The MAX BISECTION problem seeks a maximum-size cut that evenly divides the vertices of a given undirected graph. An open problem raised by Austrin, Benabbas, and Georgiou [SODA'13, TALG'16] is whether MAX BISECTION can be approximated as well as MAX CUT, i.e., to within \(\alpha_{\mathrm{GW}} \approx 0.8785672\ldots\), which is the approximation ratio achieved by the celebrated Goemans-Williamson algorithm for MAX CUT, which is best possible assuming the Unique Games Conjecture (UGC). They conjectured that the answer is yes.

The input to the Multiway Cut problem is a weighted undirected graph, with nonnegative edge weights, and k designated terminals. The goal is to partition the vertices of the graph into k parts, each containing exactly one of the terminals, such that the sum of weights of the edges connecting vertices in different parts of the partition is minimized. The problem is APX-hard for k≥3. The currently best known approximation algorithm for the problem for arbitrary k, obtained by Sharma and Vondrák [STOC 2014] more than a decade ago, has an approximation ratio of 1.2965. We present an algorithm with an improved approximation ratio of 1.2787. Also, for small values of k ≥ 4 we obtain the first improvements in 25 years over the currently best approximation ratios obtained by Karger, Klein, Stein, Thorup, and Young [STOC 1999]. (For k=3 an optimal approximation algorithm is known.)

Abstract. Assuming the unique games conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the max cut problem is [Formula: see text], obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. The current best approximation algorithm for max di-cut, i.e., the max cut problem in directed graphs, achieves a ratio of about 0.87401, leaving open the question of whether max di-cut can be approximated as well as max cut. We obtain a slightly improved algorithm for max di-cut and a new UGC-hardness for it, showing that [Formula: see text], where [Formula: see text] is the best approximation ratio that can be obtained in polynomial time for max di-cut under UGC. Our new upper bound shows that max di-cut cannot be approximated as well as max cut, which separates max di-cut from max cut and resolves a question raised by Feige and Goemans. A natural generalization of max di-cut is the max [Formula: see text]-and problem in which each constraint is of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are literals, i.e., variables or their negations (in max di-cut each constraint is of the form [Formula: see text] where [Formula: see text] and [Formula: see text] are variables). Austrin separated max [Formula: see text]-and from max cut by showing that [Formula: see text] and conjectured that max [Formula: see text]-and and max di-cut have the same approximation ratio. Our new lower bound on max di-cut refutes this conjecture, completing the separation of the three problems max [Formula: see text]-and, max di-cut, and max cut. We also obtain a new lower bound for max [Formula: see text]-and, showing that [Formula: see text]. Our upper bound on max di-cut is achieved via a simple, analytical proof. The new lower bounds on max di-cut and max [Formula: see text]-and, i.e., the new approximation algorithms, use experimentally discovered distributions of rounding functions which are then verified via computer-assisted proofs. Code for the project is available at https://github.com/jbrakensiek/max-dicut .

We prove a hardness of sampling result for the anti-ferromagnetic Ising model on random graphs of average degree d for large constant d, proving that when the normalized inverse temperature satisfies β > 1 (asymptotically corresponding to the condensation threshold), then w.h.p. over the random graph there is no stable sampling algorithm that can output a sample close in W₂ distance to the Gibbs measure. The results also apply to a fixed-magnetization version of the model, showing that there are no stable sampling algorithms for low but positive temperature max and min bisection distributions. These results show a gap in the tractability of search and sampling problems: while there are efficient algorithms to find near optimizers, stable sampling algorithms cannot access the Gibbs distribution concentrated on such solutions. Our techniques involve extensions of the interpolation technique relating behavior of the mean field Sherrington-Kirkpatrick model to behavior of Ising models on random graphs of average degree d for large d. While previous interpolation arguments compared the free energies of the two models, our argument compares the average energies and average overlaps in the two models.

Austrin showed that the approximation ratio β ≈ 0.94016567 obtained by the MAX 2-SAT approximation algorithm of Lewin, Livnat and Zwick (LLZ) is optimal modulo the Unique Games Conjecture (UGC) and modulo a Simplicity Conjecture that states that the worst performance of the algorithm is obtained on so called simple configurations. We prove Austrin's conjecture, thereby showing the optimality of the LLZ approximation algorithm, relying only on the Unique Games Conjecture. Our proof uses a combination of analytic and computational tools.

Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is $\alpha_{\text {CUT}} \simeq 0.87856$, obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. Currently, the best approximation algorithm for MAX DI-CUT, i.e., the MAX CUT problem in directed graphs, achieves a ratio of about 0.87401, leaving open the question whether MAX DI-CUT can be approximated as well as MAX CUT. We obtain a slightly improved algorithm for MAX DI-CUT and a new UG-Chardness result for it, showing that $0.87446 \leq \alpha_{\text {DI-CUT}} \leq 0.87461$, where $\alpha_{\text {DI-CUT}}$ is the best approximation ratio that can be obtained in polynomial time for MAX DI-CUT under UGC. The new upper bound separates MAX DI-CUT from MAX CUT, i.e., shows that MAX DI-CUT cannot be approximated as well as MAX CUT, resolving a question raised by Feige and Goemans. A natural generalization of MAX DI-CUT is the MAX 2-AND problem in which each constraint is of the form $z_{1} \wedge {z_{2}}$, where $z_{1}$ and ${z_{2}}$ are literals, i.e., variables or their negations. (In MAX DI-CUT each constraint is of the form $\bar{x}_{1} \wedge {x_{2}}$, where $x_{1}$ and ${x_{2}}$ are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that $\alpha_{2 \mathrm{AND}} \leq 0.87435$ and conjectured that MAX 2-AND and MAX DI-CUT have the same approximation ratio. Our new lower bound on MAX DI-CUT refutes this conjecture, completing the separation of the three problems MAX 2-AND, MAX DI-CUT and MAX CUT. We also obtain a new lower bound for MAX 2-AND showing that $0.87414 \leq \alpha_{2 \text {AND}} \leq 0.87435$. Our upper bound on MAXDI-CUT is achieved via a simple analytical proof. The new lower bounds on MAX DI-CUT and MAX 2-AND, i.e., the new approximation algorithms, use experimentally-discovered distributions of rounding functions which are then verified via computer-assisted proofs.11Code for the project: https://github.com/jbrakensiek/max-dicut

Abstract. 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 [Formula: see text] for some [Formula: see text]. We refer to this problem as MAX NAE-[Formula: see text]-SAT. For [Formula: see text], it is a slight extension of the celebrated MAX CUT problem. For [Formula: see text], it is related to the MAX CUT problem in graphs that can be fractionally covered by triangles. For [Formula: see text], it is known that an approximation ratio of [Formula: see text], obtained by choosing a random assignment, is optimal, assuming [Formula: see text]. For every [Formula: see text], an approximation ratio of at least [Formula: see text] can be obtained for MAX NAE-[Formula: see text]-SAT. There was some hope, therefore, that there is also a [Formula: see text]-approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously. Our main result is that there is no [Formula: see text]-approximation algorithm for MAX NAE-SAT, assuming the Unique Games Conjecture (UGC). In fact, even for almost satisfiable instances of MAX NAE-[Formula: see text]-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 [Formula: see text]. Using calculus of variations, we extend the analysis of O’Donnell and Wu for MAX CUT to MAX NAE-[Formula: see text]-SAT. We obtain an optimal algorithm, assuming UGC, for MAX NAE-[Formula: see text]-SAT, slightly improving on previous algorithms. The approximation ratio of the new algorithm is about 0.9089. This gives a full understanding of MAX NAE-[Formula: see text]-SAT for every [Formula: see text]. Interestingly, the rounding function used by this optimal algorithm is the solution of an integral equation. We complement our theoretical results with some experimental results. We describe an approximation algorithm for almost satisfiable instances of MAX NAE-[Formula: see text]-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 further conjecture that these are essentially the best approximation ratios that can be achieved for these problems, assuming the UGC. Somewhat surprisingly, the rounding functions used by these approximation algorithms are nonmonotone step functions that assume only the values [Formula: see text].

Given a predicate P: {-1, 1}^k → {-1, 1}, let CSP(P) be the set of constraint satisfaction problems whose constraints are of the form P. We say that P is approximable if given a nearly satisfiable instance of CSP(P), there exists a probabilistic polynomial time algorithm that does better than a random assignment. Otherwise, we say that P is approximation resistant. In this paper, we analyze presidential type predicates, which are balanced linear threshold functions where all of the variables except the first variable (the president) have the same weight. We show that almost all presidential type predicates P are approximable. More precisely, we prove the following result: for any δ₀ > 0, there exists a k₀ such that if k ≥ k₀, δ ∈ (δ₀,1 - 2/k], and {δ}k + k - 1 is an odd integer then the presidential type predicate P(x) = sign({δ}k{x₁} + ∑_{i = 2}^{k} {x_i}) is approximable. To prove this, we construct a rounding scheme that makes use of biases and pairwise biases. We also give evidence that using pairwise biases is necessary for such rounding schemes.

String matching is one of the most fundamental problems in computer science. A natural problem is to determine the number of characters that need to be queried (i.e. the decision tree complexity) in a string in order to decide whether this string contains a certain pattern. Rivest showed that for every pattern p, in the worst case any deterministic algorithm needs to query at least n-|p|+1 characters, where n is the length of the string and |p| is the length of the pattern. He further conjectured that this bound is tight. By using the adversary method, Tuza disproved this conjecture and showed that more than one half of binary patterns are evasive, i.e. any algorithm needs to query all the characters (see Section 1.1 for more details). In this paper, we give a query algorithm which settles the decision tree complexity of string matching except for a negligible fraction of patterns. Our algorithm shows that Tuza's criteria of evasive patterns are almost complete. Using the algebraic approach of Rivest and Vuillemin, we also give a new sufficient condition for the evasiveness of patterns, which is beyond Tuza's criteria. In addition, our result reveals an interesting connection to Skolem's Problem in mathematics.