has an odd alternating path from some t in K intersect B to some t' in K intersect B whose symmetric difference is a feasible base, and that the search runs in O(mn) time. If the proof of [11, Lemma 3.1] relies on the original singletons being copies of all line elements rather than on the general matroid-parity exchange property, then Proposition 2 is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1 rests on Proposition 2, which asserts that the Gabow-Stallmann augmenting-path oracle works in O(mn) time for the feasible bases B of Section 2, where the 'singletons' are the terminal vectors K={t:t in T}, each with the extra twin t* = theta(t)t. Section 3 explicitly notes that this differs from Gabow-Stallmann's original setting, where singletons are copies of all individual twins in L and the base lives in span L. The paper justifies the replacement in one paragraph: the singletons are dummy vectors whose only role is to ensure that a matching plus enough singletons forms a base. What must be true is much stronger: the entire exchange-graph/augmenting-path proof of [11] must go through verbatim when the dummy singletons are replaced by terminal vectors, including the linear relations t* = theta(t)t, and the O(mn) augmenting-path search must still find a feasible base whenever E[B] is not maximum. No proof or reference for this adaptation is given. If the exchange lemma in [11] uses the special structure of the original singletons, the oracle could return a non-feasible set or fail to augment, and the per-iteration cost would not be justified. Proposition 3, the correctness of Algorithm 1, is also asserted without a formal invariant for Algorithm 3. These are addressable gaps rather than demonstrated errors, but they are load-bearing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a deterministic algorithm for Mader's S-path packing problem that runs in O(mnk) time, where n=|V|, m=|E|, and k=|T|. The algorithm uses the known reduction of the problem to linear matroid parity and the augmenting-path framework of Gabow and Stallmann, but replaces the repeated computation of the dependence matrix via matrix multiplication with a combinatorial traversal of the current feasible base. The main theorem, Theorem 1, is derived from a series of lemmas about feasible bases, an oracle for augmentation (Proposition 2), a subroutine for computing the dependence matrix (Algorithm 3), and a final subroutine that recovers an S-path packing. The claimed running time improves the previous best deterministic O(mn^omega) bound.","tokens_in":11622,"tokens_out":11184,"duration_ms":99811,"significance":"If the main theorem is correct, the O(mnk) deterministic algorithm is a genuine improvement over the previous O(mn^omega) deterministic bound for Mader's S-path packing, and it shows that the special structure of the matroid parity instance can be exploited to avoid matrix multiplication in the dependence-matrix update. The paper also contains a clean and mostly rigorous treatment of the reduction (Lemmas 1-4) and a plausible combinatorial subroutine for the dependence matrix. However, two load-bearing proof gaps must be closed before the result is established: the adaptation of the Gabow-Stallmann oracle to the modified singleton set is asserted without proof, and the correctness of Algorithm 3 is only sketched. These are addressable issues rather than demonstrated errors.","major_comments":[{"comment":"Proposition 2 is the central oracle on which both correctness and the O(mnk) bound depend, but its adaptation to the paper's setting is not proved. The manuscript notes that the original Gabow-Stallmann setting uses copies of all individual twins in L as singletons, whereas here the singletons are the terminal vectors K={t : t in T} together with the relation t* = theta(t)t, and it asserts in one paragraph that this replacement preserves every guarantee because the singletons are only dummy vectors. This is not sufficient: the correctness of the augmenting-path search and its O(mn) time bound in [11] may rely on the specific structure of the original singleton copies, including how they interact with the exchange graph and the feasibility certificate. The authors should either cite a theorem in [11] that covers exactly this setting, or provide a self-contained proof that the augmenting-path exchange lemmas remain valid when the dummy singletons are replaced by K. Since Proposition 2 is used in Proposition 4 and Theorem 1, this gap is load-bearing.","section":"Section 3, Proposition 2"},{"comment":"Proposition 3 asserts that Algorithm 1 correctly solves Mader's problem, but no formal proof is given for the correctness of Algorithm 3, the computation of the dependence matrix. Section 4.2 describes the traversal informally, but it does not state or prove the invariant satisfied by the maintained values c(w,b), nor does it prove that the transition rules in lines 19-27 and the correction formulas in lines 9-13 produce the unique representation (34) with respect to B. In particular, the two-terminal component case, where Z∩T[B]=empty, is the delicate part and is only sketched. A proper proof should show that the first traversal computes the tentative representation of v°-t1 and v•-t1, that the formula for the representation of t2 is correct, and that the second traversal propagates the correct representations. Without such an invariant, the correctness of the O(mn) dependence-matrix computation is not established, and consequently Proposition 3 does not follow from the text.","section":"Section 4 (Proposition 3) and Algorithm 3"}],"minor_comments":[{"comment":"The sentence 'repeatedly updates the feasible base B with E[B] increased by 2 (and hence T[B] decreased by 1) per iteration' is inconsistent with Eq. (18) and with the symmetric-difference equations (31)-(32), which show that |E[B]| increases by 1 and |T[B]| decreases by 2 per augmentation.","section":"Section 3, first paragraph"},{"comment":"The instruction 'R←R∪e' should read 'R←R∪{u,v}' where e={u,v}, since e is an edge and not a set of vertices.","section":"Algorithm 2, line 7"},{"comment":"The update of c(t•,b) in line 12 uses c(t◦,b) after it has already been overwritten in line 11; the authors should state explicitly whether the assignments are meant to be sequential or simultaneous, and if sequential, confirm that the new value is intended.","section":"Algorithm 3, lines 11-12"},{"comment":"The phrase 'tentative representation of v°−t1, v•−t1' is not defined formally; the authors should define exactly which vectors the entries c(w,b) represent during the first traversal from a root that is not in B.","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the claimed improvement is significant if correct. The main obstacles are the missing proof of the adapted Gabow-Stallmann oracle (Proposition 2) and the lack of a formal invariant for Algorithm 3. I would encourage the editor to request a major revision in which these two points are fully addressed; the rest of the manuscript is carefully written and the reduction lemmas appear sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper has a genuinely new idea—replacing matrix multiplication in the dependence-matrix update with tree traversals in the original graph—and a clean O(mnk) deterministic bound for Mader's S-path packing, but two load-bearing gaps need to be nailed down before I'd bet on it.\n\nWhat's new: The reduction to linear matroid parity is standard (Schrijver). The contribution is a combinatorial subroutine that computes the dependence matrix for a feasible base by traversing the forest G[B] rooted at terminals, using the fact that a feasible base corresponds to a forest. If correct, this avoids the O(mn^{ω-1}) matrix operations per augmentation and yields the O(mnk) total. Lemmas 1–4 characterize feasible bases cleanly and their proofs are solid. The complexity accounting is careful and the O(mn) space bound is a plus.\n\nSoft spots: The stress-test is on target. Proposition 2 asserts that Gabow-Stallmann's augmenting-path oracle works when the singletons are the terminal vectors t and θ(t)t rather than copies of all twins in L. The paper justifies this with a paragraph saying the singletons are 'dummy vectors' and replacing them 'preserves every guarantee in their proof.' That may be true, but it is not proved or referenced, and the proof in [11] likely uses the specific span-L setting. A referee should check whether the exchange-graph arguments go through verbatim. Second, Proposition 3 (correctness of the main algorithm) is asserted, and Algorithm 3's correctness is only sketched. No invariants are stated for the two-pass traversal, and the transition between tentative and correct representations is not fully formal. Both gaps are addressable, and I don't see an obvious error, but they are load-bearing.\n\nBottom line: If the gaps are filled, this is a solid result—the first deterministic improvement over O(mn^ω) for Mader's problem. As it stands, it is a promising manuscript that deserves a serious referee, not a desk reject. I'd send it to review and ask for the missing proofs.","headline":"A genuinely new combinatorial subroutine for dependence-matrix updates yields an O(mnk) deterministic algorithm for Mader's S-path packing, but two load-bearing gaps need proof.","tokens_in":12190,"tokens_out":5599,"would_cite":true,"duration_ms":43305,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C85","05B35","68Q25","68W40"],"pacs":[],"model":"deepseek-v4-flash","headline":"A deterministic O(mnk) algorithm solves Mader's S-path packing, removing matrix multiplication from the bottleneck.","keywords":["Mader's S-path packing","linear matroid parity","dependence matrix","augmenting path algorithm","deterministic algorithm","combinatorial optimization"],"falsifier":"Compute, on all graphs with up to about seven vertices, both the dependence matrix returned by Algorithm 3 and the true coordinate matrix obtained by Gaussian elimination over F_q; any entry mismatch, or any base on which AugmentOrMaximum fails to return an optimal base, refutes the claimed correctness and time bound.","tokens_in":11149,"feed_emoji":"🧮","tokens_out":2750,"duration_ms":25596,"temperature":0.7,"pith_summary":"Mader's S-path packing asks for the maximum number of vertex-disjoint paths whose endpoints lie in distinct blocks of a prescribed partition of the terminals. This paper claims a deterministic algorithm that runs in O(mnk) time, where n is the number of vertices, m the number of edges, and k the number of terminals. The improvement over the previous best deterministic bound O(mn^omega) comes from replacing the costly matrix computations that update an auxiliary dependence graph with a purely combinatorial subroutine that exploits the original S-path structure. Because the algorithm works over a small finite field and only performs simple arithmetic operations, it avoids fast matrix multiplication entirely.","feed_headline":"S-path packing solved deterministically in O(mnk) time","feed_subtitle":"Eliminates matrix multiplication from the per-augmentation step, beating the old O(mn^omega) bound.","key_machinery":"The dependence matrix D(B) = (d(w,b)) records, for every line element w not in the feasible base B, the unique coefficients expressing w as a linear combination of the vectors in B. The algorithm computes D(B) combinatorially: it roots each connected component of the forest G[B] at a terminal, propagates coefficient vectors c(v^circ,b) and c(v^bullet,b) along tree edges using the walk identities of Lemma 1, and assembles the edge-twin representations from the end-vertex coefficients. This traversal replaces the matrix inversion and multiplication that the generic Gabow-Stallmann update would require.","core_discovery":"The central claim, stated as Theorem 1, is that Mader's S-path packing problem admits a deterministic algorithm with running time O(mnk). The algorithm reduces the problem to linear matroid parity via Schrijver's linear representation, starts from a feasible base that contains all terminal singletons, and then repeatedly augments the base using the Gabow-Stallmann augmenting-path method. The key new step is a subroutine that computes the dependence matrix D(B) of the current base B in O(mn) time by traversing each tree of the forest G[B] at most twice, instead of solving a general linear system, so the total cost over at most k/2 augmentations is O(mnk).","pith_inferences":["The combinatorial dependence-matrix update likely extends to the weighted version of S-path packing, since weighted linear matroid parity algorithms also need such matrix information and might now avoid fast matrix multiplication.","The traversal technique may adapt to non-zero A-path packing on group-labelled graphs, where terminals carry labels similar to the block scalars theta(t), potentially yielding a comparable improvement over matrix-based approaches.","A concrete stress test is to verify Algorithm 3 on all small graphs by comparing its output against direct Gaussian elimination, because the paper asserts the traversal invariant without a formal proof."],"forward_implications":["If correct, the result improves the deterministic worst-case bound for Mader's S-path packing from O(mn^omega) to O(mnk), which is better for instances where k is small relative to n^omega.","The algorithm runs over a finite field F_q of order q = O(|S|), so its arithmetic operations are elementary and it does not depend on the matrix multiplication exponent.","Each augmentation increases the number of lines in the feasible base by exactly one, and at most floor(k/2) augmentations suffice, so the O(mn) per-iteration cost yields the claimed total.","The final S-path packing is recovered in O(n) time from the optimal base, so the whole pipeline remains linear in the graph size outside the dependence-matrix computation."],"supporting_citations":[{"why":"Supplies the augmenting-path oracle (Proposition 2) that the main algorithm calls to augment the feasible base or detect optimality.","marker":"[11]"},{"why":"Provides the linear representation of Mader's problem and the equivalence between S-path packings and feasible bases (Proposition 1).","marker":"[39]"},{"why":"Gives the algebraic linear matroid parity algorithms that yield the O(mn^{omega-1}) and O(n^omega) randomized bounds that this paper's deterministic bound is compared against.","marker":"[5]"},{"why":"Provides the matrix multiplication cost O(mn^{omega-1}) that determines the bottleneck of the naive dependence-matrix computation.","marker":"[1]"}],"fun_headline_variants":["Deterministic S-path packing in O(mnk) time","Mader's S-path packing solved in O(mnk) deterministic","Faster deterministic algorithm for Mader's S-path packing","O(mnk) beats O(mn^omega) for S-path packing","S-path packing: linear matroid parity without matrix mult"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Gabow-Stallmann oracle keeps its O(mn) guarantee when the singletons are chosen as the terminal vectors {t} instead of the original twin copies, an adaptation asserted in Section 3 without a proof.","fun_headline_variants_meta":{"raw":{"variants":["Deterministic S-path packing in O(mnk) time","Mader's S-path packing solved in O(mnk) deterministic","Faster deterministic algorithm for Mader's S-path packing","O(mnk) beats O(mn^omega) for S-path packing","S-path packing: linear matroid parity without matrix mult"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000193,"raw_usage":{"total_tokens":1327,"prompt_tokens":899,"completion_tokens":428,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":340}},"tokens_in":515,"tokens_out":428,"duration_ms":3838,"temperature":1.0,"reasoning_tokens":340,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:24:44.492498+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, on all graphs with up to about seven vertices, both the dependence matrix returned by Algorithm 3 and the true coordinate matrix obtained by Gaussian elimination over F_q; any entry mismatch, or any base on which AugmentOrMaximum fails to return an optimal base, refutes the claimed correctness and time bound.","supporting_citations":[{"cited_title":"Gabow and Matthias Stallmann","cited_arxiv_id":null,"evidence_quote":"Supplies the augmenting-path oracle (Proposition 2) that the main algorithm calls to augment the feasible base or detect optimality."},{"cited_title":"Combinatorial Optimization: Polyhedra and Efficiency","cited_arxiv_id":null,"evidence_quote":"Provides the linear representation of Mader's problem and the equivalence between S-path packings and feasible bases (Proposition 1)."},{"cited_title":"Algebraic algorit hms for linear matroid parity problems","cited_arxiv_id":null,"evidence_quote":"Gives the algebraic linear matroid parity algorithms that yield the O(mn^{omega-1}) and O(n^omega) randomized bounds that this paper's deterministic bound is compared against."},{"cited_title":"Aho, John E","cited_arxiv_id":null,"evidence_quote":"Provides the matrix multiplication cost O(mn^{omega-1}) that determines the bottleneck of the naive dependence-matrix computation."}],"review_version":1}