Optimal Low-Degree Hardness of Maximum Independent Set

Wein, Alexander S.

doi:10.48550/arxiv.2010.06563

Cited by 3 publications

(8 citation statements)

References 15 publications

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“…Multi-OGPs have the potential to show tight or nearly-tight algorithmic hardness for stable algorithms. In the problem of maximum independent set, [Wei20] recently proved, using a multi-OGP, that the class of low degree polynomial algorithms cannot attain any objective beyond the believed algorithmic limit. This generalizes previous work [RV17] showing the same impossibility result for the more restricted class of local algorithms.…”

Section: The Overlap Gap Property As a Barrier To Algorithmsmentioning

confidence: 99%

“…• Ladder OGP: several solutions, where the i-th solution (i ≥ 2) has medium "multi-overlap" with the first i − 1 solutions, for a problem-specific notion of multi-overlap of one solution with several solutions [Wei20,BH21].…”

Section: The Overlap Gap Property As a Barrier To Algorithmsmentioning

confidence: 99%

“…Specifically, in a "star" multi-OGP [RV17, GS17,GK21b] all the replicas are pairwise equidistant. For the "ladder" OGP implementations of [Wei20,BH21], the forbidden structure is defined by applying some stopping rule to choose a finite number of solutions from a "stably evolving" sequence of algorithmic outputs. In both settings it is possible that the resulting configuration is a star ultrametric with all pairwise nonzero distances equal.…”

Section: Necessity Of Full Branching Treesmentioning

confidence: 99%

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Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses

Huang¹,

Sellke²

2021

Preprint

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We study the problem of algorithmically optimizing the Hamiltonian H N of a spherical or Ising mixed p-spin glass. The maximum asymptotic value OPT of H N /N is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing algorithms efficiently optimize H N /N up to a value ALG given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property. However, ALG < OPT can also occur, and no efficient algorithm producing an objective value exceeding ALG is known.We prove that for mixed even p-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than ALG with non-negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the disorder coefficients of H N . It encompasses natural formulations of gradient descent, approximate message passing, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving ALG mentioned above. To prove this result, we substantially generalize the overlap gap property framework introduced by Gamarnik and Sudan to arbitrary ultrametric forbidden structures of solutions. Contents

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Section: The Overlap Gap Property As a Barrier To Algorithmsmentioning

confidence: 99%

Section: The Overlap Gap Property As a Barrier To Algorithmsmentioning

confidence: 99%

Section: Necessity Of Full Branching Treesmentioning

confidence: 99%

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Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses

Huang¹,

Sellke²

2021

Preprint

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“…Moreover, γ is a normalization parameter, which we think of as a large constant; this is necessary because without it, the condition that valid outputs of f are outside the interval (−1, 1) becomes meaningless because we can simply scale f by a large constant. An analogous normalization condition appears in other hardness results against low degree polynomials in the literature [43,71]. The error tolerance of our rounding scheme is a small constant η, as discussed above.…”

Section: Definition 22 (Low Degree Polynomial) a Functionmentioning

confidence: 64%

“…To improve the threshold at which algorithms are ruled out, these arguments have been generalized to consider forbidden structures consisting of several solutions, which we term multi-OGPs. Using multi-OGPs, a line of work [69,43] culminating in the paper of Wein [71] tightly identified the algorithmic phase transition of maximum independent set on a sparse Erdős-Rényi graph for low degree polynomials. Multi-OGPs have also been used in [46,44] to rule out classes of algorithms for random k-SAT and the number partitioning problem well below the point where solutions exist, although these results are not tight against the best algorithms.…”

Section: Introductionmentioning

confidence: 99%

The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials

Bresler¹,

Huang²

2021

Preprint

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Let Φ be a uniformly random k-SAT formula with n variables and m clauses. We study the algorithmic task of finding a satisfying assignment of Φ. It is known that a satisfying assignment exists with high probability at clause density m/n < 2 k log 2 − 1 2 (log 2 + 1) + o k (1), while the best polynomial-time algorithm known, the Fix algorithm of Coja-Oghlan [25], finds a satisfying assignment at the much lower clause density (1 − o k (1))2 k log k/k. This prompts the question: is it possible to efficiently find a satisfying assignment at higher clause densities?To understand the algorithmic threshold of random k-SAT, we study low degree polynomial algorithms, which are a powerful class of algorithms including Fix, Survey Propagation guided decimation (with bounded or mildly growing number of message passing rounds), and paradigms such as message passing and local graph algorithms. We show that low degree polynomial algorithms can find a satisfying assignment at clause density (1 − o k (1))2 k log k/k, matching Fix, and not at clause density (1 + o k (1))κ * 2 k log k/k, where κ * ≈ 4.911. This shows the first sharp (up to constant factor) computational phase transition of random k-SAT for a class of algorithms. Our proof establishes and leverages a new many-way overlap gap property tailored to random k-SAT.

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Large Independent Sets on Random d-Regular Graphs with Fixed Degree d

Marino,

Kirkpatrick

2023

Computation

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The maximum independent set problem is a classic and fundamental combinatorial challenge, where the objective is to find the largest subset of vertices in a graph such that no two vertices are adjacent. In this paper, we introduce a novel linear prioritized local algorithm tailored to address this problem on random d-regular graphs with a small and fixed degree d. Through exhaustive numerical simulations, we empirically investigated the independence ratio, i.e., the ratio between the cardinality of the independent set found and the order of the graph, which was achieved by our algorithm across random d-regular graphs with degree d ranging from 5 to 100. Remarkably, for every d within this range, our results surpassed the existing lower bounds determined by theoretical methods. Consequently, our findings suggest new conjectured lower bounds for the MIS problem on such graph structures. This finding has been obtained using a prioritized local algorithm. This algorithm is termed ‘prioritized’ because it strategically assigns priority in vertex selection, thereby iteratively adding them to the independent set.

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Optimal Low-Degree Hardness of Maximum Independent Set

Cited by 3 publications

References 15 publications

Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses

Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses

The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials

Large Independent Sets on Random d-Regular Graphs with Fixed Degree d

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