The parallel approximability of a subclass of quadratic programming

In this paper we deal with the parallel approximability of a special class of Quadratic Programming (QP), called Smooth Positive Quadratic Programming. This subclass of QP is obtained by imposing restrictions on the coefficients of the QP instance. The Smoothness condition restricts the magnitudes of the coefficients while the positiveness requires that all the coefficients be non-negative. Interestingly, even with these restrictions several combinatorial problems can be modeled by Smooth QP. We show NC Approximation Schemes for the instances of Smooth Positive QP. This is done by reducing the instance of QP to an instance of Positive Linear Programming, finding in NC an approximate fractional solution to the obtained program, and then rounding the fractional solution to an integer approximate solution for the original problem. Then we show how to extend the result for positive instances of bounded degree to Smooth Integer Programming problems. Finally, we formulate several important combinatorial problems as Positive Quadratic Programs (or Positive Integer Programs) in packing/covering form and show that the techniques presented can be used to obtain NC Approximation Schemes for "dense" instances of such problems.Peer ReviewedPostprint (published version

On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201

Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP

Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple semidefinite programming and a rounding scheme for it. In this paper, we show that similar results hold for constant-time approximation algorithms in the bounded-degree model. Specifically, we present the followings: (i) For every CSP, we construct an oracle that serves an access, in constant time, to a nearly optimal solution to a basic LP relaxation of the CSP. (ii) Using the oracle, we give a constant-time rounding scheme that achieves an approximation ratio coincident with the integrality gap of the basic LP. (iii) Finally, we give a generic conversion from integrality gaps of basic LPs to hardness results. All of those results are \textit{unconditional}. Therefore, for every bounded-degree CSP, we give the best constant-time approximation algorithm among all. A CSP instance is called $\epsilon$-far from satisfiability if we must remove at least an $\epsilon$-fraction of constraints to make it satisfiable. A CSP is called testable if there is a constant-time algorithm that distinguishes satisfiable instances from $\epsilon$-far instances with probability at least $2/3$. Using the results above, we also derive, under a technical assumption, an equivalent condition under which a CSP is testable in the bounded-degree model

Tensor Network States: Optimizations and Applications in Quantum Many-Body Physics and Machine Learning

Tensor network states are ubiquitous in the investigation of quantum many-body (QMB) physics. Their advantage over other state representations is evident from their reduction in the computational complexity required to obtain various quantities of interest, namely observables. Additionally, they provide a natural platform for investigating entanglement properties within a system. In this dissertation, we develop various novel algorithms and optimizations to tensor networks for the investigation of QMB systems, including classical and quantum circuits. Specifically, we study optimizations for the two-dimensional Ising model in a transverse field, we create an algorithm for the $k$-SAT problem, and we study the entanglement properties of random unitary circuits. In addition to these applications, we reinterpret renormalization group principles from QMB physics in the context of machine learning to develop a novel algorithm for the tasks of classification and regression, and then utilize machine learning architectures for the time evolution of operators in QMB systems

Computational aspects of combinatorial pricing problems

Combinatorial pricing encompasses a wide range of natural optimization problems that arise in the computation of revenue maximizing pricing schemes for a given set of goods, as well as the design of profit maximizing auctions in strategic settings. We consider the computational side of several different multi-product and network pricing problems and, as most of the problems in this area are NP-hard, we focus on the design of approximation algorithms and corresponding inapproximability results. In the unit-demand multi-product pricing problem it is assumed that each consumer has different budgets for the products she is interested in and purchases a single product out of her set of alternatives. Depending on how consumers choose their products once prices are fixed we distinguish the min-buying, max-buying and rank-buying models, in which consumers select the affordable product with smallest price, highest price or highest rank according to some predefined preference list, respectively. We prove that the max-buying model allows for constant approximation guarantees and this is true even in the case of limited product supply. For the min-buying model we prove inapproximability beyond the known logarithmic guarantees under standard complexity theoretic assumptions. Surprisingly, this result even extends to the case of pricing with a price ladder constraint, i.e., a predefined relative order on the product prices. Furthermore, similar results can be shown for the uniform-budget version of the problem, which corresponds to a special case of the unit-demand envy-free pricing problem, under an assumption about the average case hardness of refuting random 3SAT-instances. Introducing the notion of stochastic selection rules we show that among a large class of selection rules based on the order of product prices the maxbuying model is in fact the only one allowing for sub-logarithmic approximation guarantees. In the single-minded pricing problem each consumer is interested in a single set of products, which she purchases if the sum of prices does not exceed her budget. It turns out that our results on envyfree unit-demand pricing can be extended to this scenario and yield inapproximability results for ratios expressed in terms of the number of distinct products, thereby complementing existing hardness results. On the algorithmic side, we present an algorithm with approximation guarantee that depends only on the maximum size of the sets and the number of requests per product. Our algorithm’s ratio matches previously known results in the worst case but has significantly better provable performance guarantees on sparse problem instances. Viewing single-minded as a network pricing problem in which we assign prices to edges and consumers want to purchase paths in the network, it is proven that the problem remains APX-hard even on extremely sparse instances. For the special case of pricing on a line with paths that are nested, we design an FPTAS and prove NP-hardness. In a Stackelberg network pricing game a so-called leader sets the prices on a subset of the edges of a network, the remaining edges have associated fixed costs. Once prices are fixed, one or more followers purchase min-cost subnetworks according to their requirements and pay the leader for all pricable edges contained in their networks. We extend the analysis of the known single-price algorithm, which assigns the same price to all pricable edges, from cases in which the feasible subnetworks of a follower form the basis of a matroid to the general case, thus, obtaining logarithmic approximation guarantees for general Stackelberg games. We then consider a special 2-player game in which the follower buys a min-cost vertex cover in a bipartite graph and the leader sets prices on a subset of the vertices. We prove that this problem is polynomial time solvable in some cases and allows for constant approximation guarantees in general. Finally, we point out that results on unit-demand and single-minded pricing yield several strong inapproximability results for Stackelberg pricing games with multiple followers

Approximation Algorithms for NP-Hard Problems

The workshop was concerned with the most important recent developments in the area of efficient approximation algorithms for NP-hard optimization problems as well as with new techniques for proving intrinsic lower bounds for efficient approximation

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes

Robust Approximation of Temporal CSP

A temporal constraint language G is a set of relations with first-order definitions in (Q; = 0, given a (1-e)-satisfiable instance of CSP(G), we can compute an assignment that satisfies at least a (1-f(e))-fraction of constraints in polynomial time. Here, f(e) is some function satisfying f(0)=0 and f(e) goes 0 as e goes 0. Firstly, we give a qualitative characterization of robust approximability: Assuming the Unique Games Conjecture, we give a necessary and sufficient condition on G under which CSP(G) admits robust approximation. Secondly, we give a quantitative characterization of robust approximability: Assuming the Unique Games Conjecture, we precisely characterize how f(e) depends on e for each G. We show that our robust approximation algorithms can be run in almost linear time