## About me

Currently I am a postdoc at IDSIA, Switzerland hosted by Fabrizio Grandoni. I am broadly interested in fundamental questions in algorithms and optimisation and especially for questions arising in algorithmic game theory and combinatorial optimisation. You can find out more about me in **CV** (April 2023).

I completed PhD in mathematics at the London School of Economics where I was fortunate to have László Végh as my advisor.

## News

Oct 2022 | Announced runner-up for the Doctoral Dissertation Award for the most distinguished body of research leading to a doctorate in the field of Operational Research, from UK OR Society |

Feb 2022 | Starting a postdoc position at IDSIA (Lugano, Switzerland) under supervision of Fabrizio Grandoni. |

Nov 2021 | Successfully completed my PhD degree. |

Jun 2021 | Our paper on approximating Nash Social Welfare is featured in the Sigecom Exchanges. |

March-May 2020 | Due to COVID-19 my research visit to Columbia Univeristy (NY) and prof. Tim Roughgarden is cancelled. |

Dec 2019 | I am visiting University of Helsinki and Alexandru I. Tomescu. |

Jun 2019 | The paper A polynomial-time algorithm for the independent set problem in {P10,C4,C6}-free graphsreceived the best student paper award at WG2019. |

Feb 2019 | A news article about our work on tumor evolution: An algorithm shedding light on the evolution of tumours helps individualise cancer therapies. |

Mar 2018 | I am attending Google's Algorithms and Optimization PhD Workshop in Zurich from 10.04 until 12.04. |

## Manuscripts

Click on the title to see the abstract.

## Publications

Unless marked with a '*', author names are ordered alphabetically.

13. Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions

In the maximum independent set of convex polygons problem, we are given a set of n convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the problem where the edges of the polygons can take at most d fixed directions. We present an 8d/3-approximation algorithm for this problem running in polynomial time for fixed d. Our result builds on, and generalizes the recent constant factor approximation algorithms for the maximum independent set of axis-parallel rectangles problem (which is a special case of our problem with d=2) by Mitchell [FOCS21] and Gálvez, Khan, Mari, Mömke, Reddy, and Wiese [SODA22].

**SoCG 2024**| arXiv 12. Approximating Nash Social Welfare by Matching and Local Search

We give a simple, deterministic (4+eps)-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations improving on the best approximation factor 380 via a randomized algorithm. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents valuations, and give an (w+2+eps)e-approximation if the ratio between the largest weight and the average weight is at most w.

We also show that the 1/2-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time which is both 1/2-EFX and a (8+eps)-approximation to the symmetric NSW problem under submodular valuations. The previous best approximation factor under 1/2-EFX was linear in the number of agents.

We also show that the 1/2-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time which is both 1/2-EFX and a (8+eps)-approximation to the symmetric NSW problem under submodular valuations. The previous best approximation factor under 1/2-EFX was linear in the number of agents.

**STOC 2023**| arXiv | video-STOC 11. On the Correlation Gap of Matroids

A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms and mechanism design settings. It is known that the correlation gap of a monotone submodular function is 1−1/e, and this is tight even for simple matroid rank functions.

We initiate a fine-grained study of correlation gaps of matroid rank functions. In particular, we present improved lower bounds on the correlation gap as parametrized by the rank and the girth of the matroid. We also show that the worst correlation gap of a weighted matroid rank function is achieved under uniform weights. Such improved lower bounds have direct applications for submodular maximization under matroid constraints, mechanism design, and contention resolution schemes. Previous work relied on implicit correlation gap bounds for problems such as list decoding and approval voting.

We initiate a fine-grained study of correlation gaps of matroid rank functions. In particular, we present improved lower bounds on the correlation gap as parametrized by the rank and the girth of the matroid. We also show that the worst correlation gap of a weighted matroid rank function is achieved under uniform weights. Such improved lower bounds have direct applications for submodular maximization under matroid constraints, mechanism design, and contention resolution schemes. Previous work relied on implicit correlation gap bounds for problems such as list decoding and approval voting.

**IPCO 2023**| arXiv 10. Tractable Fragments of the Maximum Nash Welfare Problem

We design a PTAS for finding an MNW allocation for the case of asymmetric agents with identical, additive valuations, thus generalizing a similar result for symmetric agents. Our techniques can also be adapted to give a PTAS for the problem of computing the optimal p-mean welfare. We also show that an MNW allocation can be computed exactly in polynomial time for identical agents with k-ary valuations when k is a constant, where everyagent has at most k different values for the goods. Next, we consider the special case where every agent finds at most two goods valuable, and show that this class admits an efficient algorithm, even for general monotone valuations. In contrast, we note that when agents can value three or more goods, maximizing Nash welfare is NP-hard, even when agents are symmetric and have additive valuations, showing our algorithmic result is essentially tight.

**WINE 2022**| arXiv 9. FPT Algorithms for Finding Near-Cliques in c-Closed Graphs

Finding large cliques or cliques missing a few edges is a fundamental algorithmic task in the study of real-world graphs, with applications in community detection, pattern recognition, and clustering. A number of effective backtracking-based heuristics for these problems have emerged from recent empirical work in social network analysis. Given the NP-hardness of variants of clique counting, these results raise a challenge for beyond worst-case analysis of these problems. Inspired by the triadic closure of real-world graphs, Fox et al. (SICOMP 2020) introduced the notion of c-closed graphs and proved that maximal clique enumeration is fixed-parameter tractable with respect to c.

In practice, due to noise in data, one wishes to actually discover "near-cliques", which can be characterized as cliques with a sparse subgraph removed. In this work, we prove that many different kinds of maximal near-cliques can be enumerated in polynomial time (and FPT in c) for c-closed graphs. We study various established notions of such substructures, including k-plexes, complements of bounded-degeneracy and bounded-treewidth graphs. Interestingly, our algorithms follow relatively simple backtracking procedures, analogous to what is done in practice. Our results underscore the significance of the c-closed graph class for theoretical understanding of social network analysis.

In practice, due to noise in data, one wishes to actually discover "near-cliques", which can be characterized as cliques with a sparse subgraph removed. In this work, we prove that many different kinds of maximal near-cliques can be enumerated in polynomial time (and FPT in c) for c-closed graphs. We study various established notions of such substructures, including k-plexes, complements of bounded-degeneracy and bounded-treewidth graphs. Interestingly, our algorithms follow relatively simple backtracking procedures, analogous to what is done in practice. Our results underscore the significance of the c-closed graph class for theoretical understanding of social network analysis.

**ITCS 2022**| arXiv | video by Shweta 8. On complete classes of valuated matroids

We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. We refute the refinement of a 2003 conjecture by Frank, exhibiting valuated matroids that are not R-minor. The family of counterexamples is based on sparse paving matroids.

Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015) asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations. Our result also has implications in the context of Lorentzian polynomials : it reveals the limitations of known construction operations.

Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015) asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations. Our result also has implications in the context of Lorentzian polynomials : it reveals the limitations of known construction operations.

**SODA 2022**| arXiv | slidesInvited to the TALG Special Issue for SODA 2022.

7. Safety in multi-assembly via paths appearing in all path covers of a DAG

A multi-assembly problem asks to reconstruct multiple genomic sequences from mixed reads sequenced from all of them. Standard formulations of such problems model a solution as a path cover in a directed acyclic graph, namely a set of paths that together cover all vertices of the graph. Since multi-assembly problems admit multiple solutions in practice, we consider an approach commonly used in standard genome assembly - output only partial solutions (contigs, or safe paths), that appear in all path cover solutions. We study constrained path covers, a restriction on the path cover solution that incorporate practical constraints arising in multi-assembly problems. We give efficient algorithms finding all maximal safe paths for constrained path covers. We compute the safe paths of splicing graphs constructed from transcript annotations of different species. Our algorithms run in less than 15 seconds per species and report RNA contigs that are over 99% precise and are up to 8 times longer than unitigs. Moreover, RNA contigs cover over 70% of the transcripts and their coding sequences in most cases. With their increased length to unitigs, high precision, and fast construction time, maximal safe paths can provide a better base set of sequences for transcript assembly programs.

**TCBB** 6. Approximating Nash Social Welfare under Rado Valuations

We present the first constant-factor approximation algorithm for the symmetric Nash social welfare problem under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems, including as special cases assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first constant-factor approximation algorithm for the asymmetric case under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant.

**STOC 2021**| SIGecom Exchanges (overview) | arXiv | slides | video-STOC 5. Auction Algorithms for Market Equilibrium with Weak Gross Substitute Demands

We present a simple auction algorithm that obtains an approximate market equilibrium for the exchange market model with divisible goods where the demands of the agents satisfy the weak gross substitutes (WGS) property. As an application of our result, we obtain an efficient algorithm to find an approximate spending-restricted market equilibrium for WGS demands, a model that has been recently introduced as a continuous relaxation of the Nash Social Welfare problem.

**STACS 2021**| TEAC | arXiv | slides | video-STACS 4. The independent set problem is FPT for even-hole-free graphs

The class of even-hole-free graphs is very similar to the class of perfect graphs, and was indeed a cornerstone in the tools leading to the proof of the Strong Perfect Graph Theorem. However, the complexity of computing a maximum independent set (MIS) is a long-standing open question in even-hole-free graphs. From the hardness point of view, MIS is W[1]-hard in the class of graphs without induced 4-cycle (when parameterized by the solution size). Halfway of these, we show in this paper that MIS is FPT when parameterized by the solution size in the class of even-hole-free graphs. The main idea is to apply twice the well-known technique of augmenting graphs to extend some initial independent set.

**IPEC 2019**| arXiv 3. A polynomial-time algorithm for the independent set problem in {P10,C4,C6}-free graphs

We consider the independent set problem, a classical NP-hard optimization problem that remains hard even under substantial restrictions on the input graphs. The complexity status of the problem is unknown for the classes of Pk-free graphs for all k ≥ 7 and for the class of even-hole-free graphs, that is, graphs not containing any even induced cycles. Using the technique of augmenting graphs we show that the independent set problem is solvable in polynomial time in the class of even-hole-free graphs not containing an induced path on 10 vertices. Our result is developed in the context of the more general class of {P10,C4,C6}-free graphs.

**WG 2019**| slidesBest Student Paper Award at WG 2019.

2. MIPUP : Minimum perfect unmixed phylogenies for multi-sampled tumors via branchings and ILP

Discovering the evolution of a tumor may help identify driver mutations and provide a more comprehensive view on the history of the tumor. Recent studies have tackled this problem using multiple samples sequenced from a tumor, and due to clinical implications, this has attracted great interest. However, such samples usually mix several distinct tumor subclones, which confounds the discovery of the tumor phylogeny. We study a natural problem formulation requiring to decompose the tumor samples into several subclones with the objective of forming a minimum perfect phylogeny. We propose an Integer Linear Programming formulation for it, and implement it into a method called MIPUP. We tested the ability of MIPUP and of four popular tools LICHeE, AncesTree, CITUP, Treeomics to reconstruct the tumor phylogeny. On simulated data, MIPUP shows up to a 34% improvement under the ancestor-descendant relations metric. On four real datasets, MIPUP’s reconstructions proved to be generally more faithful than those of LICHeE.

**Bioinformatics** 1. Perfect phylogenies via branchings in acyclic digraphs and a generalization of Dilworth's theorem

Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem; split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdurović et al. proved that both problems are NP-hard, gave a related characterization of transitively orientable graphs, and proposed a polynomial-time heuristic algorithm for the MCRS problem based on coloring cocomparability graphs. We give new, more transparent formulations of the two problems, showing that the problems are equivalent to two optimization problems on branchings in a derived directed acyclic graph. Building on these formulations, we obtain new results on the two problems, including (i) a strengthening of the heuristic by Hujdurović et al. via a new min-max result in digraphs generalizing Dilworth's theorem, which may be of independent interest, (ii) APX-hardness results for both problems, (iii) approximation algorithms, and (iv) exponential-time algorithms solving the two problems to optimality faster than the naïve brute-force approach. Our work relates to several well studied notions in combinatorial optimization; chain partitions in partially ordered sets, laminar hypergraphs, and (classical and weighted) colorings of graphs.

**WG 2017**| ACM Transactions on Algorithms | arXiv