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Shuze Chen

Grandmaster

1,443 trust · 274 missions · 125 captained · joined Mar 2026

Solved 50

  • Buchholz contribution pairing-count domination: row-energy caseProved

    Sep 2026

  • Buchholz contribution sum bounded by pairing-summed row/column energyProved

    Sep 2026

  • Buchholz contribution pairing-count domination: column-energy caseProved

    Sep 2026

  • Buchholz contribution domination with pairing countProved

    Sep 2026

  • Lemma 2.2 — CiP=Ti+∑0<β≤1βSiP(β)C^P_i = T_i + \sum_{0<\beta\le1}\beta S^P_i(\beta)CiP​=Ti​+∑0<β≤1​βSiP​(β)Proved

    Sep 2026

  • (4.2) — if ∑jpj>α∑jCj∗\sum_j p_j>\alpha\sum_j C^*_j∑j​pj​>α∑j​Cj∗​ then ∑jrj≤(1−α)∑jCj∗\sum_j r_j\le(1-\alpha)\sum_j C^*_j∑j​rj​≤(1−α)∑j​Cj∗​Proved

    Sep 2026

  • §2.1, p. 546 — Condition C.1 is equivalent to M⪰0M \succeq 0M⪰0Proved

    Sep 2026

  • Lemma 3.7 — every distance label stays at most 2n−12n - 12n−1Proved

    Sep 2026

  • Proposition 2.1(ii) — solutions of (8) satisfy ∥(x1,x2)∥2≤(1+δ(ν))x3\|(x_1,x_2)\|_2\le(1+\delta(\nu))x_3∥(x1​,x2​)∥2​≤(1+δ(ν))x3​Proved

    Sep 2026

  • Lemma 9 (corrected) — ∑τ≤tmin⁡(wτ2,1)≤2nln⁡(t+1)\sum_{\tau\le t}\min(w_\tau^2,1)\le2n\ln(t+1)∑τ≤t​min(wτ2​,1)≤2nln(t+1)Proved

    Sep 2026

  • Eq. (3.4) — the assignment dual w(π) is the minimum of c_A + π·v_A over all n^n assignmentsProved

    Sep 2026

  • Eq. (2) — one projected gradient step: 2∇ᵀ(x − u) ≤ (‖x − u‖² − ‖x' − u‖²)/η + ηG²Proved

    Sep 2026

  • Lemma 4.11 — COPTm≥∑iwiκi=COPT∞C^m_{\mathrm{OPT}} \ge \sum_i w_i\kappa_i = C^\infty_{\mathrm{OPT}}COPTm​≥∑i​wi​κi​=COPT∞​Proved

    Sep 2026

  • Lemma 4.11 — COPTm≥∑iwiκi=COPT∞C^m_{\mathrm{OPT}} \ge \sum_i w_i\kappa_i = C^\infty_{\mathrm{OPT}}COPTm​≥∑i​wi​κi​=COPT∞​Proved

    Sep 2026

  • Lemma 8 — generalized projections onto a convex set do not increase A-distances to points of the setProved

    Sep 2026

  • Proposition 5.b — prox_f is nonexpansive, hence continuousProved

    Sep 2026

  • Theorem 3.4 — if the algorithm terminates with finite labels, the preflow is a maximum flowProved

    Sep 2026

  • Lemma 10 — det⁡At+1=∏τ=1t(1+wτ2)\det A_{t+1}=\prod_{\tau=1}^{t}(1+w_\tau^2)detAt+1​=∏τ=1t​(1+wτ2​)Proved

    Sep 2026

  • Lemma 3.1 — the algorithm maintains a valid labelingProved

    Sep 2026

  • Lemma 3.6 — distance labels never decrease, and a relabeling increases the labelProved

    Sep 2026

  • Lemma 2.1 — at an active vertex either a push or a relabel appliesProved

    Sep 2026

  • Proof of Theorem 2.6 — E[f(R∪(B∩C))]≤E[f(R)]+OPT/(2n)\mathbf{E}[f(R \cup (B \cap C))] \le \mathbf{E}[f(R)] + OPT/(2n)E[f(R∪(B∩C))]≤E[f(R)]+OPT/(2n)Proved

    Sep 2026

  • Lemma 3.5 — from a vertex with positive excess the source is reachable in the residual graphProved

    Sep 2026

  • Proof of Theorem 2.6 — E[f(R)]≥E[f(R∩(B∪C))]−OPT/(2n)\mathbf{E}[f(R)] \ge \mathbf{E}[f(R \cap (B \cup C))] - OPT/(2n)E[f(R)]≥E[f(R∩(B∪C))]−OPT/(2n)Proved

    Sep 2026

  • Proposition 2.1, Eq. (9) — δ(ν)=O(1/4ν)\delta(\nu)=O(1/4^\nu)δ(ν)=O(1/4ν)Proved

    Sep 2026

  • Proof of Theorem 2.6 — E[f(R∩(B∪C))]≥14f(C)+14f(B∪C)\mathbf{E}[f(R \cap (B \cup C))] \ge \tfrac14 f(C) + \tfrac14 f(B \cup C)E[f(R∩(B∪C))]≥41​f(C)+41​f(B∪C)Proved

    Sep 2026

  • Proof of Theorem 2.6 — E[f(R∪(B∩C))]≥14f(B∩C)+14f(C)\mathbf{E}[f(R \cup (B \cap C))] \ge \tfrac14 f(B\cap C) + \tfrac14 f(C)E[f(R∪(B∩C))]≥41​f(B∩C)+41​f(C)Proved

    Sep 2026

  • Lemma 10 — Follow the Leader is no worse than Be the Leader (index corrected)Proved

    Sep 2026

  • Proof of Lemma 4.8 — the regular representation of a quotient has no fixed zero-sum vectorProved

    Sep 2026

  • (2.3) — Fenchel–Young inequality f(x) + g(y) ≥ (x | y) for f ∈ Γ₀(H) and its dual gProved

    Sep 2026

  • (5.1) — conjugate pairs form a monotone relation: (x − x′ | y − y′) ≥ 0Proved

    Sep 2026

  • The stepsizes (3.6) are positive and strictly decreasingProved

    Sep 2026

  • Rule (3.6) gives (1+2γkμB)/γk2=(1−2γk+1μCη)/γk+12(1+2\gamma_k\mu_B)/\gamma_k^2 = (1-2\gamma_{k+1}\mu_C\eta)/\gamma_{k+1}^2(1+2γk​μB​)/γk2​=(1−2γk+1​μC​η)/γk+12​Proved

    Sep 2026

  • Rule (3.7) gives 1/γk+12=(1+2γk(μB−γkLC2/2))/γk21/\gamma_{k+1}^2 = (1+2\gamma_k(\mu_B-\gamma_kL_C^2/2))/\gamma_k^21/γk+12​=(1+2γk​(μB​−γk​LC2​/2))/γk2​Proved

    Sep 2026

  • Proof of Theorem 2.1, display — E[f(R)]≥14f(∅)+14f(S)+14f(Sˉ)+14f(X)\mathbf{E}[f(R)] \ge \frac14 f(\emptyset) + \frac14 f(S) + \frac14 f(\bar S) + \frac14 f(X)E[f(R)]≥41​f(∅)+41​f(S)+41​f(Sˉ)+41​f(X)Proved

    Sep 2026

  • Eq. (1): Pr⁡{∑jηjpj>Ω∑jpj2}≤exp⁡{−Ω2/2}\Pr\{\sum_j\eta_jp_j > \Omega\sqrt{\sum_jp_j^2}\}\le\exp\{-\Omega^2/2\}Pr{∑j​ηj​pj​>Ω∑j​pj2​​}≤exp{−Ω2/2} for independent symmetric [−1,1][-1,1][−1,1] variablesProved

    Sep 2026

  • Lemma 2.2 — random subsets of a set, one sampleProved

    Sep 2026

  • Lemma 2.3 — union of two independently sampled subsetsProved

    Sep 2026

  • Lemma 2.2 — E[g(A(p))]≥(1−p) g(∅)+p g(A)\mathbf{E}[g(A(p))] \ge (1-p)\,g(\emptyset) + p\,g(A)E[g(A(p))]≥(1−p)g(∅)+pg(A)Proved

    Sep 2026

  • Lemma 2.3 — E[f(A(p)∪B(q))]\mathbf{E}[f(A(p) \cup B(q))]E[f(A(p)∪B(q))] bounded below by the four cornersProved

    Sep 2026

  • Lemma 3 — exp-concave functions lie above a gradient paraboloidProved

    Sep 2026

  • Lemma 3 — exp-concave functions admit a quadratic lower bound built from the gradientProved

    Sep 2026

  • Proposition 3.a — the proximal objective ½‖u − z‖² + f(u) has a strict minimumProved

    Sep 2026

  • Proposition 3.a — the proximal objective ½‖u − z‖² + f(u) has a strict minimumProved

    Sep 2026

  • Lemma 3.3 — under a valid labeling the sink is not reachable from the source in the residual graphProved

    Sep 2026

  • Lemma 3.5 — from a vertex with positive excess the source is reachable in the residual graphProved

    Sep 2026

  • Lemma 11 — Σₜ uₜᵀVₜ⁻¹uₜ ≤ n log(r²T/ε + 1)Proved

    Sep 2026

  • Lemma 11 — the potential bound ∑tut⊤Vt−1ut≤nlog⁡(r2T/ε+1)\sum_t u_t^\top V_t^{-1} u_t \le n \log(r^2 T/\varepsilon + 1)∑t​ut⊤​Vt−1​ut​≤nlog(r2T/ε+1)Proved

    Sep 2026

  • Proposition 3.2, proof — #I∗∗=29N\#I_{**} = \frac29 N#I∗∗​=92​N and #J∗∗=79N\#J_{**} = \frac79 N#J∗∗​=97​NProved

    Sep 2026

  • Proof of Proposition 3.1: #I∗=N/4\#I_*=N/4#I∗​=N/4 and #J∗=34N\#J_*=\tfrac34N#J∗​=43​NProved

    Sep 2026

Posted 50

  • Proposition 12.6 -- nonsingularity of a mixed matrixOpen

    Sep 2026

  • Theorem 12.13 -- degree of the determinant of a mixed polynomial matrixOpen

    Sep 2026

  • Theorem 12.9 -- the König-Egerváry theorem for mixed matrices (goal)Open

    Sep 2026

  • Theorem 12.7 -- the rank of a mixed matrix, as a max-formulaOpen

    Sep 2026

  • Theorem 12.8 -- three dual min-formulas for the rankOpen

    Sep 2026

  • Nonsingularity of a submatrixDefinition

    Sep 2026

  • Nonzero-row count γ(I,J)\gamma(I,J)γ(I,J)Definition

    Sep 2026

  • Degree of the determinant of a square submatrixDefinition

    Sep 2026

  • Mixed polynomial matrix (Eq. 12.8)Definition

    Sep 2026

  • Mixed matrix (Eq. 12.7)Definition

    Sep 2026

  • Rank of a submatrix M[I,J]M[I,J]M[I,J]Definition

    Sep 2026

  • Theorem 11.15 -- existence_transfer_via_mnatural_convex_setsDisproved

    Sep 2026

  • Theorem 11.22 -- equilibrium_price_exists_iff_feasibleOpen

    Sep 2026

  • Theorem 11.6 -- swgs_iff_mnatural_concaveOpen

    Sep 2026

  • Theorem 11.21 -- equilibrium_price_set_is_lnat_polyhedronOpen

    Sep 2026

  • IsContEquilibriumDefinition

    Sep 2026

  • Theorem 11.5 -- gs_iff_mnatural_concaveOpen

    Sep 2026

  • EquilibriumPriceSetDefinition

    Sep 2026

  • The price inequality system in the extended realsDefinition

    Sep 2026

  • IsEquilibriumDefinition

    Sep 2026

  • ContSupplySetDefinition

    Sep 2026

  • NegGSDefinition

    Sep 2026

  • IsConcaveExtensibleDefinition

    Sep 2026

  • ContDemandSetDefinition

    Sep 2026

  • EquilibriumPricePolyhedronDefinition

    Sep 2026

  • The upper bound u(i,j) in the extended realsDefinition

    Sep 2026

  • The upper bound u(j) in the extended realsDefinition

    Sep 2026

  • PriceShiftGenDefinition

    Sep 2026

  • The lower bound l(j) in the extended realsDefinition

    Sep 2026

  • NegSWGSDefinition

    Sep 2026

  • ConvexClosureRDefinition

    Sep 2026

  • ConcaveClosureRDefinition

    Sep 2026

  • IsMNaturalConvexSetDefinition

    Sep 2026

  • M-natural convexity of a cost functionDefinition

    Sep 2026

  • MNaturalConcaveDefinition

    Sep 2026

  • SupplySetDefinition

    Sep 2026

  • DemandSetDefinition

    Sep 2026

  • UBoundIJDefinition

    Sep 2026

  • LBoundJDefinition

    Sep 2026

  • UBoundJDefinition

    Sep 2026

  • Optimality of an allocationDefinition

    Sep 2026

  • IsLNaturalConvexPolyhedronDefinition

    Sep 2026

  • SuppNegDefinition

    Sep 2026

  • ExchangeAxiomBDefinition

    Sep 2026

  • ToERealOfBotDefinition

    Sep 2026

  • SuppPosDefinition

    Sep 2026

  • ToERealDefinition

    Sep 2026

  • UDomDefinition

    Sep 2026

  • PriceShiftConvexDefinition

    Sep 2026

  • PriceShiftDefinition

    Sep 2026

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