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Contention Resolution Schemes and Multilinear Relaxation Rounding for Constrained Submodular Maximization

Algorithms AIMech:ContentionResolutionSchemesForSubmodularMaximization DRAFT

Aliases: Contention Resolution Scheme, Monotone (b, c)-Balanced Contention Resolution Scheme, Chekuri-Vondrak-Zenklusen Contention Resolution Framework

A (b, c)-balanced Contention Resolution (CR) scheme pi for a polyhedral relaxation P_I subseteq [0, 1]^N of a downward-closed independence family I subseteq 2^N, introduced and analyzed by Chekuri, Vondrak, and Zenklusen (2011/2014), is a (possibly randomized) rounding procedure that for every fractional vector x in b * P_I and subset A subseteq N returns a feasible random subset pi_x(A) subseteq A cap support(x) such that (i) pi_x(A) in I with probability 1, and (ii) for every element i in support(x), Pr[i in pi_x(R(x)) | i in R(x)] >= c when R(x) includes each element j in N independently with probability x_j (Definition 1.2). The CR scheme is monotone if Pr[i in pi_x(A_1)] >= Pr[i in pi_x(A_2)] whenever i in A_1 subseteq A_2. Coupled with a linear-time sequential pruning operation eta_f(I) that retains i in I only if f(J + i) - f(J) > 0, any monotone (b, c)-balanced CR scheme rounds x in b * P_I to a feasible integral solution satisfying E[f(eta_f(pi_x(R(x))))] >= c * F(x) for any non-negative (monotone or non-monotone) submodular function f with multilinear extension F (Theorem 1.3).

Identity Notes & Concept Scope Distinct from AIMech:ContinuousGreedyAndMultilinearExtensionForMatroidSubmodularMaximization (Round 349) and AIMech:MeasuredContinuousGreedySubmodularMaximization (Round 368), which are continuous optimization algorithms that find a fractional point x in P_I maximizing F(x) (and rely on matroid-specific pipage rounding), whereas Contention Resolution Schemes provide a general, modular dependent-rounding framework across intersections of matroids, matchoids, knapsacks, sparse packing systems, and tree unsplittable flows via LP duality and the correlation gap, alongside Fractional Local Search for maximizing F(x) over arbitrary solvable down-monotone polytopes.

Claims & Grounded Evidence

3 Claims

Mechanism #r369-claim-fkg-pruning-expectation-bound-and-intersection-composition

For any non-negative submodular function f with multilinear extension F, point x in b * P_I, and monotone (b, c)-balanced CR scheme pi producing I = pi_x(R(x)), defining the pruned set J = eta_f(I) by sequentially including i in I iff f_{J cap [i-1]}(i) > 0 guarantees via submodularity and the FKG inequality on the product space conditioned on i in R(x) that E[f(J cap [i]) - f(J cap [i-1])] >= c * E[f(R cap [i]) - f(R cap [i-1])] for each i, summing to E[f(eta_f(I))] >= c * F(x) (Theorem 1.3, Section 4.1 Eqs. 8-10); furthermore, given h independence systems I_i with monotone (b, c_i)-balanced CR schemes pi^i on P_{I_i}, their intersection scheme pi_x(A) = cap_{i=1}^h pi_x^i(A) on P_I = cap_i P_{I_i} is a monotone (b, prod_{i=1}^h c_i)-balanced CR scheme by the FKG inequality (Lemma 1.6, Section 4.1).

supports Source: Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes · Locator: Section 1.2-1.3, Definition 1.2, Theorems 1.3 and 1.5, and Lemma 1.6, pp. 5-7; Section 4.1, Equations (8)-(10), Theorem 4.1, and Proof of Lemma 1.6, pp. 16-18

Mechanism #r369-claim-correlation-gap-lp-duality-and-optimal-matroid-crs

By formulating the search for a randomized (b, c)-balanced CR scheme at x in b * P_I as a linear program over distributions of valid deterministic mappings (LP1) and taking its dual (DP1), a (b, c)-balanced CR scheme exists if and only if inf_{x in b*P_I, y >= 0} E_{R(x)}[max_{S subseteq R, S in I} sum_{i in S} y_i] / sum_{i in N} x_i * y_i >= c, which equals the correlation gap kappa(I) = inf_{y >= 0} kappa(r_y) of the weighted rank functions r_y of I and can be constructed in polynomial time via the ellipsoid algorithm whenever I admits a monotone maximum-weight independent set oracle (Theorem 4.3, Theorem 4.6, Lemma 4.7, Appendix C); for any matroid M on n elements, F(b * p) >= (1 - (1 - b/n)^n) * f^+(p) (Lemmas 4.9-4.10) yields a monotone, efficiently implementable (b, (1 - (1 - b/n)^n) / b)-balanced (and (b, (1 - e^{-b}) / b)-balanced) CR scheme (Theorem 1.4, Theorem 4.8, Corollary 4.11) that is provably optimal even on rank-1 uniform matroids (Theorem 4.12), alongside a simpler deterministic (b, 1 - b)-balanced span-ordering scheme (Lemma 4.13, Theorem 4.14).

supports Source: Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes · Locator: Section 1.2, Theorem 1.4, p. 6; Section 4.2-4.4, Proposition 4.2, Theorems 4.3, 4.6, 4.8, 4.12, 4.14, Lemmas 4.7, 4.9, 4.10, 4.13, and Corollary 4.11, pp. 18-26; Appendix B-C, pp. 37-41

Mechanism #r369-claim-fractional-local-search-over-down-monotone-polytopes

For maximizing the multilinear extension F(x) of any non-negative submodular function over any solvable down-monotone polytope P subseteq [0, 1]^N, direction-wise concavity implies (y - x) . nabla F(x) >= F(x vee y) + F(x wedge y) - 2*F(x) (Lemma 3.2); Discretized Fractional Local Search (Algorithm 3.6) maintains a convex combination of q = n^a vertices of P and terminates in O(n^{2a-1}) steps at an approximate local optimum x satisfying 2*F(x) >= F(x vee y) + F(x wedge y) - 5*delta*n for all y in P (Lemmas 3.7-3.8), so running it on P and on the complement-restricted polytope Q = {y in P : y <= 1 - x} (Algorithm 3.9) gives 2*F(x) + 2*F(z) >= OPT - 10*delta*n and a (0.25 - o(1))-approximation to max_{x in P} F(x) (Lemma 3.12, Corollary 3.13), while a single Restricted Local Search on P cap [0, t]^N with t = (3 - sqrt(5))/2 (Algorithm 3.15) achieves F(x) >= ((2*t - t^2)/2)*f(C) - o(1) = ((-1 + sqrt(5))/4 - o(1))*OPT_int approx 0.309*OPT_int (Theorem 1.1, Lemma 3.19, Corollary 3.21).

supports Source: Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes · Locator: Section 1.1, Theorem 1.1, p. 4; Section 3.1-3.4, Algorithms 3.1, 3.6, 3.9, 3.15, Lemmas 3.2, 3.7, 3.8, 3.11, 3.12, 3.16, 3.19, and Corollaries 3.13, 3.21, pp. 9-15

⚠️ Curated Limitations & Negative Results

Achieving constant-factor approximations for maximizing F(x) requires the solvable polytope P to be down-monotone (no constant-factor approximation exists in the value-oracle model over general non-down-monotone polytopes such as the matroid base polytope when fractional base packing number nu -> 1, Section 3, p. 9; Appendix A), and the optimal matroid CR scheme requires solving linear programs via the ellipsoid method and Monte Carlo sampling whereas the faster deterministic span-ordering scheme achieves a weaker (b, 1 - b) balance (Section 4.4, p. 25).

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:Clustering` Clustering	related_match	Provides polyhedral relaxation and contention-resolution rounding algorithms for constrained submodular maximization, facility location, and Max-k-Cover.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`r369-primary-pdf`	Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes Chandra Chekuri, Jan Vondrak, Rico Zenklusen	primary paper full_text	ID: https://doi.org/10.1137/110839655 sha256:c8c972e4e4f9…
`r369-arxiv-abs`	Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes (arXiv abstract) Chandra Chekuri, Jan Vondrak, Rico Zenklusen	primary paper abstract_only	ID: https://arxiv.org/abs/1105.4593 sha256:a51a7955188c…
`r369-metadata-cr`	Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes (Crossref metadata) Chandra Chekuri, Jan Vondrak, Rico Zenklusen	primary paper abstract_only	ID: https://doi.org/10.1137/110839655 sha256:94f75f2015cf…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini

Created: 2026-10-19T07:32:00Z

Updated: 2026-10-19T07:32:00Z

Source File: data/curated/ContentionResolutionSchemesForSubmodularMaximization.yaml · Schema: curated_concept.yaml

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