Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph

We study the complexity of estimating the optimum value of a Boolean 2CSP (arity two constraint satisfaction problem) in the single-pass streaming setting, where the algorithm is presented the constraints in an arbitrary order. We give a streaming algorithm to estimate the optimum within a factor approaching 2/5 using logarithmic space, with high probability. This beats the trivial factor 1/4 estimate obtained by simply outputting 1/4-th of the total number of constraints. The inspiration for our work is a lower bound of Kapralov, Khanna, and Sudan (SODA\u2715) who showed that a similar trivial estimate (of factor 1/2) is the best one can do for Max CUT. This lower bound implies that beating a factor 1/2 for Max DICUT (a special case of Max 2CSP), in particular, to distinguish between the case when the optimum is m/2 versus when it is at most (1/4+eps)m, where m is the total number of edges, requires polynomial space. We complement this hardness result by showing that for DICUT, one can distinguish between the case in which the optimum exceeds (1/2+eps)m and the case in which it is close to m/4. We also prove that estimating the size of the maximum acyclic subgraph of a directed graph, when its edges are presented in a single-pass stream, within a factor better than 7/8 requires polynomial space

Streaming Approximation Resistance of Every Ordering CSP

An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $\Pi$ mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. Given an instance of OCSP$(\Pi)$ on $n$ variables and $m$ constraints, the goal is to find an ordering of the $n$ variables that maximizes the number of constraints that are satisfied, where a constraint specifies a sequence of $k$ distinct variables and the constraint is satisfied by an ordering on the $n$ variables if the ordering induced on the $k$ variables in the constraint satisfies $\Pi$. OCSPs capture natural problems including "Maximum acyclic subgraph (MAS)" and "Betweenness". In this work we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, where an instance is presented as a stream of constraints. We show that for every $\Pi$, OCSP$(\Pi)$ is approximation-resistant to $o(n)$-space streaming algorithms. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every $\epsilon>0$, MAS is not $1/2+\epsilon$-approximable in $o(n)$ space. The previous best inapproximability result only ruled out a $3/4$-approximation in $o(\sqrt n)$ space. Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and Velusamy who show tight, linear-space inapproximability results for a broad class of (non-ordering) constraint satisfaction problems over arbitrary (finite) alphabets. We design a family of appropriate CSPs (one for every $q$) from any given OCSP, and apply their work to this family of CSPs. We show that the hard instances from this earlier work have a particular "small-set expansion" property. By exploiting this combinatorial property, in combination with the hardness results of the resulting families of CSPs, we give optimal inapproximability results for all OCSPs.Comment: 23 pages, 1 figure. Replaces earlier version with $o(\sqrt{n})$ lower bound, using new bounds from arXiv:2106.13078. To appear in APPROX'2

Fair allocation of a multiset of indivisible items

We study the problem of fairly allocating a multiset $M$ of $m$ indivisible items among $n$ agents with additive valuations. Specifically, we introduce a parameter $t$ for the number of distinct types of items and study fair allocations of multisets that contain only items of these $t$ types, under two standard notions of fairness: 1. Envy-freeness (EF): For arbitrary $n$, $t$, we show that a complete EF allocation exists when at least one agent has a unique valuation and the number of items of each type exceeds a particular finite threshold. We give explicit upper and lower bounds on this threshold in some special cases. 2. Envy-freeness up to any good (EFX): For arbitrary $n$, $m$, and for $t\le 2$, we show that a complete EFX allocation always exists. We give two different proofs of this result. One proof is constructive and runs in polynomial time; the other is geometrically inspired.Comment: 34 pages, 6 figures, 1 table, 1 algorith

Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes

We consider the bit-probe complexity of the set membership problem: represent an n-element subset S of an m-element universe as a succinct bit vector so that membership queries of the form "Is x ? S" can be answered using at most t probes into the bit vector. Let s(m,n,t) (resp. s_N(m,n,t)) denote the minimum number of bits of storage needed when the probes are adaptive (resp. non-adaptive). Lewenstein, Munro, Nicholson, and Raman (ESA 2014) obtain fully-explicit schemes that show that s(m,n,t) = ?((2^t-1)m^{1/(t - min{2?log n?, n-3/2})}) for n ? 2,t ? ?log n?+1 . In this work, we improve this bound when the probes are allowed to be superlinear in n, i.e., when t ? ?(nlog n), n ? 2, we design fully-explicit schemes that show that s(m,n,t) = ?((2^t-1)m^{1/(t-{n-1}/{2^{t/(2(n-1))}})}), asymptotically (in the exponent of m) close to the non-explicit upper bound on s(m,n,t) derived by Radhakrishan, Shah, and Shannigrahi (ESA 2010), for constant n. In the non-adaptive setting, it was shown by Garg and Radhakrishnan (STACS 2017) that for a large constant n?, for n ? n?, s_N(m,n,3) ? ?{mn}. We improve this result by showing that the same lower bound holds even for storing sets of size 2, i.e., s_N(m,2,3) ? ?(?m)

Streaming complexity of CSPs with randomly ordered constraints

We initiate a study of the streaming complexity of constraint satisfaction problems (CSPs) when the constraints arrive in a random order. We show that there exists a CSP, namely $\textsf{Max-DICUT}$, for which random ordering makes a provable difference. Whereas a $4/9 \approx 0.445$ approximation of $\textsf{DICUT}$ requires $\Omega(\sqrt{n})$ space with adversarial ordering, we show that with random ordering of constraints there exists a $0.48$-approximation algorithm that only needs $O(\log n)$ space. We also give new algorithms for $\textsf{Max-DICUT}$ in variants of the adversarial ordering setting. Specifically, we give a two-pass $O(\log n)$ space $0.48$-approximation algorithm for general graphs and a single-pass $\tilde{O}(\sqrt{n})$ space $0.48$-approximation algorithm for bounded degree graphs. On the negative side, we prove that CSPs where the satisfying assignments of the constraints support a one-wise independent distribution require $\Omega(\sqrt{n})$-space for any non-trivial approximation, even when the constraints are randomly ordered. This was previously known only for adversarially ordered constraints. Extending the results to randomly ordered constraints requires switching the hard instances from a union of random matchings to simple Erd\"os-Renyi random (hyper)graphs and extending tools that can perform Fourier analysis on such instances. The only CSP to have been considered previously with random ordering is $\textsf{Max-CUT}$ where the ordering is not known to change the approximability. Specifically it is known to be as hard to approximate with random ordering as with adversarial ordering, for $o(\sqrt{n})$ space algorithms. Our results show a richer variety of possibilities and motivate further study of CSPs with randomly ordered constraints

Streaming beyond sketching for Maximum Directed Cut

We give an $\widetilde{O}(\sqrt{n})$-space single-pass $0.483$-approximation streaming algorithm for estimating the maximum directed cut size ($\textsf{Max-DICUT}$) in a directed graph on $n$ vertices. This improves over an $O(\log n)$-space $4/9 < 0.45$ approximation algorithm due to Chou, Golovnev, Velusamy (FOCS 2020), which was known to be optimal for $o(\sqrt{n})$-space algorithms. $\textsf{Max-DICUT}$ is a special case of a constraint satisfaction problem (CSP). In this broader context, our work gives the first CSP for which algorithms with $\widetilde{O}(\sqrt{n})$ space can provably outperform $o(\sqrt{n})$-space algorithms on general instances. Previously, this was shown in the restricted case of bounded-degree graphs in a previous work of the authors (SODA 2023). Prior to that work, the only algorithms for any CSP were based on generalizations of the $O(\log n)$-space algorithm for $\textsf{Max-DICUT}$, and were in particular so-called "sketching" algorithms. In this work, we demonstrate that more sophisticated streaming algorithms can outperform these algorithms even on general instances. Our algorithm constructs a "snapshot" of the graph and then applies a result of Feige and Jozeph (Algorithmica, 2015) to approximately estimate the $\textsf{Max-DICUT}$ value from this snapshot. Constructing this snapshot is easy for bounded-degree graphs and the main contribution of our work is to construct this snapshot in the general setting. This involves some delicate sampling methods as well as a host of "continuity" results on the $\textsf{Max-DICUT}$ behaviour in graphs.Comment: 57 pages, 2 figure