Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT

AbstractThe maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly(L)·2K/5, where K is the number of clauses and L is their total length. In fact, the running time is only poly(L)·2K2/5, where K2 is the number of clauses containing two literals. This bound implies the bound poly(L)·2L/10. Our results significantly improve previous bounds: poly(L)·2K/2.88 (J. Algorithms 36 (2000) 62–88) and poly(L)·2K/3.44 (implicit in Bansal and Raman (Proceedings of the 10th Annual Conference on Algorithms and Computation, ISAAC’99, Lecture Notes in Computer Science, Vol. 1741, Springer, Berlin, 1999, pp. 247–258.))As an application, we derive upper bounds for the (MAX-SNP-complete) maximum cut problem (MAX-CUT), showing that it can be solved in time poly(M)·2M/3, where M is the number of edges in the graph. This is of special interest for graphs with low vertex degree

Computational Results for Extensive-Form Adversarial Team Games

We provide, to the best of our knowledge, the first computational study of extensive-form adversarial team games. These games are sequential, zero-sum games in which a team of players, sharing the same utility function, faces an adversary. We define three different scenarios according to the communication capabilities of the team. In the first, the teammates can communicate and correlate their actions both before and during the play. In the second, they can only communicate before the play. In the third, no communication is possible at all. We define the most suitable solution concepts, and we study the inefficiency caused by partial or null communication, showing that the inefficiency can be arbitrarily large in the size of the game tree. Furthermore, we study the computational complexity of the equilibrium-finding problem in the three scenarios mentioned above, and we provide, for each of the three scenarios, an exact algorithm. Finally, we empirically evaluate the scalability of the algorithms in random games and the inefficiency caused by partial or null communication

Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks

Deep neural networks have emerged as a widely used and effective means for tackling complex, real-world problems. However, a major obstacle in applying them to safety-critical systems is the great difficulty in providing formal guarantees about their behavior. We present a novel, scalable, and efficient technique for verifying properties of deep neural networks (or providing counter-examples). The technique is based on the simplex method, extended to handle the non-convex Rectified Linear Unit (ReLU) activation function, which is a crucial ingredient in many modern neural networks. The verification procedure tackles neural networks as a whole, without making any simplifying assumptions. We evaluated our technique on a prototype deep neural network implementation of the next-generation airborne collision avoidance system for unmanned aircraft (ACAS Xu). Results show that our technique can successfully prove properties of networks that are an order of magnitude larger than the largest networks verified using existing methods.Comment: This is the extended version of a paper with the same title that appeared at CAV 201

Non-polynomial Worst-Case Analysis of Recursive Programs

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201

On the extension complexity of combinatorial polytopes

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.Comment: 15 pages, 3 figures, 2 table

On product, generic and random generic quantum satisfiability

We report a cluster of results on k-QSAT, the problem of quantum satisfiability for k-qubit projectors which generalizes classical satisfiability with k-bit clauses to the quantum setting. First we define the NP-complete problem of product satisfiability and give a geometrical criterion for deciding when a QSAT interaction graph is product satisfiable with positive probability. We show that the same criterion suffices to establish quantum satisfiability for all projectors. Second, we apply these results to the random graph ensemble with generic projectors and obtain improved lower bounds on the location of the SAT--unSAT transition. Third, we present numerical results on random, generic satisfiability which provide estimates for the location of the transition for k=3 and k=4 and mild evidence for the existence of a phase which is satisfiable by entangled states alone.Comment: 9 pages, 5 figures, 1 table. Updated to more closely match published version. New proof in appendi

Ground state of the Bethe-lattice spin glass and running time of an exact optimization algorithm

We study the Ising spin glass on random graphs with fixed connectivity z and with a Gaussian distribution of the couplings, with mean \mu and unit variance. We compute exact ground states by using a sophisticated branch-and-cut method for z=4,6 and system sizes up to N=1280 for different values of \mu. We locate the spin-glass/ferromagnet phase transition at \mu = 0.77 +/- 0.02 (z=4) and \mu = 0.56 +/- 0.02 (z=6). We also compute the energy and magnetization in the Bethe-Peierls approximation with a stochastic method, and estimate the magnitude of replica symmetry breaking corrections. Near the phase transition, we observe a sharp change of the median running time of our implementation of the algorithm, consistent with a change from a polynomial dependence on the system size, deep in the ferromagnetic phase, to slower than polynomial in the spin-glass phase.Comment: 10 pages, RevTex, 10 eps figures. Some changes in the tex

Hardness measures and resolution lower bounds

Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this report we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. We also extend these measures to all clause-sets (possibly satisfiable).Comment: 43 pages, preliminary version (yet the application part is only sketched, with proofs missing