We study the single-pass streaming complexity of deciding satisfiability of Constraint Satisfaction Problems (CSPs). A CSP is specified by a constraint language Γ, that is, a finite set of k-ary relations over the domain [q] = {0, … , q-1}. An instance of CSP(Γ) consists of m constraints over n variables x₁, …, x_n taking values in [q]. Each constraint C_i is of the form {R_i,(x_{i_1} + λ_{i_1}, …, x_{i_k} + λ_{i_k})}, where R_i ∈ Γ and λ_{i_1}, …, λ_{i_k} ∈ [q] are constants; it is satisfied if and only if (x_{i_1} + λ_{i_1}, …, x_{i_k} + λ_{i_k}) ∈ R_i, where addition is modulo q. In the streaming model, constraints arrive one by one, and the goal is to determine, using minimum memory, whether there exists an assignment satisfying all constraints. For k-SAT, Vu (TCS 2024) proves an optimal Ω_k(n^k) space lower bound, while for general CSPs, Chou, Golovnev, Sudan, and Velusamy (JACM 2024) establish an Ω(n) lower bound; a complete characterization has remained open. We close this gap by showing that the single-pass streaming space complexity of CSP(Γ) is precisely governed by its non-redundancy, a structural parameter introduced by Bessiere, Carbonnel, and Katsirelos (AAAI 2020). The non-redundancy NRD_n(Γ) is the maximum number of constraints over n variables such that every constraint is non-redundant, i.e., omitting it strictly expands the set of satisfying assignments. We prove that the single-pass streaming complexity of CSP(Γ) is characterized, up to a logarithmic factor, by NRD_n(Γ). We also extend this characterization to positive Boolean CSPs, i.e., instances in which no additive shifts are applied, a class that includes graph 2-colorability (equivalently, bipartiteness) as a canonical example. A key ingredient in our lower bound proof is a binary relation ℰ on the set of assignments [q]ⁿ, where (a, b) ∈ ℰ if every instance satisfied by a is also satisfied by b. While it is immediate that ℰ is reflexive and transitive, its symmetry, which would make it an equivalence relation, is non-trivial. We show that ℰ is an equivalence relation for general CSPs, and for positive Boolean CSPs when excluding the two constant assignments (0ⁿ and 1ⁿ). We believe this equivalence structure could be of independent interest.

We explore the use of local algorithms in the design of streaming algorithms for the Maximum Directed Cut problem. Specifically, building on the local algorithm of (Buchbinder, Feldman, Seffi, and Schwartz [14] and Censor-Hillel, Levy, and Shachnai [16]), we develop streaming algorithms for both adversarially and randomly ordered streams that approximate the value of maximum directed cut in bounded-degree graphs. In n-vertex graphs, for adversarially ordered streams, our algorithm uses O (n1-Ω(1)) (sub-linear) space and for randomly ordered streams, our algorithm uses logarithmic space. Moreover, both algorithms require only one pass over the input stream. With a constant number of passes, we give a logarithmic-space algorithm which works even on graphs with unbounded degree on adversarially ordered streams. Our algorithms achieve any fixed constant approximation factor less than 1/2. In the single-pass setting, this is tight: known lower bounds show that obtaining any constant approximation factor greater than 1/2 is impossible without using linear space in adversarially ordered streams (Kapralov and Krachun [37]) and space in randomly ordered streams, even on bounded degree graphs (Kapralov, Khanna, and Sudan [35]).

Abstract An ordering constraint satisfaction problem (OCSP) is defined by a family $$\mathcal F$$ F of predicates mapping permutations on $$\{1,\ldots,k\}$$ {1,…,k} to $$\{0,1\}$$ {0,1} . An instance of ( $$\mathcal F$$ F ) onnvariables consists of a list of constraints, each consisting of a predicate from $$\mathcal F$$ F applied onkdistinct variables. The goal is to find an ordering of thenvariables that maximizes the number of constraints for which the induced ordering on thekvariables satisfies the predicate. OCSPs capture well-studied problems including ‘maximum acyclic subgraph’ () and “maximum betweenness”. In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every $$\mathcal F$$ F , ( $$\mathcal F$$ F ) is approximation-resistant to o(n)-space streaming algorithms, i.e., algorithms using o(n) space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of , our result shows that for every $$\epsilon>0$$ ϵ>0 , is not $$(1/2+\epsilon)$$ (1/2+ϵ) -approximable in o(n) space. The previous best inapproximability result, due to Guruswami & Tao (2019), only ruled out 3/4-approximations in $$o(\sqrt n)$$ o(n) space. Our results build on recent works of Chou et al. (2022b, 2024) who provide a tight, linear-space inapproximability theorem for a broad class of “standard” (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. Our results are obtained by building a family of appropriate standard CSPs (one for every alphabet sizeq) from any given OCSP and applying their theorem to this family of CSPs. To convert the resulting hardness results for standard CSPs back to our OCSP, we show that the hard instances from this earlier theorem have the following “partition expansion” property with high probability: For every partition of thenvariables into small blocks, for most of the constraints, all variables are in distinct blocks.

A constraint satisfaction problem (CSP), \(\textsf {Max-CSP}(\mathcal {F})\) , is specified by a finite set of constraints \(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\) for positive integers q and k . An instance of the problem on n variables is given by m applications of constraints from \(\mathcal {F}\) to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family \(\mathcal {F}\) and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in \(o(\sqrt {n})\) space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of q = k =2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of \(\mathcal {F}\) support a distribution on \([q]^k\) with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables q = 2, binary constraints k =2, and singleton families \(|\mathcal {F}|=1\) and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [ 47 ] and [ 41 ], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [ 56 ], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general q -ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

We give an $\widetilde{O}(\sqrt{n})$-space single-pass 0.483-approximation streaming algorithm for estimating the maximum directed cut size (Max-DICUT) in a directed graph on n vertices. This improves over an $O(\log n)$-space $4 / 9\lt 0.45$ approximation algorithm due to Chou, Golovnev, and Velusamy (FOCS 2020), which was known to be optimal for $o(\sqrt{n})$-space algorithms. Max-DICUT is a special case of a constraint satisfaction problem (CSP). In this broader context, we give the first CSP for which algorithms with $\widetilde{O}(\sqrt{n})$- space can provably outperform $o(\sqrt{n})$- space algorithms. The key technical contribution of our work is development of the notions of a first-order snapshot of a (directed) graph and of estimates of such snapshots. These snapshots can be used to simulate certain (non-streaming) Max-DICUT algorithms, including the “oblivious” algorithms introduced by Feige and Jozeph (Algorithmica, 2015), who showed that one such algorithm Previous work of the authors (SODA 2023) studied the restricted case of bounded-degree graphs, and observed that in this setting, it is straightforward to estimate the snapshot with $\ell_{1}$ errors and this suffices to simulate oblivious algorithms. But for unbounded-degree graphs, even defining an achievable and sufficient notion of estimation is subtle. We describe a new notion of snapshot estimation and prove its sufficiency using careful smoothing techniques, and then develop an algorithm which sketches such an estimate via a delicate process of intertwined vertex- and edge-subsampling. Prior to our work, the only streaming algorithms for any CSP on general instances were based on generalizations of the $O(\log n)$-space algorithm for Max-DICUT, and can roughly be characterized as based on “zeroth” order snapshots. Our work thus opens the possibility of a new class of algorithms for approximating CSPs by demonstrating that more sophisticated snapshots can outperform cruder ones in the case of Max-DICUT.

We consider prophet inequalities subject to feasibility constraints that are the intersection of $q$ matroids. The best-known algorithms achieve a $Θ(q)$-approximation, even when restricted to instances that are the intersection of $q$ partition matroids, and with i.i.d.~Bernoulli random variables. The previous best-known lower bound is $Θ(\sqrt{q})$ due to a simple construction of [Kleinberg-Weinberg STOC 2012] (which uses i.i.d.~Bernoulli random variables, and writes the construction as the intersection of partition matroids). We establish an improved lower bound of $q^{1/2+Ω(1/\log \log q)}$ by writing the construction of [Kleinberg-Weinberg STOC 2012] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with $p^p$ disjoint cliques of size $p$, using recent techniques developed in [Alon-Alweiss European Journal of Combinatorics 2020].

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 ≤ 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.

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 Max-DICUT, for which random ordering makes a provable difference. Whereas a 4/9 ≈ 0.445 approximation of DICUT requires space with adversarial ordering, we show that with random ordering of constraints there exists a 0.483-approximation algorithm that only needs O(log n) space. We also give new algorithms for Max-DICUT in variants of the adversarial ordering setting. Specifically, we give a two-pass O(log n) space 0.483-approximation algorithm for general graphs and a single-pass space 0.483-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 -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 Erdős-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 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 space algorithms. Our results show a richer variety of possibilities and motivate further study of CSPs with randomly ordered constraints. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.07158. In this version, we omit many of the proofs.

A Boolean maximum constraint satisfaction problem, Max-CSP($f$), is specified by a predicate $f:\{-1,1\}^k\to\{0,1\}$. An $n$-variable instance of Max-CSP($f$) consists of a list of constraints, each of which applies $f$ to $k$ distinct literals drawn from the $n$ variables. For $k=2$, Chou, Golovnev, and Velusamy [CGV20, FOCS 2020] obtained explicit ratios characterizing the $\sqrt n$-space streaming approximability of every predicate. For $k \geq 3$, Chou, Golovnev, Sudan, and Velusamy [CGSV21, arXiv:2102.12351] proved a general dichotomy theorem for $\sqrt n$-space sketching algorithms: For every $f$, there exists $α(f)\in (0,1]$ such that for every $ε>0$, Max-CSP($f$) is $(α(f)-ε)$-approximable by an $O(\log n)$-space linear sketching algorithm, but $(α(f)+ε)$-approximation sketching algorithms require $Ω(\sqrt{n})$ space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting $α'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}$, we show that for odd $k \geq 3$, $α(k$AND$) = α'_k$, and for even $k \geq 2$, $α(k$AND$) = 2α'_{k+1}$. We also resolve the ratio for the "at-least-$(k-1)$-$1$'s" function for all even $k$; the "exactly-$\frac{k+1}2$-$1$'s" function for odd $k \in \{3,\ldots,51\}$; and fifteen other functions. We stress here that for general $f$, according to [CGSV21], closed-form expressions for $α(f)$ need not have existed a priori. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [CGV20] while simplifying [CGSV21]. Finally, we investigate the $\sqrt n$-space streaming lower bounds in [CGSV21], and show that they are incomplete for $3$AND.

We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In~particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every $k \geq 5$, we show that CSPs where the underlying predicate is a pure monarchy function on $k$ variables have no non-trivial sketching approximation algorithm in $o(\sqrt{n})$ space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by $O(\log(n))$ space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously.

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on n variables taking values in {0,…,q−1}, we prove that improving over the trivial approximability by a factor of q requires Ω(n) space even on instances with O(n) constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires Ω(n) space. The key technical core is an optimal, q−(k−1)-inapproximability for the Max k-LIN-mod q problem, which is the Max CSP problem where every constraint is given by a system of k−1 linear equations mod q over k variables.

Abstract. We present a general framework for designing approximately revenue-optimal mechanisms for multi-item additive auctions, which applies to both truthful and nontruthful auctions. Given a (not necessarily truthful) single-item auction format [Formula: see text] satisfying certain technical conditions, we run simultaneous item auctions augmented with a personalized entry fee for each bidder that must be paid before the auction can be accessed. These entry fees depend only on the prior distribution of bidder types and in particular are independent of realized bids. We bound the revenue of the resulting two-part tariff mechanism using a novel geometric technique that enables revenue guarantees for many common nontruthful auctions that previously had none. Our approach adapts and extends the duality framework of Cai, Devanur, and Weinberg [ SIAM J. Comput., 50 (2021), pp. STOC16-160–STOC16-200] beyond truthful auctions. Our framework can be used with many common auction formats, such as simultaneous first-price, simultaneous second-price, and simultaneous all-pay auctions. Our results for first-price and all-pay are the first revenue guarantees of nontruthful mechanisms in multidimensional environments, addressing an open question in the literature [T. Roughgarden, V. Syrgkanis, and E. Tardos, J. Artificial Intelligence Res., 59 (2017), pp. 59–101]. If all-pay auctions are used, we prove that the resulting mechanism is also credible in the sense that the auctioneer cannot benefit by deviating from the stated mechanism after observing agent bids. This is the first static credible mechanism for multi-item additive auctions that achieves a constant factor of the optimal revenue. If second-price auctions are used, we obtain a truthful [Formula: see text]-approximate mechanism with fixed entry fees that are amenable to tuning via online learning techniques.

An ordering constraint satisfaction problem (OCSP) is given by a positive integer k and a constraint predicate Π mapping permutations on {1,…,k} to {0,1}. Given an instance of OCSP(Π) 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 Π. Ordering constraint satisfaction problems 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 Π, OCSP(Π) is approximation-resistant to o(n)-space streaming algorithms, i.e., algorithms using o(n) space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every ε > 0, MAS is not 1/2+ε-approximable in o(n) space. The previous best inapproximability result only ruled out a 3/4-approximation in o(√ 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 (CSPs) over arbitrary (finite) alphabets. Our results are obtained by building a family of appropriate CSPs (one for every q) from any given OCSP, and applying their work to this family of CSPs. To convert the resulting hardness results for CSPs back to our OCSP, we show that the hard instances from this earlier work have the following "small-set expansion" property: If the CSP instance is viewed as a hypergraph in the natural way, then for every partition of the hypergraph into small blocks most of the hyperedges are incident on vertices from distinct blocks. 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.

A constraint satisfaction problem (CSP),$\text{Max}-\text{CSP}(\mathcal{F})$, is specified by a finite set of constraints$\mathcal{F}\subseteq\{[q]^{k}\rightarrow\{0,1\}\}$for positive integers$q$and$k$. An instance of the problem on$n$variables is given by$m$applications of constraints from$\mathcal{F}$to subsequences of the$n$variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the ($\gamma, \beta$)-approximation version of the problem, for parameters$0\leq\beta < \gamma\leq 1$, the goal is to distinguish instances where at least$\gamma$fraction of the constraints can be satisfied from instances where at most$\beta$fraction of the constraints can be satisfied. In this work we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family$\mathcal{F}$and every$\beta < \gamma$, we show that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in$o(\sqrt{n})$space. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of$q=k=2$where we get a dichotomy and the case when the satisfying assignments of$f$support a distribution on$[q]^{k}$with uniform marginals. Prior to this work, other than sporadic examples, the only systematic class of CSPs that were analyzed considered the setting of Boolean variables$q=2$, binary constraints$k=2$, singleton families$\vert \mathcal{F}\vert =1$and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [2] and [3], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [4], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general$q$-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

We prove tight upper and lower bounds on approximation ratios of all Boolean Max-2CSP problems in the streaming model. Specifically, for every type of Max-2CSP problem, we give an explicit constant α, s.t. for any (i) there is an ( α-ε)-streaming approximation using space O(logn); and (ii) any ( α+ε)-streaming approximation requires space Ω(√n). This generalizes the celebrated work of [Kapralov, Khanna, Sudan SODA 2015; Kapralov, Krachun STOC 2019], who showed that the optimal approximation ratio for Max-CUT was 1/2. Prior to this work, the problem of determining this ratio was open for all other Max-2CSPs. Our results are quite surprising for some specific Max-2CSPs. For the Max-DICUT problem, there was a gap between an upper bound of 1/2 and a lower bound of 2/5 [Guruswami, Velingker, Velusamy APPROX 2017]. We show that neither of these bounds is tight, and the optimal ratio for Max-DICUT is 4/9. We also establish that the tight approximation for Max-2SAT is √2/2, and for Exact Max-2SAT it is 3/4. As a byproduct, our result gives a separation between space-efficient approximations for Max-2SAT and Exact Max-2SAT. This is in sharp contrast to the setting of polynomial-time algorithms with polynomial space, where the two problems are known to be equally hard to approximate. Finally, we prove that the tight streaming approximation for Max-kSAT is √2/2 for every k ≥ 2.

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) = 𝒪((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 ≥ Ω(nlog n), n ≥ 2, we design fully-explicit schemes that show that s(m,n,t) = 𝒪((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 n₀, for n ≥ n₀, 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) ≥ Ω(√m).

We present a general framework for designing approximately revenue-optimal mechanisms for multi-item additive auctions. Our approach adapts the duality framework of Cai, Devanur and Weinberg (STOC 2016) and applies to both truthful and non-truthful auctions. Given a (not necessarily truthful) single-item auction format 'A' satisfying certain technical conditions, we run simultaneous item auctions augmented with a personalized entry fee for each bidder that must be paid before the auction can be accessed. These entry fees depend only on the prior distribution of bidder types, and in particular are independent of realized bids. We bound the revenue of the resulting two-part tariff mechanism using a novel geometric technique that enables revenue guarantees for many common non-truthful auctions that previously had none. Our framework can be used with many common auction formats, such as simultaneous first-price, simultaneous second-price, and simultaneous all-pay auctions. Our results for first price and all-pay are the first revenue guarantees of non-truthful mechanisms in multi-dimensional environments, addressing an open question in the literature. If all-pay auctions are used, we prove that the resulting mechanism is also credible in the sense that the auctioneer cannot benefit by deviating from the stated mechanism after observing agent bids. This is the first static credible mechanism for multi-item additive auctions that achieves a constant factor of the optimal revenue. If second-price auctions are used, we obtain a truthful O(1)-approximate mechanism with fixed entry fees that are amenable to tuning via online learning techniques.

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'15) 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.