Bundling Equilibrium in Combinatorial auctions

This paper analyzes individually-rational ex post equilibrium in the VC (Vickrey-Clarke) combinatorial auctions. If $\Sigma$ is a family of bundles of goods, the organizer may restrict the participants by requiring them to submit their bids only for bundles in $\Sigma$. The $\Sigma$-VC combinatorial auctions (multi-good auctions) obtained in this way are known to be individually-rational truth-telling mechanisms. In contrast, this paper deals with non-restricted VC auctions, in which the buyers restrict themselves to bids on bundles in $\Sigma$, because it is rational for them to do so. That is, it may be that when the buyers report their valuation of the bundles in $\Sigma$, they are in an equilibrium. We fully characterize those $\Sigma$ that induce individually rational equilibrium in every VC auction, and we refer to the associated equilibrium as a bundling equilibrium. The number of bundles in $\Sigma$ represents the communication complexity of the equilibrium. A special case of bundling equilibrium is partition-based equilibrium, in which $\Sigma$ is a field, that is, it is generated by a partition. We analyze the tradeoff between communication complexity and economic efficiency of bundling equilibrium, focusing in particular on partition-based equilibrium

Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation

In $masked\ low-rank\ approximation$, one is given $A \in \mathbb{R}^{n \times n}$ and binary mask matrix $W \in \{0,1\}^{n \times n}$. The goal is to find a rank-$k$ matrix $L$ for which: $$cost(L) = \sum_{i=1}^{n} \sum_{j = 1}^{n} W_{i,j} \cdot (A_{i,j} - L_{i,j} )^2 \leq OPT + \epsilon \|A\|_F^2 ,$$ where $OPT = \min_{rank-k\ \hat{L}} cost(\hat L)$ and $\epsilon$ is a given error parameter. Depending on the choice of $W$, this problem captures factor analysis, low-rank plus diagonal decomposition, robust PCA, low-rank matrix completion, low-rank plus block matrix approximation, and many problems. Many of these problems are NP-hard, and while some algorithms with provable guarantees are known, they either 1) run in time $n^{\Omega(k^2/\epsilon)}$ or 2) make strong assumptions, e.g., that $A$ is incoherent or that $W$ is random. In this work, we show that a common polynomial time heuristic, which simply sets $A$ to $0$ where $W$ is $0$, and then finds a standard low-rank approximation, yields bicriteria approximation guarantees for this problem. In particular, for rank $k' > k$ depending on the $public\ coin\ partition\ number$of$W$, the heuristic outputs rank-$k'$$L$with cost$(L) \leq OPT + \epsilon \|A\|_F^2$. This partition number is in turn bounded by the$randomized\ communication\ complexity$of$W$, when interpreted as a two-player communication matrix. For many important examples of masked low-rank approximation, including all those listed above, this result yields bicriteria approximation guarantees with$k' = k \cdot poly(\log n/\epsilon)$. Further, we show that different models of communication yield algorithms for natural variants of masked low-rank approximation. For example, multi-player number-in-hand communication complexity connects to masked tensor decomposition and non-deterministic communication complexity to masked Boolean low-rank factorization.Comment: ITCS 202

Quantum vs. Classical Read-once Branching Programs

The paper presents the first nontrivial upper and lower bounds for (non-oblivious) quantum read-once branching programs. It is shown that the computational power of quantum and classical read-once branching programs is incomparable in the following sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by error-free quantum read-once branching programs of size O(n^3), while each classical randomized read-once branching program and each quantum OBDD for this function with bounded two-sided error requires size 2^{\Omega(n)}. (ii) Quantum branching programs reading each input variable exactly once are shown to require size 2^{\Omega(n)} for computing the set-disjointness function DISJ_n from communication complexity theory with two-sided error bounded by a constant smaller than 1/2-2\sqrt{3}/7. This function is trivially computable even by deterministic OBDDs of linear size. The technically most involved part is the proof of the lower bound in (ii). For this, a new model of quantum multi-partition communication protocols is introduced and a suitable extension of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to this model is presented.Comment: 35 pages. Lower bound for disjointness: Error in application of info theory corrected and regularity of quantum read-once BPs (each variable at least once) added as additional assumption of the theorem. Some more informal explanations adde

NP-hardness of circuit minimization for multi-output functions

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions

On Characterizing the Data Movement Complexity of Computational DAGs for Parallel Execution

Technology trends are making the cost of data movement increasingly dominant, both in terms of energy and time, over the cost of performing arithmetic operations in computer systems. The fundamental ratio of aggregate data movement bandwidth to the total computational power (also referred to the machine balance parameter) in parallel computer systems is decreasing. It is there- fore of considerable importance to characterize the inherent data movement requirements of parallel algorithms, so that the minimal architectural balance parameters required to support it on future systems can be well understood. In this paper, we develop an extension of the well-known red-blue pebble game to develop lower bounds on the data movement complexity for the parallel execution of computational directed acyclic graphs (CDAGs) on parallel systems. We model multi-node multi-core parallel systems, with the total physical memory distributed across the nodes (that are connected through some interconnection network) and in a multi-level shared cache hierarchy for processors within a node. We also develop new techniques for lower bound characterization of non-homogeneous CDAGs. We demonstrate the use of the methodology by analyzing the CDAGs of several numerical algorithms, to develop lower bounds on data movement for their parallel execution

Multi-criteria scheduling of pipeline workflows

Mapping workflow applications onto parallel platforms is a challenging problem, even for simple application patterns such as pipeline graphs. Several antagonist criteria should be optimized, such as throughput and latency (or a combination). In this paper, we study the complexity of the bi-criteria mapping problem for pipeline graphs on communication homogeneous platforms. In particular, we assess the complexity of the well-known chains-to-chains problem for different-speed processors, which turns out to be NP-hard. We provide several efficient polynomial bi-criteria heuristics, and their relative performance is evaluated through extensive simulations