paper_id stringclasses 10
values | title stringclasses 10
values | prompt_type stringclasses 2
values | difficulty stringclasses 2
values | topic_tags stringclasses 10
values | rubric stringclasses 10
values | proof_source stringclasses 2
values | target_level stringclasses 3
values | node_id stringclasses 9
values | problem stringclasses 10
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L_1-distortion_of_Earth_Mover_Distances_and_Transportation_Cost_Spaces_on_High_Dimensional_Grids | $L_1$-distortion of Earth Mover Distances and Transportation Cost Spaces on High Dimensional Grids | proof_writing | hard | discrete Sobolev inequalities; random signed measures; dyadic decomposition; edge isoperimetry; second-moment method | Total: 7 points
1. [1 pt] Complement reduction
Uses \(\nu_k(G)=0\) and perimeter invariance under complementation to reduce correctly to \(0<\mathrm{Vol}(A)\le1/2\).
2. [2 pt] Probabilistic moment bounds
Derives both the first-moment bound \(\mathbb E|\mu_k(A)|\le2^k\mathrm{Vol}(A)\) and the second-moment bound ... | dag | node_21 | Let \(d\ge 3\), \(n\ge 1\), and equip
\[
G=[2^n]^d=\{1,\ldots,2^n\}^d
\]
with its nearest-neighbor graph structure. For \(B\subseteq G\), set
\[
\mathrm{Vol}(B)=2^{-nd}|B|,
\qquad
\mathrm{Per}(B)=\frac{1}{d}2^{-n(d-1)}|\partial B|,
\]
where \(\partial B\) is the set of graph edges with exactly one endpoint in \(B\).
F... | First observe that \(\nu_k(G)=0\). Hence
\[
\nu_k(A^c)=\nu_k(G)-\nu_k(A)=-\nu_k(A),
\]\nwhile \(\mathrm{Per}(A^c)=\mathrm{Per}(A)\). We may therefore replace \(A\) by its complement and assume that
\[\nv:=\mathrm{Vol}(A)\le \frac12.
\]\nIf \(v=0\), the conclusion is immediate. Henceforth suppose \(v>0\).
\nChoose an in... | |
DNF_formulas_are_efficiently_testable_with_relative_error | DNF formulas are efficiently testable with relative error | proof_writing | medium | property testing; DNF formulas; relative distance; oracle algorithms; concentration bounds | Total: 7 points
1. [1 pt] Sample-based coordinate set
Defines a valid set R from positive samples, such as the coordinates on which the samples vary, rather than assuming the relevant coordinates are known.
2. [2 pt] Concentration invariant
Proves with probability at least 0.99 that head coordinates are outside ... | dag | node_15 | Let \(h:\{0,1\}^n\to\{0,1\}\) have nonempty support. A membership query \(\mathrm{MQ}(h)\) returns \(h(x)\), while \(\mathrm{SAMP}(h)\) returns a uniformly random element of \(h^{-1}(1)\). Define
\[
\operatorname{reldist}(f,g)=\frac{|f^{-1}(1)\triangle g^{-1}(1)|}{|f^{-1}(1)|}.
\]
A term is a conjunction of literals. A... | Let
\[
\Delta=\left\lceil \frac{\mu}{\xi}\ln(200\mu)\right\rceil.
\]\nThe tester first draws independent samples \(z^{(1)},\ldots,z^{(\Delta)}\) from \(\mathrm{SAMP}(h)\). It defines
\[\nR=\{i\in[n]:z_i^{(a)}\ne z_i^{(b)}\text{ for some }a,b\in[\Delta]\};
\]\nthus \(R\) consists precisely of the coordinates on which th... | |
Hilberts_Nullstellensatz_is_in_the_Counting_Hierarchy | Hilbert’s Nullstellensatz is in the Counting Hierarchy | proof_writing | hard | algebraic complexity; counting hierarchy; multivariate resultants; zero-dimensional varieties; randomized reductions; univariate gcd | Total: 7 points
1. [1 pt] Base-ring reductions
Correctly handles number fields, including the multiplicative factor \(\deg g\), clearing denominators, and enlargement of small finite constant fields without changing geometric point counts.
2. [1 pt] Two generic supersystems
Uses two independent systems of random... | dag | node_44 | Let
\[\nR\in\{\mathbb Z,\mathbb Z[y_1,\ldots,y_k],\mathbb K,\mathbb K[y_1,\ldots,y_k],\mathbb F_p,\mathbb F_p[y_1,\ldots,y_k],\mathbb F_{p^a},\mathbb F_{p^a}[y_1,\ldots,y_k]\},
\]\nwhere \(\mathbb K\) is a number field and \(p\) is prime. Let \(\mathbb L\) be an algebraic closure of \(\operatorname{Frac}(R)\). The inpu... | We first reduce to coefficient rings for which the resultant-evaluation primitive applies directly.
Suppose \(R=\mathbb K\) or \(\mathbb K[\mathbf y]\), where \(\mathbb K=\mathbb Q[\alpha]\) and the minimal polynomial \(g\) of \(\alpha\) has degree \(e\). Replace every occurrence of \(\alpha\) in the coefficients of t... | |
Parallel_Small_Vertex_Connectivity_in_Near-Linear_Work_and_Polylogarithmic_Depth | Parallel Small Vertex Connectivity in Near-Linear Work and Polylogarithmic Depth | proof_writing | medium | parallel algorithms; approximate shortest paths; vertex-weighted graphs; graph reductions | Total: 7 points
1. [2 pt] Vertex-to-edge reduction
Constructs a valid nonnegative edge-length instance, such as $w(\{u,v\})=\ell(u)+\ell(v)$ up to a common positive scaling, without increasing the graph size asymptotically.
2. [2 pt] Path relation and minimizers
Derives the endpoint-corrected identity relating e... | dag | node_06 | Let $G=(V,E)$ be a connected undirected graph with $m=|E|$, let $s\in V$, let $\ell:V\to\mathbb{R}_{\ge 0}$, and let $0<\epsilon\le 1$. For a path $P=(v_0,\ldots,v_k)$, define
$$\ell(P)=\sum_{i=0}^k\ell(v_i),$$
and let $\operatorname{dist}_{G,\ell}(s,t)$ be the minimum of $\ell(P)$ over all $(s,t)$-paths $P$ in $G$. A ... | Construct a nonnegative edge-length function on the same graph by setting
$$w(\{u,v\})=\ell(u)+\ell(v)$$
for every edge $\{u,v\}\in E$. This can be done independently for all edges, using $O(m)$ work and $O(1)$ depth.
The key relation between the two notions of path length is the following. If $P=(v_0,v_1,\ldots,v_k)$... | |
Compressed_Inverse_Suffix_Arrays | Compressed Inverse Suffix Arrays | proof_writing | hard | compressed data structures; inverse suffix arrays; word RAM; hierarchical sampling; parameter tuning | Total: 7 points
1. [1 pt] Coarse sampling setup
Chooses D=Θ(log_σ n), samples ISA at spacing g=Θ(τD) up to power-of-two rounding, and obtains the (n/τ)log σ sampled-space term.
2. [2 pt] Hierarchical transition construction
Recovers a geometric family of transition structures and a digit decomposition of the res... | dag | node_07 | All logarithms are base 2, and logarithmic factors are clipped below by 1. Work in the word-RAM model with word size Θ(log n). Let T[0,n) be a string over [0,σ), where 2 ≤ σ ≤ n, ending in a unique minimum sentinel. For a string X, let ISA_X denote the inverse of its suffix array. The packed representation of T is stor... | Let U have power-of-two length N as in the padding fact. We first construct a general tradeoff using a branching parameter Δ and then choose Δ as a function of σ and ε.
Define L = log_σ N and let D be the largest power of two not exceeding L. Since σ ≤ n ≤ N, we have L ≥ 1, and hence
D ≤ log_σ N < 2D.
In particular,... | |
Schur_complements_for_tensors_and_multilinear_commutative_rank | Schur complements for tensors and multilinear commutative rank | proof_strategy | hard | multilinear algebra; tensor rank; finite field methods; Schur complements; polynomial multiplicity | Total: 7 points
1. [1 pt] Main obstacle
Identifies both the finite-field gap between generic and evaluated rank and the failure of ordinary rational Schur elimination to preserve multilinearity or complementary-variable partition terms.
2. [1 pt] Evaluation-rank plan
Proposes a determinant-minor and multiplicity... | paper_level | node_04 | Let $d,n,a,b$ be positive integers, let $\mathbb{F}$ be a field, and write $[d]=\{1,\ldots,d\}$. For each $i\in[d]$, let $x_i=(x_{i,1},\ldots,x_{i,n})$ be a block of variables. Let $\mathcal{M}_d$ be the space of forms
$$f(\mathbf{x})=\sum_{j_1,\ldots,j_d=1}^n c_{j_1,…,j_d}\prod_{i=1}^d x_{i,j_i},\qquad c_{j_1,…,j_d}\i... | {"main_obstacle": "The generic rank $\\mathrm{CR}(M)$ is witnessed over a rational-function field or an algebraic closure, whereas $\\mathrm{MR}(M)$ only sees base-field evaluations and $\\mathrm{PR}(M)$ requires complementary-variable factorizations. Over a finite field, no evaluation need realize $k=\\mathrm{CR}(M)$.... | |
Hereditary_2-WQO_Graph_Classes_Have_Bounded_Clique-Width | Hereditary 2-WQO Graph Classes Have Bounded Clique-Width | proof_strategy | hard | structural graph theory; two-label well-quasi-ordering; clique-width; rank-width; well-linked sets; forbidden induced patterns | Total: 7 points
1. [1 pt] Main obstacle
Identifies both mismatches: labeled well-quasi-ordering does not directly control cut-rank, and a skipped-block bound on $\rho_G(L_\alpha,R_\alpha)$ is not a bound on the full cut $\rho_G(L_\alpha)$ required by well-linkedness.
2. [2 pt] Structural separator plan
Formulate... | paper_level | node_04 | Let $\mathscr{C}$ be a hereditary class of finite simple graphs. The class is two-label well-quasi-ordered, abbreviated $2$-WQO, if every infinite sequence of pairs $(G_i,\lambda_i)$, where $G_i\in\mathscr{C}$ and $\lambda_i:V(G_i)\to\{1,2\}$, contains indices $i<j$ for which there is a label-preserving induced-subgrap... | {"main_obstacle": "The property of being $2$-WQO controls labeled induced-subgraph embeddings, whereas rank-width is controlled by ranks of cuts. Even after pattern-freeness yields a partition with small $\\rho_G(L_\\alpha,R_\\alpha)$, well-linkedness in $G$ concerns the full cut from $L_\\alpha$ to $P_\\alpha\\cup R_\... | |
Hard_CNF_Instances_for_Ideal_Proof_Systems_The_ROABP_Case | Hard CNF Instances for Ideal Proof Systems: The ROABP Case | proof_strategy | hard | algebraic proof complexity; feasible interpolation; span programs; branching program decompositions; monotone computation | Total: 7 points
1. [1 pt] Main obstacle and ordering
Identifies the need to replace full-variable certificates by fixed $\overline{x}$-only generators with scalar coefficients after specialization, and explains why having all of $\overline{x}$ as a prefix is essential.
2. [2 pt] Width-based certificate compression
... | bottleneck_subproblem | node_13 | Let $\mathbb{F}$ be a field. Let $\overline{x}$, $\overline{y}$, and $\overline{z}$ be pairwise disjoint tuples of variables, where $\overline{x}$ and $\overline{y}$ are private variables and $\overline{z}$ is the tuple of shared Boolean variables. Let $P_0(\overline{x},\overline{z})$ and $P_1(\overline{y},\overline{z}... | {"main_obstacle": "The certificates $a_p$ can depend simultaneously on $\\overline{x}$, $\\overline{y}$, and $\\overline{z}$, whereas a span program needs a fixed finite family of vectors over $\\mathbb{F}$ whose availability is controlled only by literals in $\\overline{z}$. The central task is therefore to compress e... | |
Approximate_Spanning_Tree_Counting_from_Uncorrelated_Edge_Sets | Approximate Spanning Tree Counting from Uncorrelated Edge Sets | proof_strategy | hard | randomized graph algorithms; spanning tree counting; spectral graph theory; log determinant estimation; variance analysis; algorithmic proof strategy | Total: 7 points
1. [1 pt] Central obstacle
Identifies both the off-diagonal interaction created by batch deletion and the first-order error caused by approximate leverage scores, together with the need to balance error against batch size.
2. [1 pt] Safe batch supply
Uses low-degree elimination, the leverage-scor... | paper_level | node_03 | Let $G=(V,E,w)$ be a connected undirected graph with $n:=|V|$ vertices, $m:=|E|$ edges, and positive edge weights satisfying $n^{-\mathcal{O}(1)}\le w_e\le n^{\mathcal{O}(1)}$. Define
$$\mathcal{T}(G):=\sum_{\substack{T\subseteq E\\T\text{ is a spanning tree of }G}}\prod_{e\in T}w_e.$$
For unit weights this is the numb... | {"main_obstacle": "Deleting edges one at a time permits accurate determinant-ratio updates but is too slow, while deleting a large batch introduces off-diagonal electrical interactions. Even after choosing a weakly correlated batch, replacing each $\\tau_f$ by an additive approximation $\\widetilde{\\tau}_f$ creates a ... | |
Stochastic_Gradient_Meets_Randomized_Rounding_New_Algorithms_for_Node-Weighted_Steiner_Problems | Stochastic Gradient Meets Randomized Rounding: New Algorithms for Node-Weighted Steiner Problems | proof_strategy | hard | random-order online algorithms; node-weighted Steiner forest; potential method; multiplicative weights; randomized rounding | Total: 7 points
1. [1 pt] Core obstacle and progress dichotomy
Identifies both reuse of the same LP mass across demands and the positive $d_tq_t$ learning term, and explains why a rootless forest requires same-scale augmentation or an equivalent mechanism.
2. [2 pt] Geometric charging and learning drift
Proposes... | paper_level | node_22 | Consider the following random-order online problem. The input is an undirected graph $G=(V,E)$ with $n=|V|$ vertices, zero edge costs, and nonnegative vertex costs $c_v$. A fixed collection $\mathcal D$ of $k\le n^2$ unordered terminal pairs arrives in a uniformly random permutation. After pair $s_t=(a_t,b_t)$ arrives,... | {"main_obstacle": "A naive charge of every connection distance $d_t$ to $x^\\star$ can reuse the same fractional mass for many demands. A KL argument repairs this when shortcut mass is small, but its normalization cost produces a positive term proportional to $d_tq_t$; when $X_t$ is large, no useful multiplicative upda... |
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