{"id":"16785733-3a74-4b33-99b2-60071ed9c8d5","arxiv_id":"2501.11593","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"This paper shows that joint scheduling, pairing, and few-bit phase-only beamforming for ISAC can be solved exactly via a mixed-integer linear program.","lead":"Researchers formulate the joint problem of choosing which users and radar targets to serve, pairing them, and setting low-resolution phase-only beams in an integrated sensing and communication system. They convert this hard nonlinear problem into a mixed-integer linear program that can be solved to global optimality, and show it beats staged heuristics by 27 to 69 percent in simulation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 7 uses a big-M constant that bounds Tr(G_t W_u), not τ, so J1 cuts feasible solutions and the claimed exact MILP Q is a strict restriction of P.","rationale":"The paper's stated contribution is an exact MILP reformulation, and Proposition 7 is the hinge of the sensing-scheduling part. The reader's weakest assumption identifies exactly that hinge: the big-M constant D_t is computed as a bound on Tr(G_tW_u) via Lemma 1, but the linearization removes the product τλ_t, so the off-case requires D_t to bound τ, which it does not. The small example is not contrived: it satisfies C1–C13 with target 1 scheduled and target 2 omitted, and P's optimum is 10, while Q cannot exceed τ=3. Consequently, the claim that an optimal solution to Q is also optimal to P is false. The K5 issue compounds the problem, but J1 alone suffices. Because the central exactness claim collapses, the numerical comparisons solve a different, restricted problem; the qualitative observation that joint design beats heuristics may survive for Q, but the global-optimality conclusion for P is unsupported. I agree with the reader's REJECT, so no verdict adjustment is needed.","tokens_in":11967,"tokens_out":9880,"duration_ms":104304,"concrete_test":"Exhaustively enumerate P and solve Q for the tiny instance above (N=1, K=1, U=1, T=2, J=1, Ptx=10, α=[1,0.1], arbitrary angles, small Γth): P's optimum is τ=10; if Q's optimum is at most 3, the equivalence fails. Independently, re-derive Proposition 7 by substituting λ_t=0 into J1 and checking whether it reduces to the original C10 condition; it does not, because it imposes Tr(G_t W_u) ≥ τ − 2D_t. Apply the same inactive-case check to K5 with D_q replacing D_t.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 7 asserts C10 ⇔ J1 with D_t = (Ptx/K)Tr(G_t). The big-M argument requires the constant to dominate the true 'off' value of the product τλ_t. In the inactive case λ_t=0, ρ_u,t=0, J1 reduces to Tr(G_t W_u) ≥ τ − 2D_t. But D_t is an upper bound on the left-hand side, not on τ. Since τ is the maximin DPG over the scheduled targets, τ can be much larger than D_t when the weak target t is unscheduled and a stronger target is scheduled; then J1 forces the left-hand side above a positive threshold, excluding beams that are perfectly feasible in the original problem. Concretely, take N=1, K=1, Ptx=10, U=1, T=2, J=1, α1=1, α2=0.1, with the single user paired with target 1. Original P attains τ=10. In Q, unscheduled target 2 gives J1: Tr(G2 W1) ≥ τ − 2D2, i.e. 1 ≥ τ − 2, so τ≤3. Thus Q is a strict restriction and optimality does not transfer back. The same mis-scaled big-M appears in K5, where the inactive case uses D_t although Tr(G_q W_u) is bounded by D_q; when D_q > D_t, feasible beams toward an unscheduled strong target are excluded. Since the paper's central claim is exactness of the MILP reformulation, this is decisive.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies joint user/target scheduling, user-target pairing, and low-resolution phase-only beamforming for an ISAC downlink. The problem is formulated as a nonconvex mixed-integer nonlinear program (MINLP), and the authors propose an exact mixed-integer linear program (MILP) reformulation through a chain of eight equivalence propositions. The reformulation is intended to preserve the original solution space and thus yield a globally optimal solution to the MINLP. The paper also provides simulations comparing the proposed optimal design against heuristic scheduling/pairing baselines.","tokens_in":12132,"tokens_out":12466,"duration_ms":122904,"significance":"If correct, the exact MILP reformulation would be a valuable contribution: it would allow global optimality for a practically relevant ISAC resource-allocation problem and provide a benchmark for heuristic designs. The paper is clearly written and the reformulation machinery (big-M linearizations, total unimodularity arguments) is standard. However, the central claim of exactness is load-bearing, and it is false as stated. The simulation results are therefore not evidence for the optimality of the proposed method.","major_comments":[{"comment":"The big-M constant D_t = (Ptx/K)Tr(G_t) is an upper bound on Tr(G_t W_u), but the linearized constraint J1 also needs a bound on the auxiliary variable τ. In the inactive case λ_t=0 (and hence ρ_{u,t}=0 by C6), J1 reduces to Tr(G_t W_u) ≥ τ − 2D_t. Since τ can be much larger than D_t when a strong target is scheduled while a weak target t is unscheduled, this inequality is not automatically satisfied and cuts feasible solutions. Concrete counterexample: N=1, K=1, Ptx=10, U=1, T=2, J=1, α1=1, α2=0.1, with the single user paired with target 1. The original problem P achieves τ=10, but J1 for the unscheduled target 2 gives 1 ≥ τ − 2, so τ ≤ 3 in Q. Thus Q is a strict restriction of P, and the statement that an optimal solution to Q is also optimal to P is false.","section":"Proposition 7, constraint J1"},{"comment":"The same mis-scaled big-M constant appears in K5: Tr(G_q W_u) ≤ ξ_th + (2 − π_{t,q} − ρ_{u,t}) D_t. The left-hand side is bounded by D_q = (Ptx/K)Tr(G_q), not by D_t. When D_q > D_t, the constraint for an unpaired or unscheduled strong target q can become active, excluding beams that are perfectly feasible in the original problem P. For example, if λ_q=1 but ρ_{u,t}=0 or π_{t,q}=0, the right-hand side may be smaller than the maximum possible value of Tr(G_q W_u), so the reformulation further restricts the feasible set. This confirms that the error is not isolated to J1 but affects the full reformulation.","section":"Proposition 8, constraint K5"}],"minor_comments":[{"comment":"There is a typo: 'commmunications' appears in the abstract and again in the first line of Section I.","section":"Abstract and Section I"},{"comment":"The paper reports specific Cramér-Rao bound values for different phase resolutions but does not describe how these bounds are computed for the given beampatterns; a brief derivation or reference would help reproducibility.","section":"Fig. 5 and Scenario IV"},{"comment":"The proof invokes total unimodularity to justify relaxing H3 to I3, but the argument that the row/column sums force the outer product can be made directly; the current wording is slightly misleading.","section":"Appendix, proof of Proposition 6"}],"recommendation":"reject","confidential_remarks":"The counterexample in Major Comment 1 is not a corner case; it arises whenever an unscheduled target has D_t smaller than the achieved τ. Because the paper's main contribution is the claimed exact MILP reformulation, and that claim is false, the manuscript is not acceptable in its current form. The error could in principle be repaired by using a global bound on τ (e.g., max_t D_t) in J1 and D_q in K5, but the proofs, propositions, and simulations would all need to be redone. Given the centrality of the exactness claim, I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Straight to it: the paper's central claim is wrong. The MILP Q is not an exact reformulation of P. The big-M constants in Proposition 7 (J1) and Proposition 8 (K5) are per-target DPG bounds, but the inactive cases require bounds on tau and on other targets' DPGs. In J1, with lambda_t=0 and rho_{u,t}=0, the constraint becomes Tr(G_t W_u) >= tau - 2D_t. D_t bounds the left side, not tau, and tau can be much larger than D_t when target t is weak and unscheduled. The single-antenna example is decisive: Ptx=10, alpha1=1, alpha2=0.1, true optimum tau=10, MILP caps tau at 3 via the weak target's constraint. K5 has the same scaling problem with D_t instead of D_q. So Q is a strict restriction of P, and the claimed global optimality for P does not follow.\n\nWhat is genuinely new is the joint problem. The paper is the first, as far as it cites, to put user scheduling, target scheduling, pairing, and few-bit phase-only beamforming in one optimization. That is a legitimate modeling contribution, and the qualitative result that the best heuristic depends on the scenario (BL2 wins with correlated users, BL1 wins when users are separated) is believable and useful. The reformulation machinery (big-M, RLT, total unimodularity) is applied competently and the proofs are mostly complete; Proposition 6 is a sketch, but the TU argument is plausible.\n\nThe simulation section has no code, no data, no error bars, and the measured gains are all relative to heuristics whose suboptimality is expected. The real issue is that the 'optimal' numbers are not optimal for the stated problem. That kills the paper's main claim, though the framework might be salvageable as a heuristic or a relaxation.\n\nI'd send this to a serious referee because the problem is timely and the error is specific and fixable in revision. But the exactness claim as written should not be published. Not a paper I'd cite for optimality; the formulation might be worth citing for the problem statement if the authors fix the scaling.","headline":"The joint ISAC formulation is new, but the claimed exact MILP reformulation uses mis-scaled big-M constants and is a strict restriction of the original problem.","tokens_in":12806,"tokens_out":5516,"would_cite":false,"duration_ms":56215,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Joint scheduling, pairing, and low-resolution beamforming in ISAC exactly reduces to a MILP.","keywords":["ISAC","integrated sensing and communications","resource allocation","beamforming","discrete phase shifts","scheduling","user-target pairing","MILP reformulation"],"falsifier":"Solve the original problem P and the reformulated problem Q on a minimal instance with a single user, a single weak target, and $N=1$ antenna. If the optimal $\\tau$ in P can exceed $D_t = (P_{\\mathrm{tx}}/K)\\mathrm{Tr}(G_t)$ while Q's constraint J1 with $\\lambda_t=0$, $\\rho_{u,t}=0$ forces $\\tau \\le 2D_t$, the claimed solution-space preservation fails. Similarly, with two targets where $D_q > D_t$, check whether an optimal solution that illuminates target q while pairing the user with target t is excluded by K5; if so, Q is not equivalent to P.","tokens_in":11613,"feed_emoji":"📡","tokens_out":9592,"duration_ms":84035,"temperature":0.7,"pith_summary":"This paper studies a joint resource-allocation problem for an integrated sensing and communications (ISAC) base station: which users to schedule, which targets to sense, how to pair each sensed target with exactly one served user, and what few-bit constant-modulus phase-only beamforming vectors to use. The resulting optimization problem is a nonconvex mixed-integer nonlinear program (MINLP). The paper's central claim is that this MINLP can be transformed step by step into an equivalent mixed-integer linear program (MILP), so that any globally optimal solution of the MILP is also globally optimal for the original problem. This gives an exact, non-heuristic route to optimal joint design in a regime where existing approaches rely on approximations or decoupled stages. Simulations indicate that the optimal joint design consistently outperforms four heuristic stage-wise baselines, whose relative ordering changes with scenario characteristics.","feed_headline":"Nonconvex ISAC design exactly reduces to a linear program","feed_subtitle":"Eight equivalent reformulations make joint scheduling, pairing, and low-resolution beams globally solvable, beating heuristic baselines.","key_machinery":"The load-bearing device is the reformulation chain itself, and its main components are: binary phase-selection vectors $x_{u,n}$ that pick one of the $L$ constant-modulus phases per antenna element; lifted matrices $Y_{u,n,m} = x_{u,n}x_{u,m}^{T}$, whose integrality is guaranteed by total unimodularity after relaxing entries to $[0,1]$; big-M constants $D_t = (P_{\\mathrm{tx}}/K)\\mathrm{Tr}(G_t)$ and $B_u = P_{\\mathrm{tx}}\\mathrm{Tr}(\\tilde{H}_u)+1$ that switch off sensing and SINR constraints when scheduling or pairing indicators are zero; and auxiliary binary products $\\pi_{t,q}$ that linearize $\\lambda_t\\lambda_q$. Together these variables and inequalities eliminate every nonconvexity and nonlinear coupling, leaving a linear objective and linear constraints.","core_discovery":"The core discovery is a chain of eight equivalent reformulations (Propositions 1-8) that map the nonconvex MINLP P onto the MILP Q. Discrete-phase beamforming is encoded with binary phase-selection vectors; SINR constraints are linearized by a big-M bound and a rank-one lifting that is exact; products of binary variables are replaced by auxiliary matrices whose integrality follows from total unimodularity; and sensing power-gain and cross-interference constraints are rewritten with big-M constants derived from a trace inequality. At each step the original solution space is preserved, so an optimal solution of Q is claimed to be optimal for P as well. Solving Q by branch-and-cut yields the globally optimal schedules, pairings, and low-resolution beamformers.","pith_inferences":["The exactness claim implicitly requires the big-M constants to be upper bounds in all inactive cases; a natural stress test is to verify on random instances whether Q's optimum ever violates a constraint of P.","Because the reformulation is exact, the MILP optimum can serve as a certificate against which any heuristic can be measured, allowing one to quantify the suboptimality gap of stage-wise designs in a given scenario.","The lifting-and-total-unimodularity technique is transferable to other discrete-phase ISAC resource-allocation problems, such as many-to-one pairing or multiple users per target, which the paper notes as possible extensions.","The Cramér-Rao analysis suggests that phase resolution, scheduling, and pairing interact; an extension could treat phase resolution as an optimizable variable rather than a fixed parameter."],"forward_implications":["A practitioner can now compute provably optimal schedules, pairings, and low-resolution beams for ISAC systems of moderate size, replacing heuristic stage-wise designs.","The advantage of joint optimization is scenario-dependent: the best heuristic baseline changes between correlated and uncorrelated user channels, while the optimal design is consistently best.","Higher phase resolution improves attainable sensing accuracy, as shown by the Cramér-Rao bound on angle-of-arrival estimation, and the optimal design exploits this automatically.","The linear structure of the reformulated problem lets branch-and-cut solvers reach the global optimum in a small fraction of the exhaustive-search complexity."],"supporting_citations":[{"why":"Supplies the big-M method used to decouple binary scheduling and pairing indicators from the continuous beamforming variables in Propositions 2, 7, and 8.","marker":"[10]"},{"why":"Provides the total unimodularity principle that justifies relaxing the lifted binary product matrices to continuous entries without changing the solution space.","marker":"[22]"},{"why":"Provides the trace inequality $\\mathrm{Tr}(AB) \\le \\mathrm{Tr}(A)\\mathrm{Tr}(B)$ used to compute the big-M constants $D_t$ and $B_u$.","marker":"[23]"},{"why":"Supplies the one-to-one user-target pairing model and the alignment-based baseline against which the optimal joint design is compared.","marker":"[13]"},{"why":"Introduces the directional power gain as the sensing design criterion and its use in beamforming optimization.","marker":"[16]"},{"why":"Supplies the cross-interference threshold model among scheduled targets adopted in constraint C11.","marker":"[17]"},{"why":"Establishes the ISAC framework and motivates joint radar-communication resource allocation as the problem setting.","marker":"[1]"}],"fun_headline_variants":["Exact MILP reformulation globally solves joint ISAC design","Low-resolution ISAC beamforming exactly solved via MILP","Eight equivalent reformulations turn nonconvex ISAC into exact MILP","Exact MILP beats heuristics for joint ISAC scheduling and pairing","Global optimum for ISAC via exact MILP reformulation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reformulation is exact only if the big-M constants $D_t$ are large enough to deactivate the sensing constraints when a target is unscheduled or a pair is inactive; if a desired sensing level $\\tau$ exceeds $D_t$ for a weak target, or the needed bound for a cross-interference term is $D_q$ rather than $D_t$, the MILP becomes a strict restriction of the original problem.","fun_headline_variants_meta":{"raw":{"variants":["Exact MILP reformulation globally solves joint ISAC design","Low-resolution ISAC beamforming exactly solved via MILP","Eight equivalent reformulations turn nonconvex ISAC into exact MILP","Exact MILP beats heuristics for joint ISAC scheduling and pairing","Global optimum for ISAC via exact MILP reformulation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00146,"raw_usage":{"total_tokens":5816,"prompt_tokens":827,"completion_tokens":4989,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":4901}},"tokens_in":443,"tokens_out":4989,"duration_ms":31466,"temperature":1.0,"reasoning_tokens":4901,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:09:22.378268+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the original problem P and the reformulated problem Q on a minimal instance with a single user, a single weak target, and $N=1$ antenna. If the optimal $\\tau$ in P can exceed $D_t = (P_{\\mathrm{tx}}/K)\\mathrm{Tr}(G_t)$ while Q's constraint J1 with $\\lambda_t=0$, $\\rho_{u,t}=0$ forces $\\tau \\le 2D_t$, the claimed solution-space preservation fails. Similarly, with two targets where $D_q > D_t$, check whether an optimal solution that illuminates target q while pairing the user with target t is excluded by K5; if so, Q is not equivalent to P.","supporting_citations":[{"cited_title":"RadiOrchestra: Proactive management of millimeter-wave self-backhauled small cells via joint optimization of beamforming, user association, rate selection, and admission control,","cited_arxiv_id":null,"evidence_quote":"Supplies the big-M method used to decouple binary scheduling and pairing indicators from the continuous beamforming variables in Propositions 2, 7, and 8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the total unimodularity principle that justifies relaxing the lifted binary product matrices to continuous entries without changing the solution space."},{"cited_title":"A note on the trace inequality for products of Hermitian matrix power,","cited_arxiv_id":null,"evidence_quote":"Provides the trace inequality $\\mathrm{Tr}(AB) \\le \\mathrm{Tr}(A)\\mathrm{Tr}(B)$ used to compute the big-M constants $D_t$ and $B_u$."},{"cited_title":"Channel sharing aided integrated sensing and communication: An energy-efficient sensing scheduling approach,","cited_arxiv_id":null,"evidence_quote":"Supplies the one-to-one user-target pairing model and the alignment-based baseline against which the optimal joint design is compared."},{"cited_title":"Beamforming design for active IRS-aided MIMO integrated sensing and communication systems,","cited_arxiv_id":null,"evidence_quote":"Introduces the directional power gain as the sensing design criterion and its use in beamforming optimization."},{"cited_title":"Intelligent reflecting surface enabled multi-target sensing,","cited_arxiv_id":null,"evidence_quote":"Supplies the cross-interference threshold model among scheduled targets adopted in constraint C11."}],"review_version":1}