properties assignment problem

The assignment problem revisited

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• Published: 16 August 2021
• Volume 16 , pages 1531–1548, ( 2022 )

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• Carlos A. Alfaro   ORCID: orcid.org/0000-0001-9783-8587 1 ,
• Sergio L. Perez 2 ,
• Carlos E. Valencia 3 &
• Marcos C. Vargas 1  

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First, we give a detailed review of two algorithms that solve the minimization case of the assignment problem, the Bertsekas auction algorithm and the Goldberg & Kennedy algorithm. It was previously alluded that both algorithms are equivalent. We give a detailed proof that these algorithms are equivalent. Also, we perform experimental results comparing the performance of three algorithms for the assignment problem: the \(\epsilon \) - scaling auction algorithm , the Hungarian algorithm and the FlowAssign algorithm . The experiment shows that the auction algorithm still performs and scales better in practice than the other algorithms which are harder to implement and have better theoretical time complexity.

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Some results on an assignment problem variant

Integer Programming

A Full Description of Polytopes Related to the Index of the Lowest Nonzero Row of an Assignment Matrix

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Acknowledgements

This research was partially supported by SNI and CONACyT.

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Alfaro, C.A., Perez, S.L., Valencia, C.E. et al. The assignment problem revisited. Optim Lett 16 , 1531–1548 (2022). https://doi.org/10.1007/s11590-021-01791-4

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Received : 26 March 2020

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DOI : https://doi.org/10.1007/s11590-021-01791-4

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Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

Let us explore all approaches for this problem.

Solution 1: Brute Force  

We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O(n!).

Solution 2: Hungarian Algorithm  

The optimal assignment can be found using the Hungarian algorithm. The Hungarian algorithm has worst case run-time complexity of O(n^3).

Solution 3: DFS/BFS on state space tree  

A state space tree is a N-ary tree with property that any path from root to leaf node holds one of many solutions to given problem. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer. The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.

Solution 4: Finding Optimal Solution using Branch and Bound  

The selection rule for the next node in BFS and DFS is “blind”. i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an optimal solution. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost. We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly.

There are two approaches to calculate the cost function:  

• For each worker, we choose job with minimum cost from list of unassigned jobs (take minimum entry from each row).
• For each job, we choose a worker with lowest cost for that job from list of unassigned workers (take minimum entry from each column).

In this article, the first approach is followed.

Let’s take below example and try to calculate promising cost when Job 2 is assigned to worker A. 

Since Job 2 is assigned to worker A (marked in green), cost becomes 2 and Job 2 and worker A becomes unavailable (marked in red). 

Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs. Cost becomes 2 + 3 = 5 and Job 3 and worker B also becomes unavailable. 

Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker D as it is only Job left. Total cost becomes 2 + 3 + 5 + 4 = 14. 

Below diagram shows complete search space diagram showing optimal solution path in green. 

Complete Algorithm:  

Below is the implementation of the above approach:

Time Complexity: O(M*N). This is because the algorithm uses a double for loop to iterate through the M x N matrix.  Auxiliary Space: O(M+N). This is because it uses two arrays of size M and N to track the applicants and jobs.

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Facility for Rare Isotope Beams

At michigan state university, international research team uses wavefunction matching to solve quantum many-body problems, new approach makes calculations with realistic interactions possible.

FRIB researchers are part of an international research team solving challenging computational problems in quantum physics using a new method called wavefunction matching. The new approach has applications to fields such as nuclear physics, where it is enabling theoretical calculations of atomic nuclei that were previously not possible. The details are published in Nature (“Wavefunction matching for solving quantum many-body problems”) .

Ab initio methods and their computational challenges

An ab initio method describes a complex system by starting from a description of its elementary components and their interactions. For the case of nuclear physics, the elementary components are protons and neutrons. Some key questions that ab initio calculations can help address are the binding energies and properties of atomic nuclei not yet observed and linking nuclear structure to the underlying interactions among protons and neutrons.

Yet, some ab initio methods struggle to produce reliable calculations for systems with complex interactions. One such method is quantum Monte Carlo simulations. In quantum Monte Carlo simulations, quantities are computed using random or stochastic processes. While quantum Monte Carlo simulations can be efficient and powerful, they have a significant weakness: the sign problem. The sign problem develops when positive and negative weight contributions cancel each other out. This cancellation results in inaccurate final predictions. It is often the case that quantum Monte Carlo simulations can be performed for an approximate or simplified interaction, but the corresponding simulations for realistic interactions produce severe sign problems and are therefore not possible.

Using ‘plastic surgery’ to make calculations possible

The new wavefunction-matching approach is designed to solve such computational problems. The research team—from Gaziantep Islam Science and Technology University in Turkey; University of Bonn, Ruhr University Bochum, and Forschungszentrum Jülich in Germany; Institute for Basic Science in South Korea; South China Normal University, Sun Yat-Sen University, and Graduate School of China Academy of Engineering Physics in China; Tbilisi State University in Georgia; CEA Paris-Saclay and Université Paris-Saclay in France; and Mississippi State University and the Facility for Rare Isotope Beams (FRIB) at Michigan State University (MSU)—includes  Dean Lee , professor of physics at FRIB and in MSU’s Department of Physics and Astronomy and head of the Theoretical Nuclear Science department at FRIB, and  Yuan-Zhuo Ma , postdoctoral research associate at FRIB.

“We are often faced with the situation that we can perform calculations using a simple approximate interaction, but realistic high-fidelity interactions cause severe computational problems,” said Lee. “Wavefunction matching solves this problem by doing plastic surgery. It removes the short-distance part of the high-fidelity interaction, and replaces it with the short-distance part of an easily computable interaction.”

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The research team applied this new method to lattice quantum Monte Carlo simulations for light nuclei, medium-mass nuclei, neutron matter, and nuclear matter. Using precise ab initio calculations, the results closely matched real-world data on nuclear properties such as size, structure, and binding energies. Calculations that were once impossible due to the sign problem can now be performed using wavefunction matching.

“It is a fantastic project and an excellent opportunity to work with the brightest nuclear scientist s in FRIB and around the globe,” said Ma. “As a theorist , I'm also very excited about programming and conducting research on the world's most powerful exascale supercomputers, such as Frontier , which allows us to implement wavefunction matching to explore the mysteries of nuclear physics.”

While the research team focused solely on quantum Monte Carlo simulations, wavefunction matching should be useful for many different ab initio approaches, including both classical and  quantum computing calculations. The researchers at FRIB worked with collaborators at institutions in China, France, Germany, South Korea, Turkey, and United States.

“The work is the culmination of effort over many years to handle the computational problems associated with realistic high-fidelity nuclear interactions,” said Lee. “It is very satisfying to see that the computational problems are cleanly resolved with this new approach. We are grateful to all of the collaboration members who contributed to this project, in particular, the lead author, Serdar Elhatisari.”

This material is based upon work supported by the U.S. Department of Energy, the U.S. National Science Foundation, the German Research Foundation, the National Natural Science Foundation of China, the Chinese Academy of Sciences President’s International Fellowship Initiative, Volkswagen Stiftung, the European Research Council, the Scientific and Technological Research Council of Turkey, the National Natural Science Foundation of China, the National Security Academic Fund, the Rare Isotope Science Project of the Institute for Basic Science, the National Research Foundation of Korea, the Institute for Basic Science, and the Espace de Structure et de réactions Nucléaires Théorique.

Michigan State University operates the Facility for Rare Isotope Beams (FRIB) as a user facility for the U.S. Department of Energy Office of Science (DOE-SC), supporting the mission of the DOE-SC Office of Nuclear Physics. Hosting what is designed to be the most powerful heavy-ion accelerator, FRIB enables scientists to make discoveries about the properties of rare isotopes in order to better understand the physics of nuclei, nuclear astrophysics, fundamental interactions, and applications for society, including in medicine, homeland security, and industry.

The U.S. Department of Energy Office of Science is the single largest supporter of basic research in the physical sciences in the United States and is working to address some of today’s most pressing challenges. For more information, visit energy.gov/science.

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How MSNBC’s Leftward Tilt Delivers Ratings, and Complications

NBC’s leaders have been forced to grapple with how to square its cable news network’s embrace of progressive politics with the company’s straight-news operation.

By Jim Rutenberg and Michael M. Grynbaum

MSNBC placed a big bet on becoming comfort TV for liberals. Then it doubled down.

Time slots on the cable network once devoted to news programming are now occupied by Trump-bashing opinion hosts. The channel has become a landing spot for high-profile alumni of President Biden’s administration like Jen Psaki, who went from hosting White House press briefings to hosting her own show. On Super Tuesday, when producers aired a portion of a live speech by former President Donald J. Trump, Rachel Maddow chastised her bosses on the air.

The moves have been a hit with viewers. MSNBC has leapfrogged past its erstwhile rival CNN in the ratings and has seen viewership rise over the past year, securing second place in cable news behind the perennial leader, Fox News.

But MSNBC’s success has had unintended consequences for its parent company, NBC, an original Big Three broadcaster that still strives to appeal to a mass American audience.

NBC’s traditional political journalists have cycled between rancor and resignation that the cable network’s partisanship — a regular target of Mr. Trump — will color perceptions of their straight news reporting. Local NBC stations between the coasts have demanded, again and again, that executives in New York do more to preserve NBC’s nonpartisan brand, lest MSNBC’s blue-state bent alienate their red-state viewers.

Even Comcast, NBC’s corporate owner, which is loath to intervene in news coverage, took the rare step of conveying its concern to MSNBC’s leaders when some hosts and guests criticized Israel as the Hamas attack was unfolding on Oct. 7, according to three people with knowledge of the discussions. An abrupt course correction to that coverage followed.

This account of the tensions roiling NBC and its corporate overseers is based on interviews with more than two dozen people with knowledge of the company’s inner workings, almost all of whom insisted on anonymity to share details of internal discussions.

NBC declined to make its top executives available for interviews. The chairman of the NBCUniversal News Group, Cesar Conde, has said he wants his division — which encompasses MSNBC, CNBC, a digital streaming service, Telemundo and journalistic stalwarts like “Nightly News,” “Meet the Press” and “Today” — to be a big tent.

Yet his recent efforts to include more conservative voices on the airwaves generated newsroom suspicion and ultimately led to an embarrassing rebellion over the hiring of Ronna McDaniel, a former Republican Party chair who aided Mr. Trump’s attempt to overturn his 2020 election loss.

MSNBC hosts, for their part, view their role in the political debate as more important than ever. They dismiss the accusation that MSNBC is a “Fox News for Democrats” and say their message — that Mr. Trump’s candidacy represents a unique and clear threat to democracy — is an urgent one for the electorate to hear.

And executives inside NBC’s corporate suites at Rockefeller Center say they are confident that viewers know the differences between the company’s various news brands. Any related challenges, they argue, are of a high-class sort — because their cable channels give NBC an advantage in relevance and revenue over its original Big Three competitors, ABC and CBS, which have no cable presence.

“Our strategy is built on our distinct, complementary brands including NBC News, CNBC, NBC News Now, MSNBC and Telemundo,” the NBCUniversal News Group said in a statement. “That has driven our performance as the nation’s leading news organization with the largest reach.” (Comcast does not disclose the news division’s earnings in its reports to Wall Street.)

The tensions inside NBC are, in some ways, a microcosm of the challenges facing many traditional news organizations as the country hurtles toward a tense presidential election: how to maintain trust and present neutral, fact-based reporting in a fractionalized era when partisanship carries vast financial and cultural rewards.

But the company’s challenge is also unique. It must juggle a broadcast news operation bound by traditional standards of impartiality and a cable channel increasingly bound by the partisan preferences of an intensely loyal viewership. How NBC navigates these dueling imperatives will have important implications for Comcast, a Philadelphia-based conglomerate known for its aversion to the political spotlight.

It will also have consequences for coverage of the presidential campaign. Where MSNBC’s cable news opinion-makers sustain and galvanize the Democratic faithful, the NBC broadcast network reaches millions of the potentially persuadable voters critical to both parties, which have sought to turn NBC’s internal tensions to their own advantage.

Left, Right, Left

MSNBC has caused corporate headaches since its inception.

NBC formed the channel as a joint venture with Microsoft in 1996 with the hope that it would thrust “all the value of NBC News into the cable world,” as Tom Rogers, a former NBC executive who helped found the cable network, described it in an interview.

But critics mocked the new 24-hour channel for its informal approach to news, mixing NBC’s biggest stars with younger personalities on a set reminiscent of Central Perk on “Friends.” It was almost immediately outflanked by Fox News, which followed MSNBC to market that same year and rose to the top of the cable news ratings as the first 24-hour TV channel with an overt political appeal.

MSNBC struggled with its identity. It moved to the left ahead of the Iraq war — and later moved right by hiring new hosts like the former Republican congressman Joe Scarborough. Soon it shifted leftward again, as the host Keith Olbermann hit a nerve with his strident anti-Bush — and often anti-Fox — commentary.

But when Andrew Lack, a veteran producer, took over NBC’s news division in 2015, he decided the channel needed to tone down its partisan image. Under Mr. Lack — who oversaw MSNBC’s creation in an earlier NBC stint — the cable network bumped the Rev. Al Sharpton from the weekday schedule, hired the former Fox anchor Greta Van Susteren and added more straightforward news programs, including a daily version of “Meet the Press,” NBC’s flagship political show, with Chuck Todd.

Mr. Todd was game — but would come to believe that his MSNBC duties ultimately hurt the “Meet the Press” franchise, several people at NBC said in interviews. The daily version of the show fell increasingly out of step with MSNBC’s partisan slant even as Republicans used its association with the liberal cable network to deny interview requests from the flagship Sunday edition of “Meet the Press.”

Then, Mr. Trump’s ascent shocked the Democratic base and spiked viewership of Ms. Maddow and other left-leaning hosts, whose programs became a kind of televised safe space. MSNBC’s ratings surged .

Conde Faces the Messiness

Mr. Conde succeeded Mr. Lack in spring 2020. A Wharton-trained business executive who sits on the boards of Walmart and PepsiCo, he came up through the corporate side of news, having led a turnaround at Telemundo after serving as the president of Univision Networks. Accordingly, Mr. Conde was expected to impose a more disciplined and neater corporate sensibility to the division.

He was almost immediately confronted by the messiness he had inherited.

Within a few weeks of Mr. Conde’s ascension, Mr. Trump attacked NBC when it announced the hiring of a new contributor: Lisa Page, a former F.B.I. lawyer who became a lightning rod on the right for her role in the investigation into his campaign ties to Russia. After an initial MSNBC appearance she did not show up again.

A few months later, NBC faced criticism from the other direction when it booked Mr. Trump for a prime-time interview on the night of a presidential debate that he had boycotted. (Mr. Biden was appearing at the same time on ABC.) Ms. Maddow chastised her bosses about it on the air.

That sort of partisan tumult has often riled another important constituency for Mr. Conde: NBC’s affiliated regional stations, which the company relies on to carry its major news programs to markets throughout the country.

The stations tend to be deeply embedded — and deeply trusted — in their communities. Many of them operate in red states or counties and chafed whenever MSNBC, which Mr. Trump regularly calls “MSDNC,” drew conservative ire.

Over the years the affiliates, many of which would have been thrilled to see MSNBC’s leftward tilt abandoned entirely, increasingly urged NBC executives to better distinguish its content from the NBC journalism like “Today” and “Nightly News” that they carried on their stations.

At one point after Mr. Conde took over, executives talked about the possibility of doubling down on partisanship and stripping MSNBC of news altogether, defining it as a pure opinion channel. The company would use the new NBC News Now streaming service, started under Noah Oppenheim when he was NBC News president, for 24-hour news, according to two people with knowledge of the conversations.

That idea fizzled. Mr. Conde was not prepared to entirely abandon news, but he began to better distinguish the various parts of his news division — which effectively moved MSNBC and NBC News further apart.

In the Lack era, Mr. Oppenheim of NBC News and Phil Griffin, the longtime chief of MSNBC, often worked closely as they managed a collection of stars who worked for both networks, like Mr. Todd, Craig Melvin and Hallie Jackson.

Creating more distance between the cable and broadcast outlets, Mr. Conde and Mr. Griffin’s successor, Rashida Jones, moved Mr. Todd, Ms. Jackson and Mr. Melvin off MSNBC to work exclusively at NBC News and NBC News Now. MSNBC’s daytime block of hard news shrank to six hours from eight, as the cable network extended by an hour each two opinion shows with loyal followings: “Morning Joe” featuring Mr. Scarborough and his wife Mika Brzezinski, and “Deadline: White House” with Nicolle Wallace as host.

Nothing did more to signal that MSNBC was more tightly embracing its partisan direction than Ms. Jones’s decision to hire Ms. Psaki and another Biden aide, Symone D. Sanders, straight from the White House.

It was the kind of revolving-door hiring that liberal pundits used to criticize when it happened with Fox News and the Trump administration.

It also created an awkward situation for the NBC News White House team, which was caught off guard when word that Ms. Psaki was in talks for the job leaked while she was still serving as White House press secretary.

A tense, televised confrontation followed in the White House briefing room when Kristen Welker, then NBC News’s co-chief White House correspondent, asked her future colleague: “How is it ethical to have these conversations with media outlets while you continue to have a job standing behind that podium?”

Chasing a Broad Appeal

At the same time, NBC News was going through its own changes.

Early last year, Mr. Oppenheim left his post running NBC News, and Mr. Conde split his job in three. In a jigsaw-like structure, one executive now oversaw “Today,” another “Nightly News” and NBC News Now, and a third “Meet the Press,” “Dateline” and news coverage across numerous shows and platforms.

Mr. Conde said the new setup would provide “growth opportunities,” with each show acting like its own megafranchise. “Today,” for instance, includes an e-commerce business and online sites dedicated to cooking, wellness and books.

He gave his deputies another brief: making additional efforts to ensure that news coverage reflected a wider range of political viewpoints.

Mr. Conde wanted to get Republicans back onto shows.

That was in line with an industrywide recalibration. After four years of combat between the press and Mr. Trump, media companies have sought better ways to reach Trump supporters who feel alienated from mainstream news. Television executives were also concerned that Republican elected officials were shunning their shows in favor of the congenial confines of right-wing media.

It was especially thorny for NBC, as Mr. Trump continued to yoke NBC News to MSNBC while accusing them, along with Comcast, of committing “Country Threatening Treason.”

A chance for a fresh start seemed to come last September when Ms. Welker succeeded Mr. Todd as the moderator of “Meet the Press.”

According to several people with knowledge of the internal discussions, Mr. Conde and Ms. Welker agreed that she should make booking both Mr. Trump and Mr. Biden for interviews a priority. Mr. Biden declined; Mr. Trump accepted.

But when Mr. Conde said she should schedule the Trump interview for her debut episode, Ms. Welker disagreed. Questioning the mendacious former president can be a high-wire act for even the most experienced TV interviewers, and Ms. Welker did not think it was a wise way to introduce herself to viewers. She acquiesced only after coaxing from Mr. Conde and several of his deputies.

Ms. Welker worked to fact-check Mr. Trump in real time while also eliciting an admission that he ignored his own campaign lawyers when they told him there was no evidence the 2020 presidential election results were rigged. Mr. Trump steamrolled ahead with a litany of lies nonetheless. The interview was panned on social media — complete with a “#boycottmeetthepress” campaign — but was deemed a success by Mr. Conde.

Mr. Conde and Rebecca Blumenstein, a former editor at The New York Times whom Mr. Conde hired as one of his top deputies, also worked aggressively to secure a Republican primary debate in fall 2023, pitching Ms. McDaniel and other Republican officials in person.

They succeeded, but only after accepting terms that unsettled some journalists within the company. NBC agreed to include a moderator from a right-wing media company, Salem Radio, and stream the debate live on Rumble, a video site that frequently hosts pro-Nazi and other extremist content. (NBC executives have defended the decision, noting that Rumble was already the party’s official streamer and had no editorial input.)

The debate received good marks in the press. And in general, red-state affiliates felt that Mr. Conde was doing a better job of bringing balance to NBC News, according to an executive at one company that owns affiliates.

Reverberations Continue

Each network was now set on its own distinct course: MSNBC toward more partisan and progressive opinion, and NBC News toward Mr. Conde’s commitment to “presenting our audiences with a widely diverse set of viewpoints and experiences,” as he put it.

But each tripped over the limits of its approach in an election landscape already littered with ideological tripwires.

When Hamas staged its terror attack against Israel on Oct. 7, MSNBC mixed breaking news of the attacks with discussions about the historical backdrop of Israel’s treatment of Palestinians. The coverage reflected views on the left — and presaged the pro-Palestinian demonstrations that would soon grow in number — but it struck many others as discordant, or even offensive, given that the violence was still coming into view.

“I love this network, but I’ve got to ask: Who’s writing your scripts? Hamas?” Jonathan Greenblatt, the Anti-Defamation League chief executive, asked two days later on “Morning Joe.”

Some of the blowback came from within.

In a call with Mr. Conde, Michael Cavanagh, the president of Comcast, who oversees NBC, shared concerns about that initial coverage, according to three people with knowledge of the discussions. Mr. Conde harbored the same concerns, according to a person briefed on their conversation, and he directed MSNBC to be more circumspect and to focus on facts, not opinions, in those initial days.

Five months later, Mr. Conde thought he had achieved a milestone at NBC News in his efforts to integrate right-wing perspectives into its programming. At the recommendation of Ms. Blumenstein and Carrie Budoff Brown, who oversees political coverage, Mr. Conde hired Ms. McDaniel, the former Republican Party chair, as a contributor who could offer on-air commentary.

If the hiring was in service of Mr. Conde’s goal of adding balance, it came as an unwelcome surprise to NBC’s ranks of correspondents, hosts and anchors. Ms. Welker had booked Ms. McDaniel for her next episode of “Meet the Press” — as a guest, not as a colleague. In the interview, she grilled Ms. McDaniel about her role in Mr. Trump’s effort to overturn the 2020 election result, actions that many at NBC and MSNBC viewed as disqualifying for a job there.

Mr. Todd, appearing as a guest on that day’s episode, unleashed a live, on-air denunciation of his bosses after the interview that left the control room in stunned silence. His rebellion carried over the next day on MSNBC, from “Morning Joe” up through “The Rachel Maddow Show.” Under pressure, Mr. Conde broke the deal with Ms. McDaniel, a move that only served to upset the Republicans he was trying to attract.

In the aftermath, NBC’s public stumble turned into a point of contention on the presidential campaign trail. The Republican Party said it was weighing an attempt to restrict NBC News at this summer’s convention, while Mr. Trump yet again bashed “Fake News NBC.”

Aides to Mr. Biden were also perturbed about the McDaniel hire, viewing it as part of a broader attempt by NBC News to overcompensate for MSNBC’s decidedly pro-Biden stance. In private conversations with NBC correspondents, Biden aides have argued that “Nightly News,” whose huge audience is of critical political importance to the campaign, was taking it easy on Mr. Trump and treating Mr. Biden too harshly.

Executives at NBC dismissed these complaints, saying the partisan brickbats simply come with the territory. They believe that each campaign will use anything at its disposal to pressure news organizations for more favorable coverage.

The company pointed to comments made by Mr. Conde after the McDaniel imbroglio: “We will redouble our efforts to seek voices that represent different parts of the political spectrum.” It also shared data intended to show strong performance across its cable, broadcast and online operations.

The message was clear. Regardless of any turbulence, NBC has no plans to change course.

Jim Rutenberg is a writer at large for The Times and The New York Times Magazine and writes most often about media and politics. More about Jim Rutenberg

Michael M. Grynbaum writes about the intersection of media, politics and culture. He has been a media correspondent at The Times since 2016. More about Michael M. Grynbaum