Correlation clustering is a well-studied problem, first proposed by Bansal, Blum, and Chawla [Mach. Learn. '04]. The input is an unweighted, undirected graph. The problem is to cluster the vertices so as to minimize the number of edges between vertices in different clusters and missing edges between vertices inside the same cluster. This problem has a wide application in data mining and machine learning. We introduce a general framework that transforms existing static correlation clustering algorithms into fully-dynamic ones that work against an adaptive adversary. We show how to apply our framework to known efficient correlation clustering algorithms, starting from the classic 3-approximate Pivot algorithm from Ailon, Charikar and Newman [JACM'08]. Applied to the most recent sublinear $1.485$-approximation algorithm from Cao, Cohen-Addad, Lee, Li, Lolck, Newman, Thorup, Vogl, Yan and Zhang [STOC'25], we get a $1.485$-approximation fully-dynamic algorithm that works with worst-case constant update time. The original static algorithm gets its approximation factor with constant probability, and we get the same against an adaptive adversary in the sense that for any given update step, not known to our algorithm, our solution is a $1.485$-approximation with constant probability when we reach this update. Most of previous dynamic algorithms, including the celebrated result from Behnezhad, Charikar, Ma and Tan [FOCS'19], had approximation factors around $3$ in expectation, and they could only handle an oblivious adversary. A recent algorithm by Braverman, Dharangutte, Pai, Shah, and Wang [AISTATS'25] could handle an adaptive adversary, but it has a large unspecified constant approximation ratio. This contrasts with our general transformation, which works with all the best approximation factors known for the static case.

The dichromatic number χ(D) of a digraph is the minimum number k such that V(D) can be partitioned into k subsets, each inducing an acyclic digraph. The acyclic number α(D) is the cardinality of a largest induced acyclic subdigraph of D. We study these problems from an approximation point of view. We begin with establishing that even when restricted to tournaments, approximating χ and α remain as challenging as their undirected counterparts on general graphs. Specifically, we establish that for every ε > 0, it is hard to approximate both α and χ up to a factor of n^{1-ε} even when restricted to tournaments. We next consider approximate coloring of digraphs in special cases. We begin with establishing that we can color 𝓁-dicolorable digraphs using at most 𝓁 ⋅ n^{1-1/(𝓁)} colors in time O(n^{2𝓁}); in particular, we can color 2-dicolorable digraphs with 2√n colors in polynomial time. We then focus on bounding the dichromatic number of dense digraphs as a function of the independence number α of the underlying graph. We consider two special cases in this regard: digraphs with χ(D) ≤ 2 and digraphs that do not contain any directed triangle. For these cases, we present algorithms which generalize and improve existing tools and results.

Correlation Clustering is a fundamental and widely-studied problem in unsupervised learning and data mining. The input is a graph and the goal is to construct a clustering minimizing the number of inter-cluster edges plus the number of missing intra-cluster edges. CCL+24 introduced the cluster LP for Correlation Clustering, which they argued captures the problem much more succinctly than previous linear programming formulations. However, the cluster LP has exponential size, with a variable for every possible set of vertices in the input graph. Nevertheless, CCL+24 showed how to find a feasible solution for the cluster LP in time $O(n^{\text{poly}(1/ε)})$ with objective value at most $(1+ε)$ times the value of an optimal solution for the respective Correlation Clustering instance. Furthermore, they showed how to round a solution to the cluster LP, yielding a $(1.485+ε)$-approximation algorithm for the Correlation Clustering problem. The main technical result of this paper is a new approach to find a feasible solution for the cluster LP with objective value at most $(1+ε)$ of the optimum in time $\widetilde O(2^{\text{poly}(1/ε)} n)$, where $n$ is the number of vertices in the graph. We also show how to implement the rounding within the same time bounds, thus achieving a fast $(1.485+ε)$-approximation algorithm for the Correlation Clustering problem. This bridges the gap between state-of-the-art methods for approximating Correlation Clustering and the recent focus on fast algorithms.

We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color. It is an open question as to whether or not a 2-LO colorable 3-uniform hypergraph can be LO colored with 3 colors in polynomial time. Nakajima and Živný recently gave a polynomial-time algorithm to color such hypergraphs with $\widetilde{O}(n^{1/3})$ colors and asked if SDP methods can be used directly to obtain improved bounds. Our main result is to show how to use SDP-based rounding methods to produce an LO coloring with $\widetilde{O}(n^{1/5})$ colors for such hypergraphs. We show how to reduce the problem to cases with highly structured SDP solutions, which we call balanced hypergraphs. Then, we discuss how to apply classic SDP-rounding tools to obtain improved bounds.

We consider the ℓ0-Low Rank Approximation problem, where the input consists of a matrix A ∈ ℝnR×nc and an integer k, and the goal is to find a matrix B of rank at most k that minimizes ‖A — B‖0, which is the number of entries where A and B differ. For any constant k and ɛ > 0, we present a polynomial time (1 + ɛ)- approximation time for this problem, which significantly improves the previous best poly(k)-approximation.

In the classic Correlation Clustering problem introduced by Bansal, Blum, and Chawla ‍(FOCS 2002), the input is a complete graph where edges are labeled either + or −, and the goal is to find a partition of the vertices that minimizes the sum of the +edges across parts plus the sum of the -edges within parts. In recent years, Chawla, Makarychev, Schramm and Yaroslavtsev ‍(STOC 2015) gave a 2.06-approximation by providing a near-optimal rounding of the standard LP, and Cohen-Addad, Lee, Li, and Newman ‍(FOCS 2022, 2023) finally bypassed the integrality gap of 2 for this LP giving a 1.73-approximation for the problem. While introducing new ideas for Correlation Clustering, their algorithm is more complicated than typical approximation algorithms in the following two aspects: (1) It is based on two different relaxations with separate rounding algorithms connected by the round-or-cut procedure. (2) Each of the rounding algorithms has to separately handle seemingly inevitable correlated rounding errors, coming from correlated rounding of Sherali-Adams and other strong LP relaxations. In order to create a simple and unified framework for Correlation Clustering similar to those for typical approximate optimization tasks, we propose the cluster LP as a strong linear program that might tightly capture the approximability of Correlation Clustering. It unifies all the previous relaxations for the problem. It is exponential-sized, but we show that it can be (1+є)-approximately solved in polynomial time for any є > 0, providing the framework for designing rounding algorithms without worrying about correlated rounding errors; these errors are handled uniformly in solving the relaxation. We demonstrate the power of the cluster LP by presenting a simple rounding algorithm, and providing two analyses, one analytically proving a 1.49-approximation and the other solving a factor-revealing SDP to show a 1.437-approximation. Both proofs introduce principled methods by which to analyze the performance of the algorithm, resulting in a significantly improved approximation guarantee. Finally, we prove an integrality gap of 4/3 for the cluster LP, showing our 1.437-upper bound cannot be drastically improved. Our gap instance directly inspires an improved NP-hardness of approximation with a ratio 24/23 ≈ 1.042; no explicit hardness ratio was known before.

Abstract. A [Formula: see text] -coloring of a tournament is a partition of its vertices into [Formula: see text] acyclic sets. Deciding if a tournament is 2-colorable is NP -hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present a new efficient decomposition lemma for tournaments, which we use to design polynomial-time algorithms to color various classes of tournaments with few colors, notably to color a 2-colorable tournament with 10 colors. We also use this lemma to prove equivalence between the problems of coloring 3-colorable tournaments and coloring 3-colorable graphs with constantly many colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments.

A k-coloring of a tournament is a partition of its vertices into k acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present an efficient decomposition lemma for tournaments and show that it can be used to design polynomial-time algorithms to color various classes of tournaments with few colors, including an algorithm to color a 2-colorable tournament with ten colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments.

We consider the classic correlation clustering problem: Given a complete graph where edges are labelled either + or −, the goal is to find a partition of the vertices that minimizes the sum of the +edges across parts plus the sum of the −edges within parts. Recently, Cohen-Addad, Lee and Newman [CLN22] gave a 1.995-approximation for the problem using the Sherali-Adams hierarchy, hence beating the integrality gap of 2 of the classic linear program. We significantly improve upon this result by providing a 1.73-approximation for the problem. Our approach brings together a new preprocessing of correlation clustering instances that enables a new LP formulation which combined with the algorithm from [CLN22] yields the improved bound.

Given a complete graph G=(V, E) where each edge is labeled + or −, the CORRELATION CLUSTERING problem asks to partition V into clusters to minimize the number of +edges between different clusters plus the number of −edges within the same cluster. CORRELATION CLUSTERING has been used to model a large number of clustering problems in practice, making it one of the most widely studied clustering formulations. The approximability of CORRELATION CLUSTERING has been actively investigated [BBC04], [CGW05], [ACN08], culminating in a 2.06-approximation algorithm [CMSY15], based on rounding the standard LP relaxation. Since the integrality gap for this formulation is 2, it has remained a major open question to determine if the approximation factor of 2 can be reached, or even breached. In this paper, we answer this question affirmatively by showing that there exists a($1.994+\varepsilon$)-approximation algorithm based on $O(1/\varepsilon^{2})$ rounds of the Sherali-Adams hierarchy. In order to round a solution to the Sherali-Adams relaxation, we adapt the correlated rounding originally developed for CSPs [BRSII], [GSII], [RT12]. With this tool, we reach an approximation ratio of $2+\varepsilon$ for CORRELATION CLUSTERING. To breach this ratio, we go beyond the traditional triangle-based analysis by employing a global charging scheme that amortizes the total cost of the rounding across different triangles.

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We study the traveling salesman problem (TSP) in the case when the objective function of the subtour linear programming relaxation is minimized by a half-cycle point: x_e in {0,1/2,1} where the half-edges form a 2-factor and the 1-edges form a perfect matching. Such points are sufficient to resolve half-integer TSP in general and they have been conjectured to demonstrate the largest integrality gap for the subtour relaxation. For half-cycle points, the best-known approximation guarantee is 3/2 due to Christofides' famous algorithm. Proving an integrality gap of alpha for the subtour relaxation is equivalent to showing that alpha x can be written as a convex combination of tours, where x is any feasible solution for this relaxation. To beat Christofides' bound, our goal is to show that (3/2 - epsilon)x can be written as a convex combination of tours for some positive constant epsilon. Let y_e = 3/2-epsilon when x_e = 1 and y_e = 3/4 when x_e = 1/2. As a first step towards this goal, our main result is to show that y can be written as a convex combination of tours. In other words, we show that we can save on 1-edges, which has several applications. Among them, it gives an alternative algorithm for the recently studied uniform cover problem. Our main new technique is a procedure to glue tours over proper 3-edge cuts that are tight with respect to x, thus reducing the problem to a base case in which such cuts do not occur.

Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that (i) all prefix sums of the ordering are nonnegative and (ii) the maximum value of a prefix sum is minimized. Kellerer et al. call this problem the stock size problem and showed that it can be approximated to within 3/2. They also showed that an approximation ratio of 2 can be achieved via several simple algorithms. We consider a related problem, which we call the alternating stock size problem , where the numbers of positive and negative integers in the input set S are equal. The problem is the same as that shown earlier, but we are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations. We show that it can be approximated to within 1.79. Then we show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovász, in which we want to minimize the size of the gas tank necessary to go around the track. We present a 2-approximation for this problem, using a natural linear programming relaxation whose feasible solutions are doubly stochastic matrices. Our novel rounding algorithm is based on a transformation that yields another doubly stochastic matrix with special properties, from which we can extract a suitable permutation.

Mömke and Svensson presented a beautiful new approach for the traveling salesman problem on a graph metric (graph-TSP), which yields a 4/3-approximation guarantee on subcubic graphs as well as a substantial improvement over the 3/2-approximation guarantee of Christofides's algorithm on general graphs. The crux of their approach is to compute an upper bound on the minimum cost of a circulation in a particular network, $C(G,T)$, where $G$ is the input graph and $T$ is a carefully chosen spanning tree. The cost of this circulation is directly related to the number of edges in a tour output by their algorithm. Mucha subsequently improved the analysis of the circulation cost, proving that Mömke and Svensson's algorithm for graph-TSP has an approximation ratio of at most $13/9$ on general graphs. This analysis of the circulation is local, and vertices with degree four or five can contribute the most to its cost. Thus, hypothetically, there could exist a subquartic graph (a graph with degree at most four at each vertex) for which Mucha's analysis of the Mömke--Svensson algorithm is tight. We show that this is not the case and that Mömke and Svensson's algorithm for graph-TSP has an approximation guarantee of at most 25/18 on subquartic graphs. To prove this, we present different methods to upper bound the minimum cost of a circulation on the network $C(G,T)$. Our approximation guarantee holds for all graphs that have an optimal solution to a standard linear programming relaxation of graph-TSP with subquartic support.

Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. In 1982, Beck conjectured that the discrepancy of this set system is O(1). In other words, the conjecture says that each integer from 1 through n can be colored either red or blue so that the number of red and blue integers in each interval of each permutations differs only by a constant. (The discrepancy of a set system based on two permutations is at most two.) Our main result is a counterexample to this conjecture: for any positive integer n = 3k, we construct three permutations whose corresponding set system has discrepancy Ω(log n). Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This construction also disproves a generalization of Beck's conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set √ system corresponding to £ permutations has discrepancy O(√ℓ). Our work was inspired by an intriguing paper from SODA 2011 by Eisenbrand, Palvolgyi and Rothvoß, who show a surprising connection between the discrepancy of three permutations and the bin packing problem: They show that Beck's conjecture implies a constant worst-case bound on the additive integrality gap for the Gilmore-Gomory LP relaxation for bin packing in the special case when all items have sizes strictly between 1/4 and 1/2, also known as the three partition problem. Our counterexample shows that this approach to bounding the additive integrality gap for bin packing will not work. We can, however, prove an interesting implication of our construction in the reverse direction: there are instances of bin packing and corresponding optimal basic feasible solutions for the Gilmore-Gomory LP relaxation such that any packing that contains only patterns from the support of these solutions requires at least opt + Ω(log m) bins, where m is the number of items. Finally, we discuss some implications that our construction has for other areas of discrepancy theory.

In the Discrepancy problem, we are given M sets {S1, …, SM} on N elements. Our goal is to find an assignment χ of {– 1, +1} values to elements, so as to minimize the maximum discrepancy . Recently, Bansal gave an efficient algorithm for achieving O(√N) discrepancy for any set system where M = O(N) [Ban10], giving a constructive version of Spencer's proof that the discrepancy of any set system is at most O(√N) for this range of M [Spe85]. We show that from the perspective of computational efficiency, these results are tight for general set systems where M = O(N). Specifically, we show that it is NP-hard to distinguish between such set systems with discrepancy zero and those with discrepancy Ω(√ N). This means that even if the optimal solution has discrepancy zero, we cannot hope to efficiently find a coloring with discrepancy o(√N). We also consider the hardness of the Discrepancy problem on sets with bounded shatter function, and show that the upper bounds due to Matoušek [Mat95] are tight for these sets systems as well. The hardness results in both settings are obtained from a common framework: we compose a family of high discrepancy set systems with set systems for which it is NP-hard to distinguish instances with discrepancy zero from instances in which a large number of the sets (i.e. constant fraction of the sets) have non-zero discrepancy. Our composition amplifies this zero versus non-zero gap.

We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a long-standing conjecture of Bang-Jensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments.

We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the number of disagreements with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a long-standing conjecture of Bang-Jensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments.