arXiv subjects

Uriel Feige

Publications and source records attributed to Uriel Feige.

At least 55 records · Page 3Linked to original sources

arXiv2018

Tighter bounds for online bipartite matching

We study the online bipartite matching problem, introduced by Karp, Vazirani and Vazirani [1990]. For bipartite graphs with matchings of size $n$, it is known that the Ranking randomized algorithm matches at least $(1 - \frac{1}{e})n$ edges in expectation. It is also known that no online algorithm matches more than $(1 - \frac{1}{e})n + O(1)$ edges in expectation, when the input is chosen from a certain distribution that we refer to as $D_n$. This upper bound also applies to fractional matchings. We review the known proofs for this last statement. In passing we observe that the $O(1)$ additive term (in the upper bound for fractional matching) is $\frac{1}{2} - \frac{1}{2e} + O(\frac{1}{n})$, and that this term is tight: the online algorithm known as Balance indeed produces a fractional matching of this size. We provide a new proof that exactly characterizes the expected cardinality of the (integral) matching produced by Ranking when the input graph comes from the support of $D_n$. This expectation turns out to be $(1 - \frac{1}{e})n + 1 - \frac{2}{e} + O(\frac{1}{n!})$, and serves as an upper bound on the performance ratio of any online (integral) matching algorithm.

cs.DS↗

arXiv2018

Finding cliques using few probes

Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an $n$ vertex graph, and need to output a clique. We show that if the input graph is drawn at random from $G_{n,\frac{1}{2}}$ (and hence is likely to have a clique of size roughly $2\log n$), then for every $δ< 2$ and constant $\ell$, there is an $α< 2$ (that may depend on $δ$ and $\ell$) such that no algorithm that makes $n^δ$ probes in $\ell$ rounds is likely (over the choice of the random graph) to output a clique of size larger than $α\log n$.

math.CO↗

arXiv2018

A Polynomial Time Constant Approximation For Minimizing Total Weighted Flow-time

We consider the classic scheduling problem of minimizing the total weighted flow-time on a single machine (min-WPFT), when preemption is allowed. In this problem, we are given a set of $n$ jobs, each job having a release time $r_j$, a processing time $p_j$, and a weight $w_j$. The flow-time of a job is defined as the amount of time the job spends in the system before it completes; that is, $F_j = C_j - r_j$, where $C_j$ is the completion time of job. The objective is to minimize the total weighted flow-time of jobs. This NP-hard problem has been studied quite extensively for decades. In a recent breakthrough, Batra, Garg, and Kumar presented a {\em pseudo-polynomial} time algorithm that has an $O(1)$ approximation ratio. The design of a truly polynomial time algorithm, however, remained an open problem. In this paper, we show a transformation from pseudo-polynomial time algorithms to polynomial time algorithms in the context of min-WPFT. Our result combined with the result of Batra, Garg, and Kumar settles the long standing conjecture that there is a polynomial time algorithm with $O(1)$-approximation for min-WPFT.

cs.DS↗

arXiv2018

Approximate Modularity Revisited

Set functions with convenient properties (such as submodularity) appear in application areas of current interest, such as algorithmic game theory, and allow for improved optimization algorithms. It is natural to ask (e.g., in the context of data driven optimization) how robust such properties are, and whether small deviations from them can be tolerated. We consider two such questions in the important special case of linear set functions. One question that we address is whether any set function that approximately satisfies the modularity equation (linear functions satisfy the modularity equation exactly) is close to a linear function. The answer to this is positive (in a precise formal sense) as shown by Kalton and Roberts [1983] (and further improved by Bondarenko, Prymak, and Radchenko [2013]). We revisit their proof idea that is based on expander graphs, and provide significantly stronger upper bounds by combining it with new techniques. Furthermore, we provide improved lower bounds for this problem. Another question that we address is that of how to learn a linear function $h$ that is close to an approximately linear function $f$, while querying the value of $f$ on only a small number of sets. We present a deterministic algorithm that makes only linearly many (in the number of items) nonadaptive queries, by this improving over a previous algorithm of Chierichetti, Das, Dasgupta and Kumar [2015] that is randomized and makes more than a quadratic number of queries. Our learning algorithm is based on a Hadamard transform.

cs.DS↗

arXiv2018

Max-Min Greedy Matching

A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $π$ over $V$, the other player imposes a permutation $σ$ over $U$. In the greedy matching algorithm, vertices of $U$ arrive in order $σ$ and each vertex is matched to the lowest (under $π$) yet unmatched neighbor in $V$ (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals $π$, and the second (min) player responds with the worst possible $σ$ for $π$, does there exist a permutation $π$ ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time? The main result of this paper is an affirmative answer for this question: we show that there exists a polytime algorithm to compute $π$ for which for every $σ$ at least $ρ> 0.51$ fraction of the vertices of $V$ are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian. Interestingly, even for regular graphs with arbitrarily large degree (implying a large number of disjoint perfect matchings), there is no $π$ ensuring to match more than a fraction $8/9$ of the vertices. The max-min greedy matching problem solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations. In addition, it has implications for the size of the unique stable matching in markets with global preferences, subject to the graph structure.

cs.GT↗

arXiv2017

On the Probe Complexity of Local Computation Algorithms

The Local Computation Algorithms (LCA) model is a computational model aimed at problem instances with huge inputs and output. For graph problems, the input graph is accessed using probes: strong probes (SP) specify a vertex $v$ and receive as a reply a list of $v$'s neighbors, and weak probes (WP) specify a vertex $v$ and a port number $i$ and receive as a reply $v$'s $i^{th}$ neighbor. Given a local query (e.g., "is a certain vertex in the vertex cover of the input graph?"), an LCA should compute the corresponding local output (e.g., "yes" or "no") while making only a small number of probes, with the requirement that all local outputs form a single global solution (e.g., a legal vertex cover). We study the probe complexity of LCAs that are required to work on graphs that may have arbitrarily large degrees. In particular, such LCAs are expected to probe the graph a number of times that is significantly smaller than the maximum, average, or even minimum degree. For weak probes, we focus on the weak coloring problem. Among our results we show a separation between weak 3-coloring and weak 2-coloring for deterministic LCAs: $\log^* n + O(1)$ weak probes suffice for weak 3-coloring, but $Ω\left(\frac{\log n}{\log\log n}\right)$ weak probes are required for weak 2-coloring. For strong probes, we consider randomized LCAs for vertex cover and maximal/maximum matching. Our negative results include showing that there are graphs for which finding a \emph{maximal} matching requires $Ω(\sqrt{n})$ strong probes. On the positive side, we design a randomized LCA that finds a $(1-ε)$ approximation to \emph{maximum} matching in regular graphs, and uses $\frac{1}{ε}^{O\left( \frac{1}{ε^2}\right)}$ probes, independently of the number of vertices and of their degrees.

cs.DS↗

arXiv2016

Random walks with the minimum degree local rule have $O(n^2)$ cover time

For a simple (unbiased) random walk on a connected graph with $n$ vertices, the cover time (the expected number of steps it takes to visit all vertices) is at most $O(n^3)$. We consider locally biased random walks, in which the probability of traversing an edge depends on the degrees of its endpoints. We confirm a conjecture of Abdullah, Cooper and Draief [2015] that the min-degree local bias rule ensures a cover time of $O(n^2)$. For this we formulate and prove the following lemma about spanning trees. Let $R(e)$ denote for edge $e$ the minimum degree among its two endpoints. We say that a weight function $W$ for the edges is feasible if it is nonnegative, dominated by $R$ (for every edge $W(e) \le R(e)$) and the sum over all edges of the ratios $W(e)/R(e)$ equals $n-1$. For example, in trees $W(e) = R(e)$, and in regular graphs the sum of edge weights is $d(n-1)$. {\bf Lemma:} for every feasible $W$, the minimum weight spanning tree has total weight $O(n)$. For regular graphs, a similar lemma was proved by Kahn, Linial, Nisan and Saks [1989].

math.PR↗

arXiv2016

Shotgun Assembly of Random Jigsaw Puzzles

In a recent work, Mossel and Ross considered the shotgun assembly problem for a random jigsaw puzzle. Their model consists of a puzzle - an $n\times n$ grid, where each vertex is viewed as a center of a piece. They assume that each of the four edges adjacent to a vertex, is assigned one of $q$ colors (corresponding to "jigs", or cut shapes) uniformly at random. Mossel and Ross asked: how large should $q = q(n)$ be so that with high probability the puzzle can be assembled uniquely given the collection of individual tiles? They showed that if $q = ω(n^2)$, then the puzzle can be assembled uniquely with high probability, while if $q = o(n^{2/3})$, then with high probability the puzzle cannot be uniquely assembled. Here we improve the upper bound and show that for any $\eps > 0$, the puzzle can be assembled uniquely with high probability if $q \geq n^{1+\eps}$. The proof uses an algorithm of $n^{Θ(1/\eps)}$ running time.

math.CO↗

arXiv2016

On the effect of randomness on planted 3-coloring models

We present the hosted coloring framework for studying algorithmic and hardness results for the $k$-coloring problem. There is a class ${\cal H}$ of host graphs. One selects a graph $H\in{\cal H}$ and plants in it a balanced $k$-coloring (by partitioning the vertex set into $k$ roughly equal parts, and removing all edges within each part). The resulting graph $G$ is given as input to a polynomial time algorithm that needs to $k$-color $G$ (any legal $k$-coloring would do -- the algorithm is not required to recover the planted $k$-coloring). Earlier planted models correspond to the case that ${\cal H}$ is the class of all $n$-vertex $d$-regular graphs, a member $H\in{\cal H}$ is chosen at random, and then a balanced $k$-coloring is planted at random. Blum and Spencer [1995] designed algorithms for this model when $d=n^δ$ (for $0<δ\le1$), and Alon and Kahale [1997] managed to do so even when $d$ is a sufficiently large constant. The new aspect in our framework is that it need not involve randomness. In one model within the framework (with $k=3$) $H$ is a $d$ regular spectral expander (meaning that except for the largest eigenvalue of its adjacency matrix, every other eigenvalue has absolute value much smaller than $d$) chosen by an adversary, and the planted 3-coloring is random. We show that the 3-coloring algorithm of Alon and Kahale [1997] can be modified to apply to this case. In another model $H$ is a random $d$-regular graph but the planted balanced $3$-coloring is chosen by an adversary, after seeing $H$. We show that for a certain range of average degrees somewhat below $\sqrt{n}$, finding a 3-coloring is NP-hard. Together these results (and other results that we have) help clarify which aspects of randomness in the planted coloring model are the key to successful 3-coloring algorithms.

cs.CC↗

arXiv2016

Contagious Sets in Random Graphs

math.PR↗

arXiv2014

NP-hardness of hypercube 2-segmentation

The hypercube 2-segmentation problem is a certain biclustering problem that was previously claimed to be NP-hard, but for which there does not appear to be a publicly available proof of NP-hardness. This manuscript provides such a proof.

cs.CC↗

arXiv2014

A Unifying Hierarchy of Valuations with Complements and Substitutes

We introduce a new hierarchy over monotone set functions, that we refer to as $\mathcal{MPH}$ (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, $\mathcal{MPH}$-$m$ (where $m$ is the total number of items) captures all monotone functions. The lowest level, $\mathcal{MPH}$-$1$, captures all monotone submodular functions, and more generally, the class of functions known as $\mathcal{XOS}$. Every monotone function that has a positive hypergraph representation of rank $k$ (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in $\mathcal{MPH}$-$k$. Every monotone function that has supermodular degree $k$ (in the sense defined by Feige and Izsak [ITCS 2013]) is in $\mathcal{MPH}$-$(k+1)$. In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of $\mathcal{MPH}$-$k$. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the $\mathcal{MPH}$ hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of $k+1$ if all players hold valuation functions in $\mathcal{MPH}$-$k$. The other is an upper bound of $2k$ on the price of anarchy of simultaneous first price auctions. Being in $\mathcal{MPH}$-$k$ can be shown to involve two requirements -- one is monotonicity and the other is a certain requirement that we refer to as $\mathcal{PLE}$ (Positive Lower Envelope). Removing the monotonicity requirement, one obtains the $\mathcal{PLE}$ hierarchy over all non-negative set functions (whether monotone or not), which can be fertile ground for further research.

cs.GT↗

arXiv2014

Chasing Ghosts: Competing with Stateful Policies

We consider sequential decision making in a setting where regret is measured with respect to a set of stateful reference policies, and feedback is limited to observing the rewards of the actions performed (the so called "bandit" setting). If either the reference policies are stateless rather than stateful, or the feedback includes the rewards of all actions (the so called "expert" setting), previous work shows that the optimal regret grows like $Θ(\sqrt{T})$ in terms of the number of decision rounds $T$. The difficulty in our setting is that the decision maker unavoidably loses track of the internal states of the reference policies, and thus cannot reliably attribute rewards observed in a certain round to any of the reference policies. In fact, in this setting it is impossible for the algorithm to estimate which policy gives the highest (or even approximately highest) total reward. Nevertheless, we design an algorithm that achieves expected regret that is sublinear in $T$, of the form $O( T/\log^{1/4}{T})$. Our algorithm is based on a certain local repetition lemma that may be of independent interest. We also show that no algorithm can guarantee expected regret better than $O( T/\log^{3/2} T)$.

cs.LG↗

arXiv2014

Why are images smooth?

It is a well observed phenomenon that natural images are smooth, in the sense that nearby pixels tend to have similar values. We describe a mathematical model of images that makes no assumptions on the nature of the environment that images depict. It only assumes that images can be taken at different scales (zoom levels). We provide quantitative bounds on the smoothness of a typical image in our model, as a function of the number of available scales. These bounds can serve as a baseline against which to compare the observed smoothness of natural images.

cs.CV↗

arXiv2014

Contagious Sets in Expanders

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors, where $r>1$ is the activation threshold. A \emph{contagious set} is a set whose activation results with the entire graph being active. Given a graph $G$, let $m(G,r)$ be the minimal size of a contagious set. Computing $m(G,r)$ is NP-hard. It is known that for every $d$-regular or nearly $d$-regular graph on $n$ vertices, $m(G,r) \le O(\frac{nr}{d})$. We consider such graphs that additionally have expansion properties, parameterized by the spectral gap and/or the girth of the graphs. The general flavor of our results is that sufficiently strong expansion (e.g., $λ(G)=O(\sqrt{d})$, or girth $Ω(\log \log d)$) implies that $m(G,2) \le O(\frac{n}{d^2})$ (and more generally, $m(G,r) \le O(\frac{n}{d^{r/(r-1)}})$). Significantly weaker expansion properties suffice in order to imply that $m(G,2)\le O(\frac{n \log d}{d^2})$. For example, we show this for graphs of girth at least~7, and for graphs with $λ(G)<(1-ε)d$, provided the graph has no 4-cycles. Nearly $d$-regular expander graphs can be obtained by considering the binomial random graph $G(n,p)$ with $p \simeq \frac{d}{n}$ and $d > \log n$. For such graphs we prove that $Ω(\frac{n}{d^2 \log d}) \le m(G,2) \le O(\frac{n\log\log d}{d^2\log d})$ almost surely. Our results are algorithmic, entailing simple and efficient algorithms for selecting contagious sets.

cs.DM↗

arXiv2014

On giant components and treewidth in the layers model

Given an undirected $n$-vertex graph $G(V,E)$ and an integer $k$, let $T_k(G)$ denote the random vertex induced subgraph of $G$ generated by ordering $V$ according to a random permutation $π$ and including in $T_k(G)$ those vertices with at most $k-1$ of their neighbors preceding them in this order. The distribution of subgraphs sampled in this manner is called the \emph{layers model with parameter} $k$. The layers model has found applications in studying $\ell$-degenerate subgraphs, the design of algorithms for the maximum independent set problem, and in bootstrap percolation. In the current work we expand the study of structural properties of the layers model. We prove that there are $3$-regular graphs $G$ for which with high probability $T_3(G)$ has a connected component of size $Ω(n)$. Moreover, this connected component has treewidth $Ω(n)$. This lower bound on the treewidth extends to many other random graph models. In contrast, $T_2(G)$ is known to be a forest (hence of treewidth~1), and we establish that if $G$ is of bounded degree then with high probability the largest connected component in $T_2(G)$ is of size $O(\log n)$. We also consider the infinite two-dimensional grid, for which we prove that the first four layers contain a unique infinite connected component with probability $1$.

cs.DM↗

arXiv2014

On robustly asymmetric graphs

O'Donnell, Wright, Wu and Zhou [SODA 2014] introduced the notion of robustly asymmetric graphs. Roughly speaking, these are graphs in which for every $0 \le ρ\le 1$, every permutation that permutes a $ρ$ fraction of the vertices maps a $Θ(ρ)$ fraction of the edges to non-edges. We show that there are graphs for which the constant hidden in the $Θ$ notation is roughly~1.

cs.DM↗

arXiv2012

Musical chairs

In the {\em Musical Chairs} game $MC(n,m)$ a team of $n$ players plays against an adversarial {\em scheduler}. The scheduler wins if the game proceeds indefinitely, while termination after a finite number of rounds is declared a win of the team. At each round of the game each player {\em occupies} one of the $m$ available {\em chairs}. Termination (and a win of the team) is declared as soon as each player occupies a unique chair. Two players that simultaneously occupy the same chair are said to be {\em in conflict}. In other words, termination (and a win for the team) is reached as soon as there are no conflicts. The only means of communication throughout the game is this: At every round of the game, the scheduler selects an arbitrary nonempty set of players who are currently in conflict, and notifies each of them separately that it must move. A player who is thus notified changes its chair according to its deterministic program. As we show, for $m\ge 2n-1$ chairs the team has a winning strategy. Moreover, using topological arguments we show that this bound is tight. For $m\leq 2n-2$ the scheduler has a strategy that is guaranteed to make the game continue indefinitely and thus win. We also have some results on additional interesting questions. For example, if $m \ge 2n-1$ (so that the team can win), how quickly can they achieve victory?

math.CO↗