On the discrepancy of random low degree set systems

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((tlogt)1/2) discrepancy bound in the regime when n ≤ m and an O(1) bound when n≫mt. In this paper, we give a tight O(√t) bound for the entire range of n and m, under a mild assumption that t=Ω(log log m)2. The result is based on two steps. First, applying the partial coloring method to the case when n=mlogO(1)m and using the properties of the random set system we show that the overa

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((tlogt)1/2) discrepancy bound in the regime when n ≤ m and an O(1) bound when n≫mt. In this paper, we give a tight O(√t) bound for the entire range of n and m, under a mild assumption that t=Ω(log log m)2. The result is based on two steps. First, applying the partial coloring method to the case when n=mlogO(1)m and using the properties of the random set system we show that the overall discrepancy incurred is at most O(√t). Second, we reduce the general case to that of n≤mlogO(1)

A Unified Approach to Discrepancy Minimization

We study a unified approach and algorithm for constructive discrepancy minimization based on a stochastic process. By varying the parameters of the process, one can recover various state-of-the-art results. We demonstrate the flexibility of the method by deriving a discrepancy bound for smoothed instances, which interpolates between known bounds for worst-case and random instances

Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of $T$ unit vectors in $\mathbb{R}^d$, find $\pm$ signs for each of them such that the signed sum vector along any prefix has a small $\ell_\infty$-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Koml\'os problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes. Banaszczyk gave an $O(\sqrt{\log d+ \log T})$ bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk's bound and consider natural generalizations of prefix discrepancy: We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in $T$ compared to Banaszczyk's bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of $O(\sqrt{\log d+ \log\!\log T})$ with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk's bound in the worst case. We also introduce a generalization of the prefix discrepancy problem where the discrepancy constraints correspond to paths on a DAG on $T$ vertices. We show that an analog of Banaszczyk's $O(\sqrt{\log d+ \log T})$ bound continues to hold in this setting for adversarially given unit vectors and that the $\sqrt{\log T}$ factor is unavoidable for DAGs. We also show that the dependence on $T$ cannot be improved significantly in the smoothed case for DAGs. We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.Comment: 22 pages. Appear in ITCS 202

MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems

Let $V$ be a set of $n$ vertices, ${\cal M}$ a set of $m$ labels, and let $\mathbf{R}$ be an $m \times n$ matrix of independent Bernoulli random variables with success probability $p$. A random instance $G(V,E,\mathbf{R}^T\mathbf{R})$ of the weighted random intersection graph model is constructed by drawing an edge with weight $[\mathbf{R}^T\mathbf{R}]_{v,u}$ between any two vertices $u,v$ for which this weight is larger than 0. In this paper we study the average case analysis of Weighted Max Cut, assuming the input is a weighted random intersection graph, i.e. given $G(V,E,\mathbf{R}^T\mathbf{R})$ we wish to find a partition of $V$ into two sets so that the total weight of the edges having one endpoint in each set is maximized. We initially prove concentration of the weight of a maximum cut of $G(V,E,\mathbf{R}^T\mathbf{R})$ around its expected value, and then show that, when the number of labels is much smaller than the number of vertices, a random partition of the vertices achieves asymptotically optimal cut weight with high probability (whp). Furthermore, in the case $n=m$ and constant average degree, we show that whp, a majority type algorithm outputs a cut with weight that is larger than the weight of a random cut by a multiplicative constant strictly larger than 1. Then, we highlight a connection between the computational problem of finding a weighted maximum cut in $G(V,E,\mathbf{R}^T\mathbf{R})$ and the problem of finding a 2-coloring with minimum discrepancy for a set system $\Sigma$ with incidence matrix $\mathbf{R}$. We exploit this connection by proposing a (weak) bipartization algorithm for the case $m=n, p=\frac{\Theta(1)}{n}$ that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in $\Sigma$. Finally, we prove that, whp this 2-coloring corresponds to a bipartition with maximum cut-weight in $G(V,E,\mathbf{R}^T\mathbf{R})$.Comment: 18 page

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((t log t)1/2) discrepancy bound in the regime when n ≤ m and an O(1) bound when n ≫ mt. In this paper, we give a tight bound for the entire range of n and m, under a mild assumption that t = Ω(log log m)2. The result is based on two steps. First, applying the partial coloring method to the case when n = m logO(1) m and using the properties of the random set system we show that the overall discrepancy incurred is at most . Second, we reduce the general case to that of n ≤ m logO(1) m using LP duality and a careful counting argument

On the discrepancy of random low degree set systems

\u3cp\u3eMotivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((tlog t)\u3csup\u3e1\u3c/sup\u3e/\u3csup\u3e2\u3c/sup\u3e) discrepancy bound in the regime when n ≤ m and an O(1) bound when n m\u3csup\u3et\u3c/sup\u3e. In this paper, we give a tight O(t) bound for the entire range of n and m, under a mild assumption that t = Ω(log log m)\u3csup\u3e2\u3c/sup\u3e. The result is based on two steps. First, applying the partial coloring method to the case when n = mlog\u3csup\u3eO\u3c/sup\u3e\u3csup\u3e(1)\u3c/sup\u3em and using the properties of the random set system we show that the overall discrepancy incurred is at most O(t). Second, we reduce the general case to that of n ≤ mlog\u3csup\u3eO\u3c/sup\u3e\u3csup\u3e(1)\u3c/sup\u3em using LP duality and a careful counting argument.\u3c/p\u3

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((tlog t)1/2) discrepancy bound in the regime when n ≤ m and an O(1) bound when n mt. In this paper, we give a tight O(t) bound for the entire range of n and m, under a mild assumption that t = Ω(log log m)2. The result is based on two steps. First, applying the partial coloring method to the case when n = mlogO(1)m and using the properties of the random set system we show that the overall discrepancy incurred is at most O(t). Second, we reduce the general case to that of n ≤ mlogO(1)m using LP duality and a careful counting argument

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((tlog t)1/2) discrepancy bound in the regime when n ≤ m and an O(1) bound when n mt. In this paper, we give a tight O(t) bound for the entire range of n and m, under a mild assumption that t = Ω(log log m)2. The result is based on two steps. First, applying the partial coloring method to the case when n = mlogO(1)m and using the properties of the random set system we show that the overall discrepancy incurred is at most O(t). Second, we reduce the general case to that of n ≤ mlogO(1)m using LP duality and a careful counting argument.</p