Computer Science

We consider the task of proving integer infeasibility of a bounded convex K in R n using a general branching proof system. In a general branching proof, one constructs a branching tree by adding an integer disjunction ax≤b or ax≥b+1 , a∈ Z n , b∈Z , at each node, such that the leaves of the tree correspond to empty sets (i.e., K together with the inequalities picked up from the root to leaf is empty). Recently, Beame et al (ITCS 2018), asked whether the bit size of the coefficients in a branching proof, which they named stabbing planes (SP) refutations, for the case of polytopes derived from SAT formulas, can be assumed to be polynomial in n . We resolve this question by showing that any branching proof can be recompiled so that the integer disjunctions have coefficients of size at most (nR ) O( n 2 ) , where R∈N such that K∈R B n 1 , while increasing the number of nodes in the branching tree by at most a factor O(n) . As our second contribution, we show that Tseitin formulas, an important class of infeasible SAT instances, have quasi-polynomial sized cutting plane (CP) refutations, disproving the conjecture that Tseitin formulas are (exponentially) hard for CP. As our final contribution, we give a simple family of polytopes in [0,1 ] n requiring branching proofs of length 2 n /n .

In this paper we consider the Ideal Membership Problem (IMP for short), in which we are given real polynomials f 0 , f 1 ,…, f k and the question is to decide whether f 0 belongs to the ideal generated by f 1 ,…, f k . In the more stringent version the task is also to find a proof of this fact. The IMP underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. In the majority of such applications the IMP involves so called combinatorial ideals that arise from a variety of discrete combinatorial problems. This restriction makes the IMP significantly easier and in some cases allows for an efficient algorithm to solve it. The first part of this paper follows the work of Mastrolilli [SODA'19] who initiated a systematic study of IMPs arising from Constraint Satisfaction Problems (CSP) of the form CSP(Γ) , that is, CSPs in which the type of constraints is limited to relations from a set Γ . We show that many CSP techniques can be translated to IMPs thus allowing us to significantly improve the methods of studying the complexity of the IMP. We also develop universal algebraic techniques for the IMP that have been so useful in the study of the CSP. This allows us to prove a general necessary condition for the tractability of the IMP, and three sufficient ones. The sufficient conditions include IMPs arising from systems of linear equations over GF(p) , p prime, and also some conditions defined through special kinds of polymorphisms. Our work has several consequences and applications in terms of bit complexity of sum-of-squares (SOS) proofs and their automatizability, and studying (construction of) theta bodies of combinatorial problems.

The Quadratic Assignment Problem (QAP) is a well-known NP-hard problem that is equivalent to optimizing a linear objective function over the QAP polytope. The QAP polytope with parameter n - \qappolytope{n} - is defined as the convex hull of rank- 1 matrices x x T with x as the vectorized n×n permutation matrices. In this paper we consider all the known exponential-sized families of facet-defining inequalities of the QAP-polytope. We describe a new family of valid inequalities that we show to be facet-defining. We also show that membership testing (and hence optimizing) over some of the known classes of inequalities is coNP-complete. We complement our hardness results by showing a lower bound of 2 Ω(n) on the extension complexity of all relaxations of \qappolytope{n} for which any of the known classes of inequalities are valid.

Given a set X of finite strings, one interesting question to ask is whether there exists a member of X which is simple conditional to all other members of X. Conditional simplicity is measured by low conditional Kolmogorov complexity. We prove the affirmative to this question for sets that have low mutual information with the halting sequence. There are two results with respect to this question. One is dependent on the maximum conditional complexity between two elements of X, the other is dependent on the maximum expected value of the conditional complexity of a member of X relative to each member of X.

For every constant c > 0, we show that there is a family {P_{N, c}} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, that satisfies the following properties: * For every family {f_n} of polynomials in VP, where f_n is an n variate polynomial of degree at most n^c with bounded integer coefficients and for N = \binom{n^c + n}{n}, P_{N,c} \emph{vanishes} on the coefficient vector of f_n. * There exists a family {h_n} of polynomials where h_n is an n variate polynomial of degree at most n^c with bounded integer coefficients such that for N = \binom{n^c + n}{n}, P_{N,c} \emph{does not vanish} on the coefficient vector of h_n. In other words, there are efficiently computable equations for polynomials in VP that have small integer coefficients. In fact, we also prove an analogous statement for the seemingly larger class VNP. Thus, in this setting of polynomials with small integer coefficients, this provides evidence \emph{against} a natural proof like barrier for proving algebraic circuit lower bounds, a framework for which was proposed in the works of Forbes, Shpilka and Volk (2018), and Grochow, Kumar, Saks and Saraf (2017). Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes. Our proofs also extend to finite fields of small size.

In this work, we show the first worst-case to average-case reduction for the classical k -SUM problem. A k -SUM instance is a collection of m integers, and the goal of the k -SUM problem is to find a subset of k elements that sums to 0 . In the average-case version, the m elements are chosen uniformly at random from some interval [−u,u] . We consider the total setting where m is sufficiently large (with respect to u and k ), so that we are guaranteed (with high probability) that solutions must exist. Much of the appeal of k -SUM, in particular connections to problems in computational geometry, extends to the total setting. The best known algorithm in the average-case total setting is due to Wagner (following the approach of Blum-Kalai-Wasserman), and achieves a run-time of u O(1/logk) . This beats the known (conditional) lower bounds for worst-case k -SUM, raising the natural question of whether it can be improved even further. However, in this work, we show a matching average-case lower-bound, by showing a reduction from worst-case lattice problems, thus introducing a new family of techniques into the field of fine-grained complexity. In particular, we show that any algorithm solving average-case k -SUM on m elements in time u o(1/logk) will give a super-polynomial improvement in the complexity of algorithms for lattice problems.

Red-blue pebble games model the computation cost of a two-level memory hierarchy. We present various hardness results in different red-blue pebbling variants, with a focus on the oneshot model. We first study the relationship between previously introduced red-blue pebble models (base, oneshot, nodel). We also analyze a new variant (compcost) to obtain a more realistic model of computation. We then prove that red-blue pebbling is NP-hard in all of these model variants. Furthermore, we show that in the oneshot model, a δ -approximation algorithm for δ<2 is only possible if the unique games conjecture is false. Finally, we show that greedy algorithms are not good candidates for approximation, since they can return significantly worse solutions than the optimum.

MAX NAE-SAT is a natural optimization problem, closely related to its better-known relative MAX SAT. The approximability status of MAX NAE-SAT is almost completely understood if all clauses have the same size k , for some k≥2 . We refer to this problem as MAX NAE- {k} -SAT. For k=2 , it is essentially the celebrated MAX CUT problem. For k=3 , it is related to the MAX CUT problem in graphs that can be fractionally covered by triangles. For k≥4 , it is known that an approximation ratio of 1− 1 2 k−1 , obtained by choosing a random assignment, is optimal, assuming P≠NP . For every k≥2 , an approximation ratio of at least 7 8 can be obtained for MAX NAE- {k} -SAT. There was some hope, therefore, that there is also a 7 8 -approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously. Our main result is that there is no 7 8 -approximation algorithm for MAX NAE-SAT, assuming the unique games conjecture (UGC). In fact, even for almost satisfiable instances of MAX NAE- {3,5} -SAT (i.e., MAX NAE-SAT where all clauses have size 3 or 5 ), the best approximation ratio that can be achieved, assuming UGC, is at most 3( 21 √ −4) 2 ≈0.8739 . Using calculus of variations, we extend the analysis of O'Donnell and Wu for MAX CUT to MAX NAE- {3} -SAT. We obtain an optimal algorithm, assuming UGC, for MAX NAE- {3} -SAT, slightly improving on previous algorithms. The approximation ratio of the new algorithm is ≈0.9089 . We complement our theoretical results with some experimental results. We describe an approximation algorithm for almost satisfiable instances of MAX NAE- {3,5} -SAT with a conjectured approximation ratio of 0.8728, and an approximation algorithm for almost satisfiable instances of MAX NAE-SAT with a conjectured approximation ratio of 0.8698.

We introduce a family of graph parameters, called induced multipartite graph parameters, and study their computational complexity. First, we consider the following decision problem: an instance is an induced multipartite graph parameter p and a given graph G , and for natural numbers k≥2 and ℓ , we must decide whether the maximum value of p over all induced k -partite subgraphs of G is at most ℓ . We prove that this problem is W[1]-hard. Next, we consider a variant of this problem, where we must decide whether the given graph G contains a sufficiently large induced k -partite subgraph H such that p(H)≤ℓ . We show that for certain parameters this problem is para-NP-hard, while for others it is fixed-parameter tractable.

Modeling of real-world systems with Petri nets allows to benefit from their generic concepts of parallelism, synchronisation and conflict, and obtain a concise yet expressive system representation. Algorithms for synthesis of a net from a sequential specification enable the well-developed theory of Petri nets to be applied for the system analysis through a net model. The problem of τ -synthesis consists in deciding whether a given directed labeled graph A is isomorphic to the reachability graph of a Boolean Petri net N of type τ . In case of a positive decision, N should be constructed. For many Boolean types of nets, the problem is NP-complete. This paper deals with a special variant of τ -synthesis that imposes restrictions for the target net N : we investigate dependency d -restricted tau-synthesis (DR τ S) where each place of N can influence and be influenced by at most d transitions. For a type τ , if tau-synthesis is NP-complete then DR τ S is also NP-complete. In this paper, we show that DR τ S parameterized by d is in XP. Furthermore, we prove that it is W[2]-hard, for many Boolean types that allow unconditional interactions set and reset.