arXiv subjects

Gregory B. Sorkin

Publications and source records attributed to Gregory B. Sorkin.

At least 19 recordsLinked to original sources

arXiv2025

Reverse Littlewood--Offord problems with parity conditions

We consider the probability that the random signed sum $ξ_1 v_1 + \dotsb + ξ_n v_n$ lies within a given distance $r$ of the origin, where $v_1,\dotsc,v_n \in \mathbb{R}^d$ are fixed unit vectors and $ξ_1,\dotsc,ξ_n$ are independently and uniformly distributed on $\{-1,+1\}$. In particular, our results demonstrate that, for certain values of $r$, the infimum of this probability is very sensitive to the parity of $n$. We prove that, for any $d\geq 3$, there is some $\varepsilon = \varepsilon(d) > 0$ such that for any $n \not\equiv d \mod 2$ and unit vectors $v_1,\dotsc,v_n\in \mathbb{R}^d$, there are signs $η_1,\dotsc,η_n \in \{-1,+1\}$ such that $\|\sum_{i=1}^n η_i v_i\| \leq \sqrt{d - \varepsilon}$, and so $\mathbb{P}(\| ξ_1 v_1 + \dotsb + ξ_n v_n \| \leq \sqrt{d-\varepsilon}) > 0$. This is in contrast to the case of $n\equiv d \mod 2$, wherein the above probability can be zero. More is known if $d=2$ and $n$ is odd, and in this case we present a construction demonstrating that $\mathbb{P}(\|ξ_1 v_1 + \dotsb + ξ_n v_n\| \leq 1)$ can decay exponentially as $n$ increases.

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arXiv2025

Optimally building spanning graphs in semirandom graph processes

The semirandom graph process constructs a graph $G$ in a series of rounds, starting with the empty graph on $n$ vertices. In each round, a player is offered a vertex $v$ chosen uniformly at random, and chooses an edge on $v$ to add to $G$. The player's aim is to make $G$ satisfy some property as quickly as possible. Our interest is in the property that $G$ contain a given $n$-vertex graph $H$ with maximum degree $Δ$. In 2021, Ben-Eliezer, Gishboliner, Hefetz and Krivelevich showed that there is a semirandom strategy that achieves this, with probability tending to 1 as $n$ tends to infinity, in $(1 + o_Δ(1)) \frac{3 Δn}{2}$ rounds, where $o_Δ(1)$ is a function that tends to $0$ as $Δ$ tends to infinity. We improve this to $(1 + o_Δ(1)) \frac{Δn}{2}$, which can be seen to be asymptotically optimal in $Δ$. We show the same result for a variant of the semirandom graph process, namely the semirandom tree process introduced by Burova and Lichev, where in each round the player is offered the edge set of a uniformly chosen tree on $n$ vertices, and chooses one edge to keep.

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arXiv2025

Belief Propagation Guided Decimation on Random k-XORSAT

We analyse the performance of Belief Propagation Guided Decimation, a physics-inspired message passing algorithm, on the random $k$-XORSAT problem. Specifically, we derive an explicit threshold up to which the algorithm succeeds with a strictly positive probability $Ω(1)$ that we compute explicitly, but beyond which the algorithm with high probability fails to find a satisfying assignment. In addition, we analyse a thought experiment called the decimation process for which we identify a (non-) reconstruction and a condensation phase transition. The main results of the present work confirm physics predictions from [RTS: J. Stat. Mech. 2009] that link the phase transitions of the decimation process with the performance of the algorithm, and improve over partial results from a recent article [Yung: Proc. ICALP 2024].

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arXiv2024

Perfect matchings and loose Hamilton cycles in the semirandom hypergraph model

We study the 2-offer semirandom 3-uniform hypergraph model on $n$ vertices. At each step, we are presented with 2 uniformly random vertices. We choose any other vertex, thus creating a hyperedge of size 3. We show a strategy that constructs a perfect matching, and another that constructs a loose Hamilton cycle, both succeeding asymptotically almost surely within $Θ(n)$ steps. Both results extend to $s$-uniform hypergraphs. The challenges with hypergraphs, and our methods, are qualitatively different from what has been seen for semirandom graphs. Much of our analysis is done on an auxiliary graph that is a uniform $k$-out subgraph of a random bipartite graph, and this tool may be useful in other contexts.

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arXiv2022

Hamilton cycles in a semi-random graph model

We show that with high probability we can build a Hamilton cycle after at most $1.85 n$ rounds in a particular semi-random model. In this model, in one round, we are given a {uniform random} $v\in[n]$ and then we can add an {arbitrary} edge $\{v,w\}$. Our result improves on $2.016 n$ in a recent paper of Gao, MacRury, and Pralat.

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arXiv2022

Snakes and Ladders and Intransitivity, or what mathematicians do in their time off

This recreational mathematics article shows that the game of Snakes and Ladders is intransitive: square 69 has a winning edge over 79, which in turn beats 73, which beats 69. Analysis of the game is a nice illustration of Markov chains, simulations of different sorts, and size-biased sampling. Connecting this to "intransitive dice" illustrates the power of a name, and the joy of working with colleagues. When draws do not count, we show a minimal example of intransitive dice, with one die having just a single "face" and two dice each having two faces.

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arXiv2020

The Ising antiferromagnet and max cut on random regular graphs

The Ising antiferromagnet is an important statistical physics model with close connections to the {\sc Max Cut} problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the {\sc Max Cut} of random regular graphs predicted by Zdeborová and Boettcher [Journal of Statistical Mechanics 2010]. As an application we prove that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten-Stigum bound.

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arXiv2020

Successive shortest paths in complete graphs with random edge weights

Consider a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$. The weight of the shortest (minimum-weight) path $P_1$ between two given vertices is known to be $\ln n / n$, asymptotically. Define a second-shortest path $P_2$ to be the shortest path edge-disjoint from $P_1$, and consider more generally the shortest path $P_k$ edge-disjoint from all earlier paths. We show that the cost $X_k$ of $P_k$ converges in probability to $2k/n+\ln n/n$ uniformly for all $k \leq n-1$. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterise the collectively cheapest $k$ edge-disjoint paths, i.e., a minimum-cost $k$-flow. We also obtain the expectation of $X_k$ conditioned on the existence of $P_k$.

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arXiv2019

Successive minimum spanning trees

In a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$ (or alternatively an exponential distribution $\operatorname{Exp}(1)$), let $T_1$ be the MST (the spanning tree of minimum weight) and let $T_k$ be the MST after deletion of the edges of all previous trees $T_i$, $i<k$. We show that each tree's weight $w(T_k)$ converges in probability to a constant $γ_k$ with $2k-2\sqrt k <γ_k<2k+2\sqrt k$, and we conjecture that $γ_k = 2k-1+o(1)$. The problem is distinct from that of Frieze and Johansson (2018), finding $k$ MSTs of combined minimum weight, and for $k=2$ ours has strictly larger cost. Our results also hold (and mostly are derived) in a multigraph model where edge weights for each vertex pair follow a Poisson process; here we additionally have $\mathbb E(w(T_k)) \to γ_k$. Thinking of an edge of weight $w$ as arriving at time $t=n w$, Kruskal's algorithm defines forests $F_k(t)$, each initially empty and eventually equal to $T_k$, with each arriving edge added to the first $F_k(t)$ where it does not create a cycle. Using tools of inhomogeneous random graphs we obtain structural results including that $C_1(F_k(t))/n$, the fraction of vertices in the largest component of $F_k(t)$, converges in probability to a function $ρ_k(t)$, uniformly for all $t$, and that a giant component appears in $F_k(t)$ at a time $t=σ_k$. We conjecture that the functions $ρ_k$ tend to time translations of a single function, $ρ_k(2k+x)\toρ_\infty(x)$ as $k \to \infty$, uniformly in $x\in \mathbb R$. Simulations and numerical computations give estimated values of $γ_k$ for small $k$, and support the conjectures just stated.

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arXiv2017

Separate, Measure and Conquer: Faster Algorithms for Max 2-CSP and Counting Dominating Sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. An instance of the problem Max (r,2)-CSP, or simply Max 2-CSP, is parametrized by the domain size r (often 2), the number of variables n (vertices in the constraint graph G), and the number of constraints m (edges in G). When G is cubic, and omitting sub-exponential terms here for clarity, we give an algorithm running in time r^((1/5)n) = r^((2/15)m); the previous best was r^((1/4)n) = r^((1/6)m). By known results, this improvement for the cubic case results in an algorithm running in time r^((9/50)m) for general instances; the previous best was r^((19/100)m). We show that the analysis of the earlier algorithm was tight: our improvement is in the algorithm, not just the analysis. The new algorithm, like the old, extends to Polynomial and Ring CSP. We also give faster algorithms for #Dominating Set, counting the dominating sets of every cardinality 0,...,n for a graph G of order n. For cubic graphs, our algorithm runs in time 3^((1/6)n); the previous best was 2^((1/2)n). For general graphs, we give an unrelated algorithm running in time 1.5183^n; the previous best was 1.5673^n. The previous best algorithms for these problems all used local transformations and were analyzed by the "Measure and Conquer" method. Our new algorithms capitalize on the existence of small balanced separators for cubic graphs - a non-local property - and the ability to tailor the local algorithms always to "pivot" on a vertex in the separator. The new algorithms perform much as the old ones until the separator is empty, at which point they gain because the remaining vertices are split into two independent problem instances that can be solved recursively. It is likely that such algorithms can be effective for other problems too, and we present their design and analysis in a general framework.

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arXiv2014

The Satisfiability Threshold for k-XORSAT

We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \geq 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiability of constrained $k$-XORSAT for every $k\ge 3$, and we use standard results on the 2-core of a random $k$-uniform hypergraph to extend this result to find the threshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrow the phase transition window, showing that $m-n \to -\infty$ implies almost-sure satisfiability, while $m-n \to +\infty$ implies almost-sure unsatisfiability.

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arXiv2013

VCG Auction Mechanism Cost Expectations and Variances

We consider Vickrey-Clarke-Groves (VCG) auctions for a very general combinatorial structure, in an average-case setting where item costs are independent, identically distributed uniform random variables. We prove that the expected VCG cost is at least double the expected nominal cost, and exactly double when the desired structure is a basis of a bridgeless matroid. In the matroid case we further show that, conditioned upon the VCG cost, the expectation of the nominal cost is exactly half the VCG cost, and we show several results on variances and covariances among the nominal cost, the VCG cost, and related quantities. As an application, we find the asymptotic variance of the VCG cost of the minimum spanning tree in a complete graph with random edge costs.

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arXiv2013

The Satisfiability Threshold for $k$-XORSAT, using an alternative proof

We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \ge 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiability of constrained $k$-XORSAT for every $k \ge 3$, and we use standard results on the 2-core of a random $k$-uniform hypergraph to extend this result to find the threshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrow the phase transition window, showing that $n-m \to \infty$ implies almost-sure satisfiability, while $m-n \to \infty$ implies almost-sure unsatisfiability.

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arXiv2010

Structure of random r-SAT below the pure literal threshold

It is well known that there is a sharp density threshold for a random $r$-SAT formula to be satisfiable, and a similar, smaller, threshold for it to be satisfied by the pure literal rule. Also, above the satisfiability threshold, where a random formula is with high probability (whp) unsatisfiable, the unsatisfiability is whp due to a large "minimal unsatisfiable subformula" (MUF). By contrast, we show that for the (rare) unsatisfiable formulae below the pure literal threshold, the unsatisfiability is whp due to a unique MUF with smallest possible "excess", failing this whp due to a unique MUF with the next larger excess, and so forth. In the same regime, we give a precise asymptotic expansion for the probability that a formula is unsatisfiable, and efficient algorithms for satisfying a formula or proving its unsatisfiability. It remains open what happens between the pure literal threshold and the satisfiability threshold. We prove analogous results for the $k$-core and $k$-colorability thresholds for a random graph, or more generally a random $r$-uniform hypergraph.

cs.DM↗

arXiv2009

A universally fastest algorithm for Max 2-Sat, Max 2-CSP, and everything in between

In this paper we introduce "hybrid" Max 2-CSP formulas consisting of "simple clauses", namely conjunctions and disjunctions of pairs of variables, and general 2-variable clauses, which can be any integer-valued functions of pairs of boolean variables. This allows an algorithm to use both efficient reductions specific to AND and OR clauses, and other powerful reductions that require the general CSP setting. We use new reductions introduced here, and recent reductions such as "clause-learning" and "2-reductions" generalized to our setting's mixture of simple and general clauses. Parametrizing an instance by the fraction p of non-simple clauses, we give an exact (exponential-time) algorithm that is the fastest known polynomial-space algorithm for p=0 (which includes the well-studied Max 2-Sat problem but also instances with arbitrary mixtures of AND and OR clauses); the only efficient algorithm for mixtures of AND, OR, and general integer-valued clauses; and tied for fastest for general Max 2-CSP (p=1). Since a pure 2-Sat input instance may be transformed to a general CSP instance in the course of being solved, the algorithm's efficiency and generality go hand in hand. Our algorithm analysis and optimization are a variation on the familiar measure-and-conquer approach, resulting in an optimizing mathematical program that is convex not merely quasi-convex, and thus can be solved efficiently and with a certificate of optimality. We produce a family of running-time upper-bound formulas, each optimized for instances with a particular value of p but valid for all instances.

cs.DS↗

arXiv2009

Polynomial Constraint Satisfaction, Graph Bisection, and the Ising Partition Function

We introduce a problem class we call Polynomial Constraint Satisfaction Problems, or PCSP. Where the usual CSPs from computer science and optimization have real-valued score functions, and partition functions from physics have monomials, PCSP has scores that are arbitrary multivariate formal polynomials, or indeed take values in an arbitrary ring. Although PCSP is much more general than CSP, remarkably, all (exact, exponential-time) algorithms we know of for 2-CSP (where each score depends on at most 2 variables) extend to 2-PCSP, at the expense of just a polynomial factor in running time. Specifically, we extend the reduction-based algorithm of Scott and Sorkin; the specialization of that approach to sparse random instances, where the algorithm runs in polynomial expected time; dynamic-programming algorithms based on tree decompositions; and the split-and-list matrix-multiplication algorithm of Williams. This gives the first polynomial-space exact algorithm more efficient than exhaustive enumeration for the well-studied problems of finding a minimum bisection of a graph, and calculating the partition function of an Ising model, and the most efficient algorithm known for certain instances of Maximum Independent Set. Furthermore, PCSP solves both optimization and counting versions of a wide range of problems, including all CSPs, and thus enables samplers including uniform sampling of optimal solutions and Gibbs sampling of all solutions.

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arXiv2008

Linear-programming design and analysis of fast algorithms for Max 2-Sat and Max 2-CSP

The class $(r,2)$-CSP, or simply Max 2-CSP, consists of constraint satisfaction problems with at most two $r$-valued variables per clause. For instances with $n$ variables and $m$ binary clauses, we present an $O(n r^{5+19m/100})$-time algorithm which is the fastest polynomial-space algorithm for many problems in the class, including Max Cut. The method also proves a treewidth bound $\tw(G) \leq (13/75+o(1))m$, which gives a faster Max 2-CSP algorithm that uses exponential space: running in time $\Ostar{2^{(13/75+o(1))m}}$, this is fastest for most problems in Max 2-CSP. Parametrizing in terms of $n$ rather than $m$, for graphs of average degree $d$ we show a simple algorithm running time $\Ostar{2^{(1-\frac{2}{d+1})n}}$, the fastest polynomial-space algorithm known. In combination with ``Polynomial CSPs'' introduced in a companion paper, these algorithms also allow (with an additional polynomial-factor overhead in space and time) counting and sampling, and the solution of problems like Max Bisection that escape the usual CSP framework. Linear programming is key to the design as well as the analysis of the algorithms.

cs.DM↗

arXiv2007

A tight bound on the collection of edges in MSTs of induced subgraphs

Let $G=(V,E)$ be a complete $n$-vertex graph with distinct positive edge weights. We prove that for $k\in\{1,2,...,n-1\}$, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of $G$ with $n-k+1$ vertices has at most $nk-\binom{k+1}{2}$ elements. This proves a conjecture of Goemans and Vondrak \cite{GV2005}. We also show that the result is a generalization of Mader's Theorem, which bounds the number of edges in any edge-minimal $k$-connected graph.

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