Approximating Semi-Matchings in Streaming and in Two-Party Communication

We study the communication complexity and streaming complexity of approximating unweighted semi-matchings. A semi-matching in a bipartite graph G = (A, B, E), with n = |A|, is a subset of edges S that matches all A vertices to B vertices with the goal usually being to do this as fairly as possible. While the term 'semi-matching' was coined in 2003 by Harvey et al. [WADS 2003], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 <= \epsilon <= 1 uses space O(n^{1+\epsilon}) and computes an O(n^{(1-\epsilon)/2})-approximation to the semi-matching problem. Furthermore, with O(log n) passes it is possible to compute an O(log n)-approximation with space O(n). In the one-way two-party communication setting, we show that for every \epsilon > 0, deterministic communication protocols for computing an O(n^{1/((1+\epsilon)c + 1)})-approximation require a message of size more than cn bits. We present two deterministic protocols communicating n and 2n edges that compute an O(sqrt(n)) and an O(n^{1/3})-approximation respectively. Finally, we improve on results of Harvey et al. [Journal of Algorithms 2006] and prove new links between semi-matchings and matchings. While it was known that an optimal semi-matching contains a maximum matching, we show that there is a hierarchical decomposition of an optimal semi-matching into maximum matchings. A similar result holds for semi-matchings that do not admit length-two degree-minimizing paths.Comment: This is the long version including all proves of the ICALP 2013 pape

Approximating Semi-matchings in Streaming and in Two-Party Communication

We study the streaming complexity and communication complexity of approximating unweighted semi-matchings. A semi-matching in a bipartite graph G = ( A , B , E ) with n = | A | is a subset of edges S ⊆ E that matches all A vertices to B vertices with the goal usually being to do this as fairly as possible. While the term semi-matching was coined in 2003 by Harvey et al. [2003], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 ⩽ ϵ ⩽ 1 uses space Õ( n 1+ϵ and computes an O( n (1−ϵ)/2 )-approximation to the semi-matching problem. Furthermore, with O(log n ) passes it is possible to compute an O(log n )-approximation with space Õ( n ). In the one-way two-party communication setting, we show that for every ϵ &gt; 0, deterministic communication protocols for computing an O( n 1/(1+ϵ) c +1) -approximation require a message of size more than cn bits. We present two deterministic protocols communicating n and 2 n edges that compute an O√ n and an O(n 1/3 )-approximation, respectively. Finally, we improve on the results of Harvey et al. [2003] and prove new links between semi-matchings and matchings. While it was known that an optimal semi-matching contains a maximum matching, we show that there is a hierarchical decomposition of an optimal semi-matching into maximum matchings. A similar result holds for semi-matchings that do not admit length-two degree-minimizing paths.</jats:p

Random Sources in Private Computation

We consider multi-party information-theoretic private computation. Such computation inherently requires the use of local randomness by the parties, and the question of minimizing the total number of random bits used for given private computations has received considerable attention in the literature. In this work we are interested in another question: given a private computation, we ask how many of the players need to have access to a random source, and how many of them can be deterministic parties. We are further interested in the possible interplay between the number of random sources in the system and the total number of random bits necessary for the computation. We give a number of results. We first show that, perhaps surprisingly, $t$ players (rather than $t+1$) with access to a random source are sufficient for the information-theoretic $t$-private computation of any deterministic functionality over $n$ players for any $t<n/2$; by a result of (Kushilevitz and Mansour, PODC\u2796), this is best possible. This means that, counter intuitively, while private computation is impossible without randomness, it is possible to have a private computation even when the adversary can control all parties who can toss coins (and therefore sees all random coins). For randomized functionalities we show that $t+1$ random sources are necessary (and sufficient). We then turn to the question of the possible interplay between the number of random sources and the necessary number of random bits. Since for only very few settings in private computation meaningful bounds on the number of necessary random bits are known, we consider the AND function, for which some such bounds are known. We give a new protocol to $1$-privately compute the $n$-player AND function, which uses a single random source and $6$ random bits tossed by that source. This improves, upon the currently best known results (Kushilevitz et al., TCC\u2719), at the same time the number of sources and the number of random bits (KOPRT19 gives a $2$-source, $8$-bits protocol). This result gives maybe some evidence that for $1$-privacy, using the minimum necessary number of sources one can also achieve the necessary minimum number of random bits. We believe however that our protocol is of independent interest for the study of randomness in private computation

Reordering buffer management with advice

In the reordering buffer management problem, a sequence of colored items arrives at a service station to be processed. Each color change between two consecutively processed items generates some cost. A reordering buffer of capacity k items can be used to preprocess the input sequence in order to decrease the number of color changes. The goal is to find a scheduling strategy that, using the reordering buffer, minimizes the number of color changes in the given sequence of items. We consider the problem in the setting of online computation with advice. In this model, the color of an item becomes known only at the time when the item enters the reordering buffer. Additionally, together with each item entering the buffer, we get a fixed number of advice bits, which can be seen as information about the future or as information about an optimal solution (or an approximation thereof) for the whole input sequence. We show that for any ε>0 there is a (1+ε)-competitive algorithm for the problem which uses only a constant (depending on ε) number of advice bits per input item. This also immediately implies a (1+ε)-approximation algorithm which has 2O(nlog1/ε) running time (this should be compared to the trivial optimal algorithm which has a running time of kO(n)). We complement the above result by presenting a lower bound of Ω(logk) bits of advice per request for any 1-competitive algorithm